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NASA CR 135320j
Department of Aerospace Engineering and Applied MechanicsUniversity of Cincinnati
ANALYSIS OF THE CROSS FLOW INA
RADIAL INFLOW TURBINE SCROLL (NASA-CR-135320) ANALYSIS OF THE CROSS FLOW N78-191 53 IN A RADIAL INFLOW TURBINE SCROLL (Cincinnati Univ) 61 p HC A0MF A01
CSCL 21E Unclas G307 07350
BY
AHAMEDJ S ABDALLAH AND WTABAKOFF
Supported by
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
Grant No NSG 3066
httpsntrsnasagovsearchjspR=19780011210 2020-04-08T105027+0000Z
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Jshy
NASA CR-135320
ANALYSIS OF THE CROSS FLOW IN A
RADIAL INFLOW TURBINE SCROLL
by
A Hamed S Abdallah and W Tabakoff
Supported by
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
Grant No NSG 3066
II
1 Report No j2 Grmenrn Accesion No 3 Recipients Catalog No NASA CR 135320 1 I
4 Title and Subtitle 5 Report Cate November 19771N A RADIL 1NFLW]ANALYSIS OF THE CROSS FLOW
TURBINE SCROLL 6 Performing Orgnatin Code
7 Author(s) 8 Perfirming Organization Repot No
A Earned S Abdallah and W Tabakoff 10 Work Unit No
S Performing Organization Name and Address Departnent of Aerospace Engineering University of Cincinnati
amp Applied Mechanics II Contract or Grant No
Cincinnati Ohio 45221 NASA Grant NSG 3066
13 Type of Report and Perod Covered
12 Sporsoring Agency Name zad Address Contractor Report
National Aeronautical and Space Washington DC 20546
AamprionistratIon 14 Sponsoring Agency Code
15 Supplementary Notes
16 Abstract a-
The equations of motion are derived end a computational procedure is presented for determining the nonviscous flow characteristics in the crossshysectional planes of a curved channel due to contiruous mass discharge or mass addition The analysis ts applied to the radial inflow turbine scroll to szudy the effects of scroll geomretry and the through flow velocity profile on the flow behavior The comouted flow velocity component in the scroll crossshysectional plane together with the through flow velocity profile which can be deter=ned in a separaze analysis provide a complete description of the three diarenslonal flow in the scroll
17 Key Words (Suggested by Ajthor(s) 18 Oistributicn Statenent
Radial MachInery Unclassified - unlimited
19 Security Clsauf (of th s reportl 20 Secunty Classif (of ths poa) 21 No of Pages 22 Price Unclassified Unclass-f-ed 57
For sale by the National Technical Infornaton Service Springrield Virginia 22151
N4SA-C-168 (Rev 10-75) 1
t t
TABLE OF CONTENTS
Pa~ge
SUMMARY 1
INTRODUCTION 2
MATHEMATICAL FORMULATION 3
The Equations of Motion 3
The Boundary Conditions 6
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS 8
NUMERICAL SOLUTION i 10
ITERATIVE PROCEDURE 12
Boundary Conditions in the Iterative Scheme 12
Choice of Under-Relaxation Factor 13
RESULTS AND DICUSSION 14
CONCLUSION 17
REFERENCES 18
NOMENCLATURE 20
FIGURES 22
APPENDIX 38
ii-b
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
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=
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OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
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13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
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02190905
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= -00007b56
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I 01 tY90
19
0 19t090b
20 0P29U905
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22 -shy0 5
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= -000ot5o
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27 -000159
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00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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9 -05622026320-01 -043252955D-0t
to -O55849114OD--01 -o5073895890-0
VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
3 1 OtflO58976D-02 00 3 2 01o68033090-02 -0oI138356640-02 3 3 01306a7575D-02 -shyO343931741D-03 3 4 06979866D-03 0365976810D-03
3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
6 406b459983I0-02 -01251$420JUh02 C 5 0bt)0473019D-02 -0$2J571010-03 6 6 0 35958072U-0 -u48arJ6966-vuj
5 7 O459939b980-02 0484733781D-04 6 8 O3S0o0JTbZD-2 0810603050-01 6 9 O300I 47120-02 01 50t85310-02
7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
9 600o0230-02 -0IV024s0910-02
7 4 0907118799D-02 -0112149t4V-04 7 5 06J77915900-02 -01bZ2600-02
7 6
ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
48
7 9 046707385D-2 01124044600j 7 10 04416bOA9bO-02 D151497386U-02
8 I 01293808bD-O 00 a 2 0o10408842JD-01 -0214077844-02
a 3 01243793230-0 -023b25b7700-02 a 4 01179959200-01 -0247428632D-02 a 5 O109623464-01 -0243J727760-02 a 6 09977353230-02 -02211313630-02
a 7 O888543576D-02 -01791199$50-O O 8 0770522amp54D-02 -O1172b02370--02 8 9 O6402533090-U2 -0391268524D-03
a t0 0478355850D-02 O920211545D-03 9 1 01o33186330-03 00 9 2 1618962380-0t -0241825893-02 9 3 015b9211900-01 -02932bO6IbD-02 9 4 0t4bUS 12$D-O -0328773559D-02 9 5 0138480008D-0t -03434524870-02 9 6 012b435635D-01 -02345674420-02 9 7 01133d21640-0 -0302101119D-02
9 a 09973640940-02 -0248624346D-02 9 9 08564t1841D-02 -017976203bD-0
9 to gijiojaiSaO-oz 0486525053D-04 20 1 0203043364D-01 00
10 2 0201093709D-01 -02780Z2770-02 s0 3 09400981D-01 -0366346293D-02 20 4 O10A4340jw-Q3 -0433I404530-02 10 5 0170751999D-01 -04b78776860-02 10 6 0t557627850-I -04721097970-02
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57
NOTICE
THIS DOCUMENT HAS BEEN REPRODUCED
FROM THE BEST COPY FURNISHED US BY
THE SPONSORING AGENCY ALTHOUGH IT
LS RECOGNIZED THAT CERTAIN PORTIONS
ARE ILLEGI-BLE IT IS BEING RELEASED
IN THE INTEREST OF MAKING AVAILABLE
AS MUCH INFORMATION AS POSSIBLE
Jshy
NASA CR-135320
ANALYSIS OF THE CROSS FLOW IN A
RADIAL INFLOW TURBINE SCROLL
by
A Hamed S Abdallah and W Tabakoff
Supported by
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
Grant No NSG 3066
II
1 Report No j2 Grmenrn Accesion No 3 Recipients Catalog No NASA CR 135320 1 I
4 Title and Subtitle 5 Report Cate November 19771N A RADIL 1NFLW]ANALYSIS OF THE CROSS FLOW
TURBINE SCROLL 6 Performing Orgnatin Code
7 Author(s) 8 Perfirming Organization Repot No
A Earned S Abdallah and W Tabakoff 10 Work Unit No
S Performing Organization Name and Address Departnent of Aerospace Engineering University of Cincinnati
amp Applied Mechanics II Contract or Grant No
Cincinnati Ohio 45221 NASA Grant NSG 3066
13 Type of Report and Perod Covered
12 Sporsoring Agency Name zad Address Contractor Report
National Aeronautical and Space Washington DC 20546
AamprionistratIon 14 Sponsoring Agency Code
15 Supplementary Notes
16 Abstract a-
The equations of motion are derived end a computational procedure is presented for determining the nonviscous flow characteristics in the crossshysectional planes of a curved channel due to contiruous mass discharge or mass addition The analysis ts applied to the radial inflow turbine scroll to szudy the effects of scroll geomretry and the through flow velocity profile on the flow behavior The comouted flow velocity component in the scroll crossshysectional plane together with the through flow velocity profile which can be deter=ned in a separaze analysis provide a complete description of the three diarenslonal flow in the scroll
17 Key Words (Suggested by Ajthor(s) 18 Oistributicn Statenent
Radial MachInery Unclassified - unlimited
19 Security Clsauf (of th s reportl 20 Secunty Classif (of ths poa) 21 No of Pages 22 Price Unclassified Unclass-f-ed 57
For sale by the National Technical Infornaton Service Springrield Virginia 22151
N4SA-C-168 (Rev 10-75) 1
t t
TABLE OF CONTENTS
Pa~ge
SUMMARY 1
INTRODUCTION 2
MATHEMATICAL FORMULATION 3
The Equations of Motion 3
The Boundary Conditions 6
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS 8
NUMERICAL SOLUTION i 10
ITERATIVE PROCEDURE 12
Boundary Conditions in the Iterative Scheme 12
Choice of Under-Relaxation Factor 13
RESULTS AND DICUSSION 14
CONCLUSION 17
REFERENCES 18
NOMENCLATURE 20
FIGURES 22
APPENDIX 38
ii-b
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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OF POOR QUALITY
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57
NASA CR-135320
ANALYSIS OF THE CROSS FLOW IN A
RADIAL INFLOW TURBINE SCROLL
by
A Hamed S Abdallah and W Tabakoff
Supported by
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
Grant No NSG 3066
II
1 Report No j2 Grmenrn Accesion No 3 Recipients Catalog No NASA CR 135320 1 I
4 Title and Subtitle 5 Report Cate November 19771N A RADIL 1NFLW]ANALYSIS OF THE CROSS FLOW
TURBINE SCROLL 6 Performing Orgnatin Code
7 Author(s) 8 Perfirming Organization Repot No
A Earned S Abdallah and W Tabakoff 10 Work Unit No
S Performing Organization Name and Address Departnent of Aerospace Engineering University of Cincinnati
amp Applied Mechanics II Contract or Grant No
Cincinnati Ohio 45221 NASA Grant NSG 3066
13 Type of Report and Perod Covered
12 Sporsoring Agency Name zad Address Contractor Report
National Aeronautical and Space Washington DC 20546
AamprionistratIon 14 Sponsoring Agency Code
15 Supplementary Notes
16 Abstract a-
The equations of motion are derived end a computational procedure is presented for determining the nonviscous flow characteristics in the crossshysectional planes of a curved channel due to contiruous mass discharge or mass addition The analysis ts applied to the radial inflow turbine scroll to szudy the effects of scroll geomretry and the through flow velocity profile on the flow behavior The comouted flow velocity component in the scroll crossshysectional plane together with the through flow velocity profile which can be deter=ned in a separaze analysis provide a complete description of the three diarenslonal flow in the scroll
17 Key Words (Suggested by Ajthor(s) 18 Oistributicn Statenent
Radial MachInery Unclassified - unlimited
19 Security Clsauf (of th s reportl 20 Secunty Classif (of ths poa) 21 No of Pages 22 Price Unclassified Unclass-f-ed 57
For sale by the National Technical Infornaton Service Springrield Virginia 22151
N4SA-C-168 (Rev 10-75) 1
t t
TABLE OF CONTENTS
Pa~ge
SUMMARY 1
INTRODUCTION 2
MATHEMATICAL FORMULATION 3
The Equations of Motion 3
The Boundary Conditions 6
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS 8
NUMERICAL SOLUTION i 10
ITERATIVE PROCEDURE 12
Boundary Conditions in the Iterative Scheme 12
Choice of Under-Relaxation Factor 13
RESULTS AND DICUSSION 14
CONCLUSION 17
REFERENCES 18
NOMENCLATURE 20
FIGURES 22
APPENDIX 38
ii-b
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ABSOLUTE VELOCITY C
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WITH TE X COO
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57
1 Report No j2 Grmenrn Accesion No 3 Recipients Catalog No NASA CR 135320 1 I
4 Title and Subtitle 5 Report Cate November 19771N A RADIL 1NFLW]ANALYSIS OF THE CROSS FLOW
TURBINE SCROLL 6 Performing Orgnatin Code
7 Author(s) 8 Perfirming Organization Repot No
A Earned S Abdallah and W Tabakoff 10 Work Unit No
S Performing Organization Name and Address Departnent of Aerospace Engineering University of Cincinnati
amp Applied Mechanics II Contract or Grant No
Cincinnati Ohio 45221 NASA Grant NSG 3066
13 Type of Report and Perod Covered
12 Sporsoring Agency Name zad Address Contractor Report
National Aeronautical and Space Washington DC 20546
AamprionistratIon 14 Sponsoring Agency Code
15 Supplementary Notes
16 Abstract a-
The equations of motion are derived end a computational procedure is presented for determining the nonviscous flow characteristics in the crossshysectional planes of a curved channel due to contiruous mass discharge or mass addition The analysis ts applied to the radial inflow turbine scroll to szudy the effects of scroll geomretry and the through flow velocity profile on the flow behavior The comouted flow velocity component in the scroll crossshysectional plane together with the through flow velocity profile which can be deter=ned in a separaze analysis provide a complete description of the three diarenslonal flow in the scroll
17 Key Words (Suggested by Ajthor(s) 18 Oistributicn Statenent
Radial MachInery Unclassified - unlimited
19 Security Clsauf (of th s reportl 20 Secunty Classif (of ths poa) 21 No of Pages 22 Price Unclassified Unclass-f-ed 57
For sale by the National Technical Infornaton Service Springrield Virginia 22151
N4SA-C-168 (Rev 10-75) 1
t t
TABLE OF CONTENTS
Pa~ge
SUMMARY 1
INTRODUCTION 2
MATHEMATICAL FORMULATION 3
The Equations of Motion 3
The Boundary Conditions 6
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS 8
NUMERICAL SOLUTION i 10
ITERATIVE PROCEDURE 12
Boundary Conditions in the Iterative Scheme 12
Choice of Under-Relaxation Factor 13
RESULTS AND DICUSSION 14
CONCLUSION 17
REFERENCES 18
NOMENCLATURE 20
FIGURES 22
APPENDIX 38
ii-b
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
U 2 1909U5
0219905
0U
0
60
00
00
00
00
00
00
00 00
U 190909
OPt9u O
0219090
00
00
0U
00
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021901905
0219090b
0219B905
000
00
00
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O0
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0219903
041990D
00
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04190905
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I)19yu3b 00 00
000 00
U0
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 U0 00
00 00 00
003 V0 U0
00
00 00 o0
-0000755
I -0OOT5b
= -000ot5o
I = -- OOOVfl~f
I =30
-OUI07br
-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
0I711913123) Ott 02 00 u501) 00 0l2t9iVII0 00 D35049700 UJ25f6yJIb 00 041iJ9JIIJ3 I0 CA4076007O 00 0bOjt164440 00
I0 -06551j5])-0h -obb5lttO(tJ)-0I -0 504042520-0 -0bkV$42300-1 -0 5s0443btU-0 I -U 55-4 96Ishy 01 - 0 5409I 11 -0 f 54 I 2 01250-01 -05374011l to-ut -V 511 bI6br4u-Of -0501 9V41 Ja-uI -U40I136794t-Ot4 -3 40104442)-0 U| - A6Ofb0btO1shy - 41500l530-U -U0t7 flO 9-j - J25654 10-0 1 -Q 4rftl91o-0l | -0 1504312940-01 00 04o3J 194)-01 0 0(O3J p-U Qt1J46Ot2JU 0
011790sItonb) 00 024r) leo5I0 00 026)52 I1750 00 UJISTtp(t19) 00 0 16210540I 0U 04 06 21 t VU0 044 0l27440) 00 000074 102J 00
00 -OSI t)44I)-0I - 5iI Jt 41)- -4D-059055Qbb4tI- bampAST3tuOt-0I -J t020t17070-0 -0t deg5 27|1 -O -0Jsq0tl0totI-0l -0544J079U-ol -004136b)Ibltshy 0 I -iIb236 0W-0l -053 IpJ94AtU-0I -0oAa71 t 4u-0 1 -04 T549)-0 -N-vt0432) I 001)M-01 -044JUP0J0-Ut -0 90U4 I 90-01 -03iV0J j1-ol -0 t0 44j046t)-0t -05411110t1)t1-Ut 00 00 00 0-0
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50
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55
ORIGINAL PAGE IS
OF POOR QUALITY
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57
TABLE OF CONTENTS
Pa~ge
SUMMARY 1
INTRODUCTION 2
MATHEMATICAL FORMULATION 3
The Equations of Motion 3
The Boundary Conditions 6
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS 8
NUMERICAL SOLUTION i 10
ITERATIVE PROCEDURE 12
Boundary Conditions in the Iterative Scheme 12
Choice of Under-Relaxation Factor 13
RESULTS AND DICUSSION 14
CONCLUSION 17
REFERENCES 18
NOMENCLATURE 20
FIGURES 22
APPENDIX 38
ii-b
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
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=
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Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
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98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
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02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
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00 U0 00
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00
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= -000ot5o
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I =30
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27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ORIGINAL PAGE IS
OF POOR QUALITY
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57
SUMMARY
The equations of motion are derived and a computational
procedure is presented for determining the nonviscous flow
characteristics in the cross-sectional planes of a curved
channel due to continuous mass discharge or mass addition
The analysis is applied to the radial inflow turbine scroll
to study the effects of scroll geometry and the through flow
velocity profile on the flow behavior The computed flow
velocity component in the scroll cross-sectional plane
together with the through flow velocity profile which can be
determined in a separate analysis provide a complete descripshy
tion of the three dimensional flow in the scroll
1
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
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ORIdINAL PAGE ISOF POR QUALIIY
50
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OIGINAL PAGE 18
OF POOR QUkLI Y
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55
ORIGINAL PAGE IS
OF POOR QUALITY
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57
INTRODUCTION
Although radial turbines have been in operation for a very
long time there is a renewed interest in small radial inflow
turbines With advantages such as high efficiency over the
limited range of specific speed simplicity and ruggedness
these units are used in different applications such as autoshy
motive and diesel engines aircraft cooling systems and space
power generation
References [1] [2] and [3] are some examples of the recent
interest in radial inflow turbine design and development Several
studies dealing with the prediction of the performance of these
turbines can also be found in the literature [4-8] While many
flow studies were concerned with the flow in the impeller very
little has been done to investigate the flow in the rest of
the turbine components An experimental and theoretical study
of the losses in the nozzle vanes and the vaneless nozzle is
reported in Ref [9] It has been shown in that reference that
the performance of the vaned nozzle is greatly influenced by the
flow incidence angle at inlet to the nozzle vanes Therefore
in order to use any available loss analysis the knowledge of
the flow conditions at exit from the scroll is essential A
better understanding of the flow behavior in the scroll can thus
contribute to a more accurate performance prediction
The radial inflow turbine scroll is usually arranged in a
helical configuration as shown in Fig la to discharge uniformly
into the nozzle of the radial turbine or the impeller in the
case of nozzleless turbine The actual flow in the scroll is
three dimensional compressible and viscous Even in the absence
of viscosity effects the three dimensional flow is caused by the
continuous flow discharge from the scroll Although under real
operating conditions other factors such as secondary flow effects
may contribute to the three dimensional behavior of the flow
the effect of the radiaiomponent of the exit flux from the
scroll will have the most predominant effect over the three
dimensional flow behavior
2
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
U 2 1909U5
0219905
0U
0
60
00
00
00
00
00
00
00 00
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5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
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11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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13 10 016218710ID-0l -05269839340-U2 3 1 0 b32039b00-01 00
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ABSOLUTE VELOCITY C
2 2 2 2
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WITH TE X COO
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ORIGINAL PAGE IS
OF POOR QUALITY
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57
The present study investigates the three dimensional flow
behavior in the scroll due to the continuous flow discharge
The equations governing the flow motion in the scroll are
formulated in a special way in order to be solved -in the-c-ross
sectional plane for a given through flow profile Solutions
are obtained numerically for the typical scroll cross sections
of Figures 2 and 3 and the results are presented to show the
effects of scroll geometry on the flow behavior in the scroll
MATHEMATICAL FORMULATION
The differential equations governing the steady inviscid
flow motion in the cross-sectional plane of the scroll will
be developed It is convenient for this purpose to consider
the control volume a shown in Fig lb which is contained
between two cross-sectional planes
The Equations of Motion
Referring to the control volume of Fig lb the equation of
conservation of mass can be written as
If p qn dS = 0 (i) S
where p is the density q is the flow velocity vector in the
scroll and n is the unit vector perpendicular to the surface
S of the control volume The surface S is the union of the
scroll surface S2 the cylindrical surface of the flow exit
from the scroll Sl and the two cross sectional planes A1
and A2
On the two surfaces S1 and $2 qn is equal to V-n where
V is the velocity vector in the axial radial plane Equation
(1) can therefore be rewritten as
If pVn dS = Q (2) Sl+S ORIGINAL PAGE IS
OF POOR QUALITY
where Q is the net mass influx through the planes A1 and A2
3
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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=
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
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= -00007b56
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
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3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
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Ol00OOOOO0 01 00
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08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
The difference between the mass flux on the surfaces A1
and A2 due to the through flow velocity can be represented by
a continuous source distribution g in the control volume 0
according to the following relation
Q = ff1 gdv (3)
Substituting equation (3) into equation (2) and using
Gauss divergence theorem we can write
ff1 div(p V)dv = If gdv (4)SI 2
Since this equation is correct for any arbitrary control volume
a of incremental depth we can write
V(p V) = g (5)
For irrotational flow the conservation of momentum is
satisfied by introducing the velocity potential function such that
V = V (6)
Substituting equation (6) into equation (5) we obtain the
following relation
VpVO) = g (7)
The following isentropic relation is used to determine the
density p of the adiabatic irrotational nonviscous flow
1
+ Y-= 2 qO+ T (8)
where p0 is the flow stagnation density and T is the flow temshy
perature which is a function of the stagnation temperature To
and the magnitude of the flow velocity q according to the
following relation
4
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ORIGINAL PAGE IS
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57
T = T (y-l) q2 (9) 0 2y R
g
For given inlet stagnation conditions equations- C6- through
(9) can be solved to determine the flow density temperature and
velocity components in the cross sectional planes of a scroll
The solution will naturally depend on the source distribution
which corresponds to the through flow profile at each cross
section
Equation (7) is rewritten in a different form which will
be used in the iterative numerical solution later
V = - V-Vp + gp (10)p
It can be seen from this form that the velocity potential in the
cross sectional plane is governed by Poissons equation The
first term on the right hand side of equation (10) is a source
or sink term contributed by the flow compressibility and the
second term in the same equation represents the contribution
of the through flow velocity to the source
A very limited amount of experimental data is available
from flow measurements in the scroll Reference [8] reports only
five points pressure measurements at one radial location in
the scroll On the other hand the pressure measurements of
Ref [10] in the centrifugal pump were obtained at the volute
center At the present time detailed measurements through
scroll cross sections are not available Several factors
such as the scroll radius of curvature and the location of the
particular cross section under consideration can affect the
through flow velocity profile
The condition of constant angular momentum has been applied
to the flow in the scroll and vaneless nozzles [3 10] In
the absence of any blades to exert circumferential force on
the flow the product (pwr) remains independent of the radial
distance r from the machine axis where w is the through flow
velocity A variation in g across the scroll cross section
that is proportional to lr is one of the cases considered in the
5
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
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98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
U 2 1909U5
0219905
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0
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0219090b
0219B905
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00
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00 U0 00
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00
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I -0OOT5b
= -000ot5o
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I =30
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-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
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29 29 30 30
31 31 32
32
2 2
3 3 4
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5 5
o 6
7 7
8 8
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04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
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57
results For a given mass flow in the scroll the proportionality
constant will depend on the geometry of the cross section and
on the ratio of the particular cross sectional dimensions to
the scroll radius of curvature at that section A uniform g
distribution is also considered which corresponds to the case
of a scroll with infinite radius of curvature or a straight
variable area tube with continuous uniform mass ejection The
boundary layer development will naturally cause variation from
these profiles near the scroll walls with the effects being
more pronounced in the latter scroll sections A third type
of g distribution namely that of a paraboloid of revolution
in the circular part of the scroll cross section was also
considered to study this viscous effect
The Boundary Conditions
The equations governing the flow motion in the crossshy
sectional plane of the scroll have been derived in the previous
section The boundary conditions will generally depend on the
scroll geometry and the mass flow rate in the scroll
Figures 2 and 3 show some typical scroll cross sectional
geometries Referringto these figures the flow boundary
conditions can be expressed as follows
Vn = V on S (11)
Vn = 0 on S2 (12)
where V1 is the radial flow velocity at exit from the scroll
The boundary conditions given by equations (11) and (12) are
of the Neumann type when expressed in terms of the velocity
potential function 0
-2 = V on S (13)3n 1 1
Lt 10 on S2 (14)3n n 2
6
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
-068889924bD-03
04121553900-03
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
0339770359D-O2 025402769002o Uz43905u001 -O1494910U
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ABSOLUTE VELOCITY C
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WITH TE X COO
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
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57
Equations (13) and (14) provide the boundary conditions
which are necessary to solve the governing equations (6) (8)
(9) and (10) Two of the parameters appearing in these equations
are not really independent Both the source strength d-istrishy
bution g due to the through flow and the outlet velocity V1
are related to the total mass flow rate in the scroll
Assuming axisymmetric flow conditions at the scroll exit
and referring to Figures 2 and 3 we can write
f p1 V1 ds = m2w R1 (15) B
where m is the rate of total mass flow in the scroll The
integral on the left hand side of equation (15) is carried over
the scroll exit width B of radial location RI of which the
density and velocity are equal to p1 and V1 respectively
The mean value of g can also be expressed in terms of the
total mass flow rate m as follows
If g dA m2w R1 (16) A
where the integral on the left hand side is carried over the
scroll cross sectional area Equations (15) and (16) can be
combined to give the following relation
If g dA = I P1 V1 dS (17) A B
It is well known that a solution to Poisson equation with
Neumann boundary conditions exists and is unique within an
arbitrary constant if Greens first integral theorem is
satisfied Closer examination of equations (10) and (13) show
that in the special case of incompressible flow equation (17)
represents Greens first integral condition
PAGE ISORIGINAL Op pOOR QUALITY
7
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
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021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
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13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
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= -000ot5o
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27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
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12 13
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is
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16 16
17
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20 21 21 22
22 23 23
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25
25 26 26
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1 2 1 2
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000917463199W 00
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00 00 01580443260-02 2-41823
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00 00 0233992848D-02 3b8700
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56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
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4 2 4 1 5
2 5
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04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
NONDIMENSIONAL FORM OF THE GOVERNING EQUATIONS
It is more convenient to express the flow variables in the
following dimensionless form
= 4V1 D V VV 1
T = TT 0 P = PP
9 D
g - poVl g
where D is a characteristic dimension of the scroll cross-section
which can be taken as the diameter in the typical sections of
Figures 2 and 3 The coordinates are also normalized with
respect to D In the above equations and in the following
derivations the radial exit velocity component VI will be
taken as constant across the scroll exit width B Using
equation (18) one can write the flow governing equations in
nondimensional form as follows
Equations of Motion
2cent 1 -V = - (V V p) + g p ( )
P
V =V (20)
T Ck q2 (21)
1 p ( + Ck q 2 T) y- (22)
Boundary Conditions
- 1 on S (23) an
= 0 on S 2 (24) 9n
8
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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55
ORIGINAL PAGE IS
OF POOR QUALITY
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57
where(j)(Y-1) V22
Ck yRT (25)yg~o
Equations (23) and (24) show that the boundary conditions of
the nondimensional problem are homogeneous thus leading to
self similar solutions in the case of geometrically similar
scroll cross sections with similar through flow profiles
The above equations are not tractable to analytical solutions
in scroll cross sections such as the typical ones shown in
Figures 2 and 3 A numerical solution which is based on the
finite difference method is used to determine the two dimensional
velocity potential distribution on the scroll cross-sectional
plane Before writing the governing equations in finite
difference form equations (19) (20) (23) and (24) are rewritten
using the Cartesian coordinates as follows
2 2 =2 1 (ua P+) a- plusmna u+ av- + g p (26)2 aXy p 3x 8
u and v - (27) ax ay
1 on S1 (28)ax
n + ny -0 on S2 (29)-x
axa2
where u v are the x and y components of the nondimensional
velocity vector in the cross-sectional plane and nx ny are
the direction cosines of the unit vector which is perpendicular
to S2 According to equation (28) the two perpendicular
axes will be taken such that y is in the direction of the
machine axis
9Ql-ALI2y
9
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
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=
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Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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-
02l911905 14
021911905 U 2190905 0219690 b
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98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
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= -000ot5o
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I =30
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27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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ORIGINAL PAGE IS
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57
NUMERICAL SOLUTION
The derivations of the finite difference equations is
followed by a description of the iterative procedure used to
obtain the numerical solution
The Finite Difference Equations
Referring to Figure 4 the finite difference representation
of equation (26) is obtained using standard five-point Laplace
difference operator
F 2 euroo [613(11+433)+ BB6 2 4 (4 2 2 + 4 )- D ](C1 3+C 2 4 ) (30)Dx
where22= (DxDy)BB
C13 16163 C2 4 = BB6 2 54 (31)
613 1(S1+63) 624 = i(62+64)
6i i = 1 2 3 4 (32)
1 (U p 9P( S=g p -- v - (33)
P ax ay
The first order derivatives needed in the u and v calculashy
tions as well as the derivatives of the density in equation
(33) are evaluated using central difference formulas at the
interior points Forward or backward difference schemes were
used in the first order derivative computations at the boundaries
of the scroll cross section The boundary conditions can
generally be written in the following form
n3x + - L (34) 3y ORIGINXLPkG1I
O poR QUALY
10
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
U 2 1909U5
0219905
0U
0
60
00
00
00
00
00
00
00 00
U 190909
OPt9u O
0219090
00
00
0U
00
00
00
00
00
021901905
0219090b
0219B905
000
00
00
00
00
00
O0
00
00
U0
0219903
041990D
00
00
00
00
00
00
U0
00
00
04190905
U0
00
1
O9
00
00
00
00
O
00
U0
I)19yu3b 00 00
000 00
U0
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 U0 00
00 00 00
003 V0 U0
00
00 00 o0
-0000755
I -0OOT5b
= -000ot5o
I = -- OOOVfl~f
I =30
-OUI07br
-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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ORIGINL pAGE IS 54 OF pOOR QU TY
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
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57
where n and n are the direction cosines of the normal to the
boundary and L is a constant which is equal to one or zero as
in equations (28) and (29)
Referring to Fig 5 and for a general boundary -po-nt-4jshy-bne can writethe boundary condition in the following finite
difference form [11
n n 6Dy64Dy6 - 6 L (35)4 =(1 + n- -4n 4 Dx 3 n
Similar equations can be written for the boundary points
on the other three quarters by proper permutation The solution
to equations (30) and (35) is obtained using a successive
relaxation iterative method which is explained later In the
following sections the superscript star will be omitted for
convenience when referring to the nondimensional variables
Interdependence Between the Source Distribution and the
Boundary Conditions
It is well known that in the case of Poisson equation with
Neumann boundary conditions a solution exists and is unique
within an arbitrary constant if the following condition is
satisfied
ff F(xy) dA = f ds (36)snA
where F(xy) is the source strength in Poisson equation which
is equal to the sum of two terms on the right hand side of
equation (10) The integrals in the above equation are carried
over the solution domain A and its boundary s respectively
In case the source distribution F(xy) in the differential
equation is arbitrarily chosen the above condition will not
necessarily be satisfied This was the case in Ref [15) in
which Briley before proceeding with the numerical solution of
Poisson equation added a uniform correction to the source
distribution He ccomplished that by computing numerically
the difference E betweentthe right hand and left hand sides of
equation (36) then subtracting the area averaged amount EA
11
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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ORIdINAL PAGE ISOF POR QUALIIY
50
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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55
ORIGINAL PAGE IS
OF POOR QUALITY
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57
fromthe value of F(xy) at each grid point In the case of
incompressible flow such procedure is unnecessary since the
source strength F is equal to g which already satisfies the
conservation of mass given by the integral condition of
equation (17) For compressible flow solution in which case
F(xy) is also dependent on the velocity and density at the
grid points the average value of the difference between the
two sides of equation (36) is calculated numerically and
subtracted from the value of F at each interior grid point
before every iteration
ITERATIVE PROCEDURE
A successive relaxation method is used to obtain the numerishy
cal solution The following recurrence formula is used
n+l (1[ 33]+ nn = Cl-Wf)con + n+li+ n BB 624[ 22 + $44n+lII
F 311 2 4 37 - Dx2)(C3 + C24) (37)
The computations of the contribution of the compressibility to
the source term were allowed to lag the potential function
calculations by one iteration Taking the value of the density
from the previous iteration is known to improve the stability
of the numerical solution to compressible flow problems [12]
Boundary Conditions in the Iterative Scheme
The general recurrence formula for the iterative solution
is given by equation (37) If this is used to determine the
value of $ at all interior points and equations similar to (35) are to calculate explicitly the boundary values of thesolution will not converge but will drift slowly and endlessly
[15 161 Miyakoda [161 therefore recommended that the boundary
conditions be implicitly introduced into the difference
equations at the interior points next to the boundary such as
point 0 in Fig 5 Referring to the same figure the following
form of finite difference equation is- used at the interior
points adjacent to the boundary
12
nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
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02190905
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= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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24 -000055
Rb
021990
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00
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00 U0 00
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00
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I -0OOT5b
= -000ot5o
I = -- OOOVfl~f
I =30
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-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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8 -0)5b294e01b0-0l -0458071744001L
9 -05622026320-01 -043252955D-0t
to -O55849114OD--01 -o5073895890-0
VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
3 1 OtflO58976D-02 00 3 2 01o68033090-02 -0oI138356640-02 3 3 01306a7575D-02 -shyO343931741D-03 3 4 06979866D-03 0365976810D-03
3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
6 406b459983I0-02 -01251$420JUh02 C 5 0bt)0473019D-02 -0$2J571010-03 6 6 0 35958072U-0 -u48arJ6966-vuj
5 7 O459939b980-02 0484733781D-04 6 8 O3S0o0JTbZD-2 0810603050-01 6 9 O300I 47120-02 01 50t85310-02
7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
9 600o0230-02 -0IV024s0910-02
7 4 0907118799D-02 -0112149t4V-04 7 5 06J77915900-02 -01bZ2600-02
7 6
ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
48
7 9 046707385D-2 01124044600j 7 10 04416bOA9bO-02 D151497386U-02
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ORIGINAL PAGE IS
OF POOR QUALITY
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nln n+l n n (_n+l - ( 1 0 )4 deg + W(61 3 [4iI + 3 3 ] + BB 624[422 + 1
n n -shy
+Z -pDy - -n+-l -- nx Dy 4-h -_ F n Dx o ny Dx 3 n L Dx2)(C13+C24 )y (38)
Similar expressions can be written for such points in the
other three quarters n+lof the domain These equations are solved
algebraically for o appearing in both sides The solutions
to the difference equations consisting of equation (37) at the
regular interior points and equation (38) at the interior
points adjacent to the boundary was found to converge The
numerical values of the potential function at the interior
grid points calculated in this fashion arethen used in
difference equations (35) to compute the values of 4 at the boundary points
Choice of Under-Relaxation Factor
The convergence of the numerical solution to Poisson
equation is known to be sensitive to the relaxation factor
In the matrix solution to compressible flow equation in turboshy
machines using the relaxation methods the under-relaxation
is known to get stronger as the Mach number approaches unity
The boundary conditions in those problems are different however
from those involved in the present study In addition to
Dirichlet type at parts of the domain boundary they include
periodicity and specified flow angles over the rest of the
boundary Furthermore solutions to the Reynolds equation
which is similar to equation (7) were also easily obtained
using relaxation methods in both lubrication problems [131
and magnetostatic fields [14] when Dirichlet type boundary
conditions were involved In such cases an optimum value of
the under-relaxation factor can be used at all grid points
In the present solution an optimum value was determined by
trial and error and was used in equation (37) at all the regular
interior points that are removed at least one grid point from
the boundary The same value of w was not necessarily used
13
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
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13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
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00 U0 00
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27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
0oamp1989797J-01 00
02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
0b97750700) 00
086584601J 00
00
O89849332 O 00 O
0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
in equation (38) at the interior grid points adjacent to the
boundary When necessary other values of w were chosen in
order that the following condition is satisfiedNn
24_ a __Z)( 0 3 + C2 4 )) 0 (39)24- n-yny D4
RESULTS AND DISCU$SION
A computer program has been developed to solve the governing
flow eqations in the cross-sectional planes of the scroll of
arbitrary cross section The input to the program consists of
the fluid properties at the scroll inlet the exit radial veloshy
city the scroll geometric dimensions and the through flow
velocity profile The typical geometric parameters needed in the
input are shown in Figures 2 and 3 for the symmetric and
nonsymmetric scroll cross sections The maximum number of allowshy
able mesh points is (50 x 30) in the x and y directions
respectively In all the results presented here the computation
time did not exceed 30 CPU seconds
The results of the computations are presented in the form
of plots of constant potential contours and arrows showing
the velocity directions in the scroll cross sectional plane
The effects of through flow velocity profile scroll cross
sectional geometry and compressibility on the flow behavior
in the cross sectional plane are investigated
Figures 6 and 7 were obtained using uniform source distrishy
bution representing uniform through flow velocity profile
The results obtained using a source variation which is proportional
to Yr are presented in Figures 8 through 11 to show the effects
of the scroll curvature Figures 12 and 13 present the results
in the case of a circular paraboloid source distribution
All the results of Figures 6 through 13 were computed for a
scroll with symmetric scroll cross section in which case it was
sufficient to carry out the computations in only half the cross
section Zero potential gradient was specified in this case
14
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
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0
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13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
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02l911905 14
021911905 U 2190905 0219690 b
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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24 -000055
Rb
021990
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27 -000159
20 -00007b56
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-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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ORIGINL pAGE IS 54 OF pOOR QU TY
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7 42938 3-01 J225107
16 16
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3221913 3268702
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17 2 09209340243-01 3472305 17 17
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1 2
01977643083 01790739bS4
00 00
00 338b677
19 3 01293576650 00 3k71264 19 4 09923S9433)-03 3062907 19 5 077018v520-0O 3038482 19 6 bJ3404b93)-01 3046152 20 I 0326627672D 00 00 20 2 02913552bO 00 3318263 20 3 0175101065) 00 2958518 20 4 01170638B3 00 2955621 20 5 Uo 759U7010-0 2464816 21 1 059786397amp 00 00 21 2 Q6Z47098850 O0 3546042 22 1 0-619091237) 00 00 22 2 09057321J8J 00 3592E51 24 I 08bb5530773 00 00 23 2 03828842123 uO 4598769 24 $ U809970bl) 00 00
22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
2 t 2
1 2 1 2
1 2
I
2
VVUX
2
I
2 1
2
1 2
1
2
t 2
1 2
1
2 1
2 I
2 t
2 1
2 1
2
I
2 1 2
2
I 2 1
2 1
2 2 2 1
2 1 2
t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
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02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
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00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
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VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
at the axis of symmetry The results obtained from computations
in a scrollwith an unsymmetric cross section are shown in
Figures 14 and 15 The compressible flow results which were
obt-a-ined -at an- exit Mach numb- of- 7-are presented in
Figures 16 and 17 All the last four figures were obtained
with a uniform source distribution ie excluding the influence
of the scrolls curvature In all the figures higher velocity
potential gradients are observed at the exit portion of the
scroll cross sections This means velocity components
in the cross sectional plane are higher in magnitude in
this region than in the rest of the scroll cross sections
Since the solution for the velocity potential is unique within
an arbitrary constant the value of 01 was always chosen at
the neck of the cross section in order to compare the different
results
Figures 6 and 7 show in the case of symmetric scroll cross
section with constant through flow velocity profile that noshy
where in the flow field does the velocity have component in the
radially outward direction The largest radial velocity comshy
ponents are observed at the scroll exit portion decreasing
constantly to zero value at the radially farthest scroll
surface point on the axis of symmetry
Figures 8 through 11 show the velocity potential distrishy
bution and the velocity direction in the case of a source
distribution which is inversely proportional to r The first
two figures correspond to a ratio of the scrolls radius of
curvature to cross sectional diameter of 30 while the last
two figures show the results for RD of 15 The influence of
the scrolls curvature on the flow behavior can be detected by
comparing Figs 9 and 11 with 7 and 10 and 12 with 8 It can
be seen that the influence of the curvature is limited to the
radially outward part of the cross section leading to a
difference in the direction of the velocity vector in the cross
sectional plane Some radially outward velocities can be
observed in Figs 9 and 12 although the magnitudes of the
ORIGINAL PAGE IS15 OF pOOR QUALITY
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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0
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04190905 0oaIU9I
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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24 -000055
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0219090b
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00 U0 00
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003 V0 U0
00
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-0000755
I -0OOT5b
= -000ot5o
I = -- OOOVfl~f
I =30
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-000075(
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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ORIGINL pAGE IS 54 OF pOOR QU TY
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7 42938 3-01 J225107
16 16
a 9
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
2 t 2
1 2 1 2
1 2
I
2
VVUX
2
I
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2
1 2
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2 t
2 1
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t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
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THETA
00 00 01580443260-02 2-41823
U0 00
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OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
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00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
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015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
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VVBY
09244b3373 00 00
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09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
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THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
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08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
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0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
velocity in this region are very small The ratios of RD in
Figs 8 through 11 are quite low however and were merely
selected for the purpose of illustration Such magnitudes
would not be encountered in radial inflow turbine scrolls
where the value of RD constantly increases from its minimum
value at the scroll inlet It can therefore be concluded
that in most scroll cross sections it is sufficient to
consider constant source distribution in the cross sectional
computations
Figures 12 and 13 show the results obtained using a
circular paraboloid source distribution which represents an
extreme case corresponding to fully developed pipe flow It
can be seen from these figures that some radially outward
velocity components even though of very small values are
observed at radially outer portion of the scroll cross section
In this case the magnitude of the velocity component in the
cross section plane as measured by the velocity potential
gradient is generally smaller in most of the scroll cross
sections compared to the case of constant source distribution
It should be pointed out that in all the cases of Figures
8 through 13 the values of the integral of the source disshy
tribution over the scroll cross section were all the same and
equal to the value for the case of constant source distrishy
bution of Figures 6 and 7
The velocity potential contours and the velocity directions
of the velocity vector in a nonsymmetric scroll cross section
are shown in Figures 14 and 15 These results were obtained
using uniform through flow profile It can be seen that the
flow behavior is generally smoother in this case which can be
a result of the more gradual change in the scroll section in
the direction of the machine axis This type of scroll cross
section can therefore be recommended over the symmetric
section
Thecompressible flow solution corresponding to an exit
Mach number of 07 is shown in Figures 16 and 17 These resultsin a symmetric scroll cross section were calculated assuming
16
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
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=
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OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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d y905 0298905 0li0905
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0
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13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
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021911905 U 2190905 0219690 b
04190905 0oaIU9I
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
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= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
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1 2 1 2
1 2
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000917463199W 00
0917922054D 00 007-4 00
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09482613523 00 00
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098243314D O0 00 09825270183 00 00904
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THETA
00 00 01580443260-02 2-41823
U0 00
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OU 00
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00 00 0233992848D-02 3b8700
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00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
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017665l413)-OI J484538 051211935I-0 00
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00
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-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
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VVBY
09244b3373 00 00
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09736138893 00 00 0973012amp473 00 00
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THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
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08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
constant through flow velocity profile By comparing these
two figures with the incompressible results of Figures 6 and 7
it can be concluded that the shape of the velocity potential
contours is unaffected by compressibility Consequently the
shape of the streamlines is not changed by compressibility
although the magnitudes of the velocities are While higher
velocities are observed in the scroll neck region lower
velocities are found in the rest of the scroll cross sectional
plane as compared to the incompressible case
CONCLUSION
The known through flow velocity which is used to provide
the source distribution and the computed flow velocity in the
cross sectional plane provide a complete description of the
three dimensional flow field in the scroll A derivation of the
equations governing the flow motion and a computational
procedure for determining the flow velocities in the crossshy
sectional plane are presented in this study Other simplified
flow models can be used in the through flow velocity compushy
tations
17
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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=
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
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I =
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= -00007b56
1
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I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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ORIdINAL PAGE ISOF POR QUALIIY
50
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ORIGINL pAGE IS 54 OF pOOR QU TY
14 3 04367394c6-ot 3468452 14 4 0405498420)-0 340 952 14 5 030 534Sl2-O-1 3363268 1 6 0329508628)-01 334386 14 7 02909q98750-01 3313507 14 8 025746043)-01 330o941 14 9 Q2224169440-01 33240Z6 14 10 0|94
4 0Zb542-0l 339092J
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7 42938 3-01 J225107
16 16
a 9
0319679421D-01 0270519699-01
3221913 3268702
17 1 09614240050-01 00
17 2 09209340243-01 3472305 17 17
3 A
0a14940890)D-0I 06998022910-01
4362E40 3278115
17 5 05922171310-01 3219232 17 amp 04960198141-01 3183018 17 7 0413731s2o-ot 316a955 17 8 034vo12708-01 3189543 18 I 0132b869040 00 00 18 2 01240279130 00 3438058 lb 3 01023099681 00 3292892 18 4 08369239940-01 3197852 i8 5 06825000680-0 3339385 t8 6 0552402945)-01 3106301 16 7 045745402D--01 3329689 18 a 03880593b40-0l 316748 19 19
1 2
01977643083 01790739bS4
00 00
00 338b677
19 3 01293576650 00 3k71264 19 4 09923S9433)-03 3062907 19 5 077018v520-0O 3038482 19 6 bJ3404b93)-01 3046152 20 I 0326627672D 00 00 20 2 02913552bO 00 3318263 20 3 0175101065) 00 2958518 20 4 01170638B3 00 2955621 20 5 Uo 759U7010-0 2464816 21 1 059786397amp 00 00 21 2 Q6Z47098850 O0 3546042 22 1 0-619091237) 00 00 22 2 09057321J8J 00 3592E51 24 I 08bb5530773 00 00 23 2 03828842123 uO 4598769 24 $ U809970bl) 00 00
22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
2 t 2
1 2 1 2
1 2
I
2
VVUX
2
I
2 1
2
1 2
1
2
t 2
1 2
1
2 1
2 I
2 t
2 1
2 1
2
I
2 1 2
2
I 2 1
2 1
2 2 2 1
2 1 2
t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
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02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
0b97750700) 00
086584601J 00
00
O89849332 O 00 O
0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
REFERENCES
1 Calvert GS and Okapuu U Design and Evaluation of a
High Temperature Radial Turbine Phase I - Final Report
USAAVLABS Technical Report 68-69
2 Watanabe I Ariga 1 and Mashimo T Effect of Dimenshy
sional Parameters of Impellers on Performance Characteristics
of a Radial Inflow Turbine ASME Paper No 70-GT-90
3 Lagneau JP Contribution to the Studytof Advanced Small
Radial Gas Turbines VKI IN 38 March 1970
4 Kastner LJ and Bhinder FS A Method for Predicting
the Performance of a Centripetal Gas Turbine Fitted With
A Nozzle-Less Volute Casing ASME Paper No 75-GT-65
5 Wallace FJ Cave PR and Miles J Performance of
Inward Radial Flow Turbines Under Steady Flow Conditions
With Special Reference to High Pressure Ratios and Partial
Admission Proc Instn Mech Engrs Vol 184 Pt 1
No 56 1969-1970
6 Samuel MF and Wasserbauer CA Off-Design Performance
Prediction With Experimental Verification for a Radial
Inflow Turbine NASA TND 2621 1965
7 Knoernschild EM The Radial Turbine for Low Specific
Speeds and Low Velocity Factors Journal of Engineering
for Power Trans ASME Series A Vol 83 No 1
January 1961 pp 1-8
8 Watanabe I and Ando T Experimental Study on Radial
Turbine With Special Reference to the Influence of the
Number of Impeller Blades on Performance Characteristics
Bulletin of JSME Vol 2 No 7 1959 pp 457-462
9 Khalil IM Tabakoff W and Hamed A Losses in Radial
Inflow Turbines Journal of Fluids Engineering September
1976 pp 364-373
10 Iversen HW Rolling RE and Carlson JJ Volute
Pressure Distribution Radial Force on the Impeller and
Volute Mixing Losses of a Radial Flow Centrifugal Pump
Journal of Engineering for Power April 1960 pp 136-144
18
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
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=
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02191190b
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0
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-
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021911905 U 2190905 0219690 b
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
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021990
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00 U0 00
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00
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I -0OOT5b
= -000ot5o
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I =30
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-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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ORIGINL pAGE IS 54 OF pOOR QU TY
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7 42938 3-01 J225107
16 16
a 9
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17 1 09614240050-01 00
17 2 09209340243-01 3472305 17 17
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17 5 05922171310-01 3219232 17 amp 04960198141-01 3183018 17 7 0413731s2o-ot 316a955 17 8 034vo12708-01 3189543 18 I 0132b869040 00 00 18 2 01240279130 00 3438058 lb 3 01023099681 00 3292892 18 4 08369239940-01 3197852 i8 5 06825000680-0 3339385 t8 6 0552402945)-01 3106301 16 7 045745402D--01 3329689 18 a 03880593b40-0l 316748 19 19
1 2
01977643083 01790739bS4
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19 3 01293576650 00 3k71264 19 4 09923S9433)-03 3062907 19 5 077018v520-0O 3038482 19 6 bJ3404b93)-01 3046152 20 I 0326627672D 00 00 20 2 02913552bO 00 3318263 20 3 0175101065) 00 2958518 20 4 01170638B3 00 2955621 20 5 Uo 759U7010-0 2464816 21 1 059786397amp 00 00 21 2 Q6Z47098850 O0 3546042 22 1 0-619091237) 00 00 22 2 09057321J8J 00 3592E51 24 I 08bb5530773 00 00 23 2 03828842123 uO 4598769 24 $ U809970bl) 00 00
22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
2 t 2
1 2 1 2
1 2
I
2
VVUX
2
I
2 1
2
1 2
1
2
t 2
1 2
1
2 1
2 I
2 t
2 1
2 1
2
I
2 1 2
2
I 2 1
2 1
2 2 2 1
2 1 2
t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
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02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
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00
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00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
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VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
11 Parker R Short Communications Normal Gradient Boundary
Conditions in Finite Difference Calculations International
Journal for Numerical Methods in Engineering Vol 7
No- 3 1974 pp 397-400
12 Smith DJL and Frost DH Calculation of the Flow Past
Turbomachine Blades Proceedings of the Institute of
Mechanical Engineers Vol 184 Part 36 (II) 1969-1970
13 Holmes AG and Ettles CMM A Study of Iterative
Solution Techniques for Elliptic Partial Differential
Equations With Particular Reference to the Reynolds Equation
Computer Methods in Applied Mechanics and Engineering Vol
5 1975 pp 309-328
14 Winslow A Numerical Solution to the Quasilinear Poisson
Equation in a Nonuniform Triangle Mesh Journal of Comshy
putational Physics Vol 2 1967 pp 149-172
15 Briley WR Numerical Method for Predicting Three-
Dimensional Steady Viscous Flow in Ducts Journal of
Computational Physics Vol 14 1974 pp 1-20
16 Miyakoda K Contribution to the Numerical Weather
Prediction Computation with Finite Difference Japanese
Journal of Jeophysics Vol 3 1962 pp 160-167
19
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
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I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
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U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
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=
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9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
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50
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15 1 0577054574-01 00 i5 2 0566647T70J-Ot 3509056 is 3 053320229b0-0l 3437370 15 4 04804844270-01 3375766 15 5 0434358825)-Di 3427086 15 6 038173425D-01 3292710 15 7 03320129810-01 3273513 Is a 0285890334)-al 32708|2 15 9 0414110540-01 3290937 15 10 0224752546)-01 3372001 16 1 0743018883-0I 00 16 2 0713052907J-01 3494632 16 3 065b8639290-0I 3407122 16 4 0-583196b6-0I 3334795 16 5 050B99818Y0-01 3280313 16 6 04300906550-01 3243860 16 7 03
7 42938 3-01 J225107
16 16
a 9
0319679421D-01 0270519699-01
3221913 3268702
17 1 09614240050-01 00
17 2 09209340243-01 3472305 17 17
3 A
0a14940890)D-0I 06998022910-01
4362E40 3278115
17 5 05922171310-01 3219232 17 amp 04960198141-01 3183018 17 7 0413731s2o-ot 316a955 17 8 034vo12708-01 3189543 18 I 0132b869040 00 00 18 2 01240279130 00 3438058 lb 3 01023099681 00 3292892 18 4 08369239940-01 3197852 i8 5 06825000680-0 3339385 t8 6 0552402945)-01 3106301 16 7 045745402D--01 3329689 18 a 03880593b40-0l 316748 19 19
1 2
01977643083 01790739bS4
00 00
00 338b677
19 3 01293576650 00 3k71264 19 4 09923S9433)-03 3062907 19 5 077018v520-0O 3038482 19 6 bJ3404b93)-01 3046152 20 I 0326627672D 00 00 20 2 02913552bO 00 3318263 20 3 0175101065) 00 2958518 20 4 01170638B3 00 2955621 20 5 Uo 759U7010-0 2464816 21 1 059786397amp 00 00 21 2 Q6Z47098850 O0 3546042 22 1 0-619091237) 00 00 22 2 09057321J8J 00 3592E51 24 I 08bb5530773 00 00 23 2 03828842123 uO 4598769 24 $ U809970bl) 00 00
22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
is IS 19
19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
-
1
2 t 2
1 2 1 2
1 2
I
2
VVUX
2
I
2 1
2
1 2
1
2
t 2
1 2
1
2 1
2 I
2 t
2 1
2 1
2
I
2 1 2
2
I 2 1
2 1
2 2 2 1
2 1 2
t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
0oamp1989797J-01 00
02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
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086584601J 00
00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
NOMENCLATURE
Ck Nondimensional expression defined in equation (25)
B Scroll exit width Figs 2 and 3
BBC1 3C 2 4 Expressions defined in equation (31)
Do Scroll inlet diameter Figure 1
DH Characteristic dimensions of the scroll crossshysection Figs 2 and 3
F Expression defined in equation (33)
g Source strength distribution in the scroll crossshysection due to the through flow
L A constant equal to one or zero on the boundary equation (34)
m Total rate of mass flow in the scroll
n Local unit vector normal to the scroll surface
nx ny Components of the unit vector n in the x andy directions respectively
q Velocity vector in the scroll
Q Difference in rate of mass flow across two radialplanes equation (2)
R1 Radius of the scroll exit Fig 1
R Gas constantg
S Surface of the control volume 0 Figure 1
s Distance along the boundary contour of the scroll cross section
T Local static gas temperature
T 0Stagnation temperature
uv Velocity components in the scroll cross sectionalplane in the x and y directions respectively
Velocity vector in the axial radial plane
V1 Velocity component in the radial direction at thescroll exit
20 ORIGINAL PAGE IS OF POOR QUALITshy
V
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
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U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
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=
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
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0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
0oamp1989797J-01 00
02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
0b97750700) 00
086584601J 00
00
O89849332 O 00 O
0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
Greek Symbols
The velocity potential function in the crossshysectional plane
p Local gas density
PO Stagnation density
Pi Flow static density at the scroll exit
6 Mesh size parameter Fig 4
6131624 Expressions defined in equation (31)
0Successive relaxation factor
9 The control volume of Fig 1
Subscripts
i=123amp4 Refer to parameters associated with the five mesh points of Fig 4
xy Refer to derivatives in the x and y directions
Superscripts
Refer to the nondimensional variables
nn+l Refer to the previous and present iterations
21
CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
o0Pf93905
1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
Rb
021990
U 2 1909U5
0219905
0U
0
60
00
00
00
00
00
00
00 00
U 190909
OPt9u O
0219090
00
00
0U
00
00
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00
021901905
0219090b
0219B905
000
00
00
00
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00
O0
00
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0219903
041990D
00
00
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U0
00
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04190905
U0
00
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O9
00
00
00
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00
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I)19yu3b 00 00
000 00
U0
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 00 00
00 U0 00
00 00 00
003 V0 U0
00
00 00 o0
-0000755
I -0OOT5b
= -000ot5o
I = -- OOOVfl~f
I =30
-OUI07br
-000075(
27 -000159
20 -00007b56
29 -00007qr(
-UOOO70
00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
OOb6 00007Wb 00 00 00 00 00 00 00 1 0
IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
0I711913123) Ott 02 00 u501) 00 0l2t9iVII0 00 D35049700 UJ25f6yJIb 00 041iJ9JIIJ3 I0 CA4076007O 00 0bOjt164440 00
I0 -06551j5])-0h -obb5lttO(tJ)-0I -0 504042520-0 -0bkV$42300-1 -0 5s0443btU-0 I -U 55-4 96Ishy 01 - 0 5409I 11 -0 f 54 I 2 01250-01 -05374011l to-ut -V 511 bI6br4u-Of -0501 9V41 Ja-uI -U40I136794t-Ot4 -3 40104442)-0 U| - A6Ofb0btO1shy - 41500l530-U -U0t7 flO 9-j - J25654 10-0 1 -Q 4rftl91o-0l | -0 1504312940-01 00 04o3J 194)-01 0 0(O3J p-U Qt1J46Ot2JU 0
011790sItonb) 00 024r) leo5I0 00 026)52 I1750 00 UJISTtp(t19) 00 0 16210540I 0U 04 06 21 t VU0 044 0l27440) 00 000074 102J 00
00 -OSI t)44I)-0I - 5iI Jt 41)- -4D-059055Qbb4tI- bampAST3tuOt-0I -J t020t17070-0 -0t deg5 27|1 -O -0Jsq0tl0totI-0l -0544J079U-ol -004136b)Ibltshy 0 I -iIb236 0W-0l -053 IpJ94AtU-0I -0oAa71 t 4u-0 1 -04 T549)-0 -N-vt0432) I 001)M-01 -044JUP0J0-Ut -0 90U4 I 90-01 -03iV0J j1-ol -0 t0 44j046t)-0t -05411110t1)t1-Ut 00 00 00 0-0
Q0 00 110 O0 0 U 410 00
3 4 00 - 0561411f oI(- 1 -0 U )a do- U I - 05-f 4130214 M3-0I - o1 t oJ 11to-O 1 - b 04b A1)shy t I
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16 0499631 7111-Q0I
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29 -0432409Z490-0l
20 -0374345ST7O-01
21 00
22 O46351049t)-oI
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Z4 01346402322) 00
25 01905188503 00
40 02239735a10 00
2 026S521150 00
28 03157l6890 00
29 03627054amp3b0 00
30 0410622121D3 00
33 04593127440 013
32 0bQ8OTT0iO 00
J payl P13Y2
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5 -Ob64SL466JJ-0t -03b7740lI~U-0l
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8 -0)5b294e01b0-0l -0458071744001L
9 -05622026320-01 -043252955D-0t
to -O55849114OD--01 -o5073895890-0
VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
3 1 OtflO58976D-02 00 3 2 01o68033090-02 -0oI138356640-02 3 3 01306a7575D-02 -shyO343931741D-03 3 4 06979866D-03 0365976810D-03
3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
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ORIdINAL PAGE ISOF POR QUALIIY
50
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OF POOR QUkLI Y
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CROSS SECTIONAL PLANE
FIG 1A
ORIGINAL PAGE IS OF POOR QUALITY
v- n
FIG 1B CONTROL VOLUME E
FIG 1 RADIAL INFLOW TURBINE SCROLL
22
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
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FIG 4 GRID POINTS INTHE FINITE
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ORIGINAL PAGE Is OF POOR QUALITY
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FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
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FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
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Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
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ORIGINAL PAGE IS
OF POOR QUALITY
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57
FIG 2 SYMMETRIC SCROLL CROSS SECTION
RC
y
FIG 3 NON SYMMETRIC SCROLL CROSS SECTION
23
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
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FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
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FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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=
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
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2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
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3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
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6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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13 10 016218710ID-0l -05269839340-U2 3 1 0 b32039b00-01 00
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16 4 05222507440-0 16 5 04318463710-01 16 b 03b61939240-01
16 7 0297027034D-01 16 a 02525984700-01
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17 6 0370412024 -01 17 7 0302335544D-Ot 17 8 02o37106080-01
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24 3 00199123J70 00 22 2 090063435D 00 23 1 06554J077U 00 23 2 OUu2aa24b9U 00 24 0t109976511 00
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ORIdINAL PAGE ISOF POR QUALIIY
50
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29 1 29 2 30 1
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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WITH TE X COO
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
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8 8
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56
27 1 27 2 28 1
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57
2 (xy+a2 Dy)
(x-61 Dxy) (xy) (x+ 3Dxy)
1 3
4 (xy-S4 Dy)
FIG 4 GRID POINTS INTHE FINITE
DIFFERENCE SCHEME
ORIGINAL PAGE Is OF POOR QUALITY
24
1 1+1
Jl
e - xJ-1
FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
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ORIdINAL PAGE ISOF POR QUALIIY
50
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
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57
1 1+1
Jl
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FIG 5 GRID SYSTEM FOR INTERIOR POINTS
25
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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ITMAX 2O0
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WP = 0 000
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2 1 2
2
I 2 1
2 1
2 2 2 1
2 1 2
t
2 1 2
000917463199W 00
0917922054D 00 007-4 00
0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
0oamp1989797J-01 00
02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
0b97750700) 00
086584601J 00
00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
003003 0004
0040
___ j o03 04 05 06l ~~01
0O7 009 008
FIG 6 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION IN A SYMMETRIC SCROLL CROSS SECTION
FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
to
~~~~~-b- -r- c - t b ~
FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
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2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
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3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
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7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
9 600o0230-02 -0IV024s0910-02
7 4 0907118799D-02 -0112149t4V-04 7 5 06J77915900-02 -01bZ2600-02
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50
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FIG 7 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONALPLANE OF A SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY
003
004
0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
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007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
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29 29 30 30
31 31 32
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57
003
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0035
0070
0102 03 04 05 06
007 009008
FIG 8 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLL
CURVATURE (RID = 3)
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
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U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
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= 0
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=
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
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11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ABSOLUTE VELOCITY C
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OF POOR QUALITY
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57
FIG 9 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 3)
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
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16 16
17
17
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19 2U
20 21 21 22
22 23 23
24 Z2
25
25 26 26
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1 2 1 2
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000917463199W 00
0917922054D 00 007-4 00
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09482613523 00 00
093226206D 00
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098243314D O0 00 09825270183 00 00904
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THETA
00 00 01580443260-02 2-41823
U0 00
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OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
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00 00 036b409420-02 235783
00 00 04664164470-02 174577
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56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
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2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
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VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
O1035
004
bull 0045 005
0 f O 06
0102 0 0405 06
007 009 008
FIG 10 NON-DIMENSIONAL VELOCITY POTENTIAL CONTOURS SHOWING THE EFFECT OF SCROLLCURVATURE (RD =15)
FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
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U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
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= 0
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=
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NO Or t NirIAT I 01Sm 250
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VELOCITY COMP UampV IN THE XZ Y DIR
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4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
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FIG 11 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS SECTION PLANE
SHOWING THE EFFECT OF SCROLL CURVATURE (RD = 15)
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
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FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
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FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
-068889924bD-03
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
0339770359D-O2 025402769002o Uz43905u001 -O1494910U
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ABSOLUTE VELOCITY C
2 2 2 2
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6 6
6 6
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WITH TE X COO
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
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12 13
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17
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00 00 01580443260-02 2-41823
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27 1 27 2 28 1
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57
0108 oi 02 03 04 05 06
FIG 12 NON-DIMENSIONAL VELOCITY POTENTIAL bISTRIBUTION INA SYMMETRICSCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGH FLOWVELOCITY PROFILE
5gc
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
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U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
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=
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9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
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0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
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11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ABSOLUTE VELOCITY C
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WITH TE X COO
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ORIGINAL PAGE IS
OF POOR QUALITY
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57
s S k A - s - a
FIG 13 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTION PLANE OFA SYMMETRIC SCROLL CROSS SECTION FOR CIRCULAR PARABOLOID THROUGHFLOW VELOCITY PROFILE
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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0932644046gt 00 O08J
09482613523 00 00
093226206D 00
0940b35498 00 00864 09608875b 00 00 096527375 00 00893
098243314D O0 00 09825270183 00 00904
099b7920369 00 OU
0995815488) 00 00901
THETA
00 00 01580443260-02 2-41823
U0 00
06869256b0)-03 5J1303
OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
00 00
0355429093)-02 300001
00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
Olb7b300860-0 354Z516 041420854O-0I 00
017665l413)-OI J484538 051211935I-0 00
0190043020-UI 3425331
0oamp1989797J-01 00
02090284b2D-01 33o4125
08240479690-01 00
025342533931-01 32-9907 4
0k09so0o ) 00 00
O33b7831ou-l 3231207
015b4946040 00 00 O37 0448-01 3155636 0240034812) 0 00
0b749304120-01 3ubSoOS 04142205J33 00 00
0858184b00J01l 29b842b U72b74200 00 00
0927v39b9 00 00
0008b561bO~b 00
0042 17t723 00 V0 0875411100J 00 Uu 088I4777000 00 00
00
011436)00 0
0b97750700) 00
086584601J 00
00
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0o 00 090194470D 00 00
-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
t0
2 10
VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
Ol00OOOOO0 01 00
0347801419)-03 007491 0307178679D 00 2815274
08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
0
005
007
0 3 04 05 6
FIG 14 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION INA NON-
SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILE
FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
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Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
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02191190b
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d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
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02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
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02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
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= -00007b56
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
00 -o9G0Si J94U-0l -05044 7bI7-01 -0 5612t30OD-u0 -0 b6 UU090(P-0 -0btvplusmnb I nO-oI -05S 3599pA FOshy 0shy -0 54 9140z(-0l -04051900t f31512 63-) -0 5203UbOJU-0 I -shy 50t5 4i v- 0 1 -0 qj69460I qu-O - 4 0 0q0-0 -0 4J3 43)0 I -0 4112740Jbt)-1 -0O OtJb0-O I -0 J I I J J uI)-0 I --02l3U bb3 9-0I -0 1 1 r1t 2qIlb0-0t V 09242 100)-P 0 Ut9t~)0-IU90113J031t)-0I Ui4t03UljI) OD
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
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3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
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7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
9 600o0230-02 -0IV024s0910-02
7 4 0907118799D-02 -0112149t4V-04 7 5 06J77915900-02 -01bZ2600-02
7 6
ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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9 a 09973640940-02 -0248624346D-02 9 9 08564t1841D-02 -017976203bD-0
9 to gijiojaiSaO-oz 0486525053D-04 20 1 0203043364D-01 00
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
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OF POOR QUkLI Y
52
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ORIGINAL PAGE IS
OF POOR QUALITY
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FIG 15 THE DIRECTION OF THE VELOCITY VECTOR INTHE CROSS-SECTIONAL PLANE OF A
NON SYMMETRIC SCROLL FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
004
01 025 0 065
008
FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
0PI9905 02[I0905 02 9o5 02198905 0 2 190905 02190905 0419b4o9 O2190905 0 vtbt05
= 0
0P190905 02190905 02191905 0 901905 0 19091)5 0 19 LOW 02190905 019d905 0419by05 0 t190
=
0229090S
9 021911905 02190905 021941905 02190905 09 09b05 021990 0219
1905 021
-8905 02 W1U0
=
02190905
Ito
OP190905 0o190905 0190905 0 19905 02198905 02191590 019b905 019905 U212 9r
I
02191190b
It 02190905 02190905 021911905 0219090b 02190905 02|9890 019
d y905 0298905 0li0905
021 90905 0 2191905 02190905 0 21911905 019U905 0 2I1Q0905 0 19090b 0 1909 0219609b 0229090
0
I 90190us
13 0I29905 00190905 0219 905 0P2 40905 0190905 0 190
) b 0 AV905 0 198905 02191190
-
02l911905 14
021911905 U 2190905 0219690 b
04190905 0oaIU9I
05 U1)91190b 0-c 1
98905 04 t91905 V2)90
02190906 0 p19090a 5 01911901 0219090 0 1490 04)90905 02)90905 0219490 0419u905 02190905
0 229090us 0219090b U21 0 02 90119 2t 91)0 0219S905 u19190 0l 190 t| 44 O 90U 01 O
= 17
1oU 4105 OS196 UI6-10u419105 0l990 0 0Uv22 Ot002 1900 02194U0o 0 199 0201 9005
is0C
wda
k4
I 021VO905
I =
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1 0219U905
02190905
I -000007
= -00007b56
1
-000075bb
I -00050-00O0l5bb
I 01 tY90
19
0 19t090b
20 0P29U905
0219H8U9
22 -shy0 5
23 -00007556
24 -000055
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00
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0219090b
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0219903
041990D
00
00
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00
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00 00 00
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00 00 00
00 00 00
00 U0 00
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00
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-0000755
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= -000ot5o
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00 -0O0i6 0O75tt 00 90 00 VA1 00 00 010
V 32-
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IPS = 0637T-05
NO Or t NirIAT I 01Sm 250
VAtLS or POTENTIAL VLItOTlYV PcJ) rull INCONPILS5r1t fLOW
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to -O55849114OD--01 -o5073895890-0
VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
3 1 OtflO58976D-02 00 3 2 01o68033090-02 -0oI138356640-02 3 3 01306a7575D-02 -shyO343931741D-03 3 4 06979866D-03 0365976810D-03
3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
6 406b459983I0-02 -01251$420JUh02 C 5 0bt)0473019D-02 -0$2J571010-03 6 6 0 35958072U-0 -u48arJ6966-vuj
5 7 O459939b980-02 0484733781D-04 6 8 O3S0o0JTbZD-2 0810603050-01 6 9 O300I 47120-02 01 50t85310-02
7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
9 600o0230-02 -0IV024s0910-02
7 4 0907118799D-02 -0112149t4V-04 7 5 06J77915900-02 -01bZ2600-02
7 6
ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
48
7 9 046707385D-2 01124044600j 7 10 04416bOA9bO-02 D151497386U-02
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a 7 O888543576D-02 -01791199$50-O O 8 0770522amp54D-02 -O1172b02370--02 8 9 O6402533090-U2 -0391268524D-03
a t0 0478355850D-02 O920211545D-03 9 1 01o33186330-03 00 9 2 1618962380-0t -0241825893-02 9 3 015b9211900-01 -02932bO6IbD-02 9 4 0t4bUS 12$D-O -0328773559D-02 9 5 0138480008D-0t -03434524870-02 9 6 012b435635D-01 -02345674420-02 9 7 01133d21640-0 -0302101119D-02
9 a 09973640940-02 -0248624346D-02 9 9 08564t1841D-02 -017976203bD-0
9 to gijiojaiSaO-oz 0486525053D-04 20 1 0203043364D-01 00
10 2 0201093709D-01 -02780Z2770-02 s0 3 09400981D-01 -0366346293D-02 20 4 O10A4340jw-Q3 -0433I404530-02 10 5 0170751999D-01 -04b78776860-02 10 6 0t557627850-I -04721097970-02
10 7 01400323310-01 -0461122370-02 10 a 01242763380-01 -0894658037U-02 10 9 01089319100-01 03246507080-0Z to 10 09420318070-02 -01092549030-02
11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
31 4 02241593420-01 -O amp5626330bU--OZ It 5 02067559540-OL -06261649940-02 11 6 0i77063430-01 -06417641210-02 11 7 01684302470-01 -0616965595)-02 it 8 0149977539D-01 -0559026550D-02 13 9 0133038511-01 -04804936070-02 II 10 0117998647D-0 -0218490347D-U 12 1 O306618840U-01 00 2 2 030248810OD-0shy -0390529683D--U0
12 3 0289576234D-01 -0597710795D-02 12 02r0OOb709D-01 -074uoogE40-02 12 5 0246b3o011-O3 -08324076700-Z 12 b 02197g128D-UI -08556amp072-02 12 7 0197909155U-01 -0643400e660-02 12 8 0I75$90163D-U -0748575390-04 12 9 O2SbUO2759D-Ol -06-4b436980-02 42 10 O
3 4110
9 1 t30-01 -03b0$4b990-2
13 1 OJ759311363D-0 00 13 z 0369567035D--01 -0 49767b78O-02 13 1 03504739OD--01 -0767970759D-02 13 4 OJ22605456-l -094519b0160--0 13 5 0290473b6W-O -011072o08O0-0 13 6 02078087070-01 -01129502600-01 13 7 02274224610-01 -0107773504U-01 I3 b 02007312000-01 - 70bg2b13U-04 13 9 01787774U60-0 -08277044200-02
13 10 016218710ID-0l -05269839340-U2 3 1 0 b32039b00-01 00
14 2 045292t9470-0I -06b3I3735100-0Z 14 3 0423490570-0 -01067S2b590-Ul t4 4 03amp2458071-01 -o1jb699o3-0
49
I 5 03375206750-0I
14 6 029301840-O 14 7 O2553956b5D-0J 14 6 024030-0I 14 9 0497137030-01
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15 2 0559b490610-0j 15 3 0511891241-01 i5 4 049T31629-0l
15 5 03860410520-0
Is 6 032828778b0-0
I5 7 02795823010-01 15 8 0240U135920-O
15 9 02071529260-01
1r 10 020720b3740-01
16 1 0F33018883D-01 36 2 0701050206D-01
16 3 0619080329D-01
16 4 05222507440-0 16 5 04318463710-01 16 b 03b61939240-01
16 7 0297027034D-01 16 a 02525984700-01
pound6 9 0226b8729b-01 17 1 0961424005D-01
17 2 O98189811D-Ul 17 3 074b9917140-01
17 4 0b92371118D-03 17 5 046u2438250-01
17 6 0370412024 -01 17 7 0302335544D-Ot 17 8 02o37106080-01
Ie I 0132606904c 00
t8 2 01191123360 00 38 3 0UT97032300-0
i8 4 063916$4790-0
2o 5 04t36576760-01
IS 6 0S59flSAOZ-01 IB 7 03116456530-01 18 8 02032686290-01
19 1 019776430D 00 t9 2 0186B38240 00 19 3 09481502260-01
19 4 66150448650-Ol pound9 043243a3110-01 19 6 036159Z760-0l
20 1 03266276740 00 O 0256057893D 00
20 0703602496-01 20 4 05052910390-0
20 5 04003809720-01 21 3 0b970639760 00 21 2 Ob20 9511430 00
24 3 00199123J70 00 22 2 090063435D 00 23 1 06554J077U 00 23 2 OUu2aa24b9U 00 24 0t109976511 00
24 2 OtJ4d554110 00
25 1 0Oyz1bTbblD 00 2s 2 0093J31930 V0
26 1 0904005680D 00 2b 2 0904692897V 00 27 1 09t74631990 00
27 2 09179212b 00
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-- 260511179 0
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00
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0001024429W-0200
01I192470a9U-0
ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
-068889924bD-03
04121553900-03
04064117160-03
0187194278D-02
0307810624D-02
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051303787D-02
05869087620-02
00
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01amp18247250-01
O12310962 01
0219299S770-01
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0 A853442IP 00 096bOB759t 00 09b52725a06 00
09$24333140 00 098252S7950 00
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VBX2
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03983631 2D-03
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00
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00
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
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29 29 30 30
31 31 32
32
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FIG 16 NON-DIMENSIONAL VELOCITY POTENTIAL DISTRIBUTION OF COMPRESSIBLE FLOW INASYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW VELOCITY PROFILE
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
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WP = 0 000
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ORIGINL pAGE IS 54 OF pOOR QU TY
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55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
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12 13
13 1
14
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19 2U
20 21 21 22
22 23 23
24 Z2
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098243314D O0 00 09825270183 00 00904
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THETA
00 00 01580443260-02 2-41823
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OU 00
05690902310-03 444212
00 00 0233992848D-02 3b8700
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00 00 036b409420-02 235783
00 00 04664164470-02 174577
00 00
05244335263-02 11l370 00 0-0
0bb98655000-02 57J92 0275o49b03J-0 00
00 00 0337S88178)-0 00
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00
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-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
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VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
09403361190 00 00 0950503630a O 00 09bO7327230 00 00
09736138893 00 00 0973012amp473 00 00
09931927493 00 00
09912391430 00 00
0991192739 00 00 0991239143D 00 U0
THEIA
0333155355 -03 264571)
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08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
01002471123 00 2935607
0102184109-02 691335 07431552660-01 2999905 02439494200-02 531303 053705Jb9 )-01 3068605
025202223D-02 821018
04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
FIG 17 THE DIRECTION OF THE COMPRESSIBLE FLOW VELOCITY VECTOR INTHE CROSS SECTION
PLANE OF A SYMMETRIC SCROLL CROSS SECTION FOR UNIFORM THROUGH FLOW
VELOCITY PROFILEI
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
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7
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
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3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
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ORIGINAL PAGE IS
OF POOR QUALITY
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57
APPENDIX
COMPUTER PROGRAM AND INPUT PREPARATION
A brief description of the input and output of the computer
program will be given in this appendix A listing of the program
which can be used to solve the equations of potential flow in
the cross-sectional planes of radial inflow turbine can be
obtained from
Description of Input and Output
The computer program requires as input the cross sectional
plane geometry the radius of curvature at the plane appropriate
shape of the through flow profile and the operating conditions
such as the inlet velocity and the inlet total temperature
Any consistant set of units can be used in the input data The
computations are carried out and the results presented in
terms of the dimensionless quantities given in equation (18)
The output obtained from the program includes crossshy
sectional plane velocities and potential velocity values throughshy
out the region of solution
Input Data
First Set (Fluid Properties) one card
READ RG GAMA TO VE
according to format (4F100)
RG Gas constant
GAMA Ratio of specific heats
TO Stagnation temperature
VE Scroll exit velocity in radial direction
Second Set (Scroll Geometry) one card
READ Ml M2 DO Rl B H R RC
according to format (2110 6F100)
Ml M2 Number of mesh lines in the circular part of thecross-section of the scroll in the x and ydirections respectively (Figure 3a)
DO Scroll inlet diameter
38
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
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000704371SV-03
0001024429W-0200
01I192470a9U-0
ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
-068889924bD-03
04121553900-03
04064117160-03
0187194278D-02
0307810624D-02
03358129260-02
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051303787D-02
05869087620-02
00
015674050D-01
0171122738D-02
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O12310962 01
0219299S770-01
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0932202206D 00 09326431040 00
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0 A853442IP 00 096bOB759t 00 09b52725a06 00
09$24333140 00 098252S7950 00
09957920860 00 09958142570 00
VBX2
-01 i4229893D-02
0549540520D-03
03983631 2D-03
0140395709D-02
01777145460-02
01466607770-02
01399249340-02
01048a6b70D-02
Ob898655000-03
00
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51
03449630470-01
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
0339770359D-O2 025402769002o Uz43905u001 -O1494910U
01593413760-02 02150bbUD-02 02130231290-01 -00317l81OD-UI
ABSOLUTE VELOCITY C
2 2 2 2
2
3 3 3 3
3
3 4
4 4 a
4 4 4
4 5
5 5
5 5 5 5
5 6
6 6
6 6
6 6
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6 7
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7 74 7 1
ITS
I
DIR
1 2
3 4
5
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5
6 1
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A 5 6 7
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WITH TE X COO
VV
OS40636473D-03 09672405090-03
03366740540-03 0136210924D-02
0709522853D-03
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0774098423-03
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02289242963-02 052852b667gt-02
0542b4812D-02 050501b7lb-z
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THETA
002966186
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3126612
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to to
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11 11
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II 12
12
12 12
12
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13
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13 13
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13 13
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14 14
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1
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ORIGINL pAGE IS 54 OF pOOR QU TY
14 3 04367394c6-ot 3468452 14 4 0405498420)-0 340 952 14 5 030 534Sl2-O-1 3363268 1 6 0329508628)-01 334386 14 7 02909q98750-01 3313507 14 8 025746043)-01 330o941 14 9 Q2224169440-01 33240Z6 14 10 0|94
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7 42938 3-01 J225107
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
o 6
7 7
8 8
9
9 10
to
It
II 12
12 13
13 1
14
is
is
16 16
17
17
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19 2U
20 21 21 22
22 23 23
24 Z2
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25 26 26
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THETA
00 00 01580443260-02 2-41823
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56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
31 2
32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
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VVBY
09244b3373 00 00
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THEIA
0333155355 -03 264571)
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04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
R1 Scroll radius of curvature Fig la
B Width of the scroll exit Figs 2 and 3
H Length of scroll exit nozzle Figs 2 and 3
R Radius of the scroll cross-section equal D2 Figs 2 and 3
RC Radius of exit portion of the unsymmetric crossshysection Fig 3
Third Set (Control Parameters) one card
READ ICOM ISYM IFG IPAP IFREVO
according to format (5110)
ICOM Parameter to control type of calculationsICOM = 0 for incompressible flow and ICOM = 1for compressible flow
ISYM Parameter to control type of scroll cross sectionFor symmetric cross-section ISYM = 1 and forunsymmetric cross section ISYM = 0
IFG IPAP Parameters to control the mass source distributionIFREVO (ie the type of through flow velocity profile)
(i) IFG = 1 IPAP = 0 IFREVO = 0 for anarbitrary source distribution In this casethe value of the mass source is fed as inputat all the interior mesh points
(ii) IPAP = 1 IFG = 0 IFREVO = 0 for a circularparaboloid source distribution
(iii) IFREVO = 1 IPAP = 0 IFG = 0 for freevortex source distribution
(iv) IFG = 0 IPAP = 0 IFREVO = 0 represents theuniform source distribution
Fourth Set (Numerical Parameters) one card
READ ITMAX EPSMAX WO WOP
according to format (110 3F100)
ITMAX Maximum allowable number of iterations
EPSMAX The largest tolerable value of the square of the sum of the absolute values of the deviations of qij from their previously computed values
WO WOP Successive relaxation factors for interior mesh points and for the points adjacent to the boundaries respectively
39
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
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ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
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NO Or t NirIAT I 01Sm 250
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
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7 7
8 8
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12 13
13 1
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THETA
00 00 01580443260-02 2-41823
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-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
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3
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2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
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VVBY
09244b3373 00 00
09249500U9J 00 00 094001907 00 00
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THEIA
0333155355 -03 264571)
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04074424200-01 3144176
04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
Fifth Set (Output Control Parameters) one card
READ IP IVPLOT IPLT
according to format (3110)
IP Frequence of intermediate printout Solutions areprinted after every IP iterations
IVPLOT Parameter to control plotting of program outputIVPLOT = 1 a plot is prepared if IVPLOT = 0 no plot is prepared
IPLT Number of contours in the output plot for thepotential velocity
Sixth Set (Values for Velocity Potential Contour Plotting) number
of cards is equal to IPLT
READ VISOBR
according to format (F100)
VISOBR Numerical values of the velocity potentialcontours to be plotted as output
Seventh Set (Arbitrary Source Distribution)
Is required only in the case of arbitrary source disshy
tribution ie IFG = 1 The value of the source distribution
F(IJ) is read in DO loop according to format (8F100)
The input data is fed starting from I = 1 to I = M3 and
marching in J direction from J = JL JU as shown in figure
1A
Output
The program output includes a printout of the pertinent flow
properties at all the grid points every IP iterations and of
two figures one for the desired velocity potential contours as
specified by IPLT and VISOBR and the second showing the velocity
direction in the cross-sectional plane
Explanation of the labels of the print output are given
below
F Vector containing the values of Poissons sourcestrength at each grid point
P Vector containing the values of the potentialfunction 6 at all grid points
40
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
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III I9046~ Il = 007447
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Id
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ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
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VELOCITY COMP UampV IN THE XZ Y DIR
I tU V
2 1 05406364730-03 00 2 2 0433514420D-03 -08646499000-03
2 3 0275906843DM03 0192937381D-03 2 4 01247729tD-0J 013b6382440-02
2 5 04809022790-03 -0-5216054200-03
3 1 OtflO58976D-02 00 3 2 01o68033090-02 -0oI138356640-02 3 3 01306a7575D-02 -shyO343931741D-03 3 4 06979866D-03 0365976810D-03
3 5 -02904816320-0J 0718392642D-03
3 6 05-6919660-03 0-6482817260-03 A 1 0339384838D-02 00
4 2 0329384990-02 -01359363910-02 4 3 0298440632D-00 -07775970450-03 4 4 0248b70142D-02 -0240974547D-03 4 5 OQ18721259b-02 02166834710-03
4 e 0144544030--02 0743053505D-03 4 7 O10b5493810-02 O14jso94190-02
4 8 02188441460-02 0b718J1307D--3 5 1 0528S26687D-02 00 5 2 0520019007D-02 -054779 90-02
5 3 0491770300D-O0 -0I149080250-02 5 4 0446b57375D-02 -074630393D-03
6 b 0390713592D-OZ -0324068478--03 5 6 0332735810D-02 017756737D1-03 5 7 0267670539D-02 Odeg80JboyeO-03 5 8 01a8073729D-2 0I2800 860-02
6 I 074816009D--u 00 6 4 073q926896D-0Q -O T26927780-02 b 3 O1 tb7b54D-UZ - L1IO02133D-02
6 406b459983I0-02 -01251$420JUh02 C 5 0bt)0473019D-02 -0$2J571010-03 6 6 0 35958072U-0 -u48arJ6966-vuj
5 7 O459939b980-02 0484733781D-04 6 8 O3S0o0JTbZD-2 0810603050-01 6 9 O300I 47120-02 01 50t85310-02
7 1 0100145109D-01 00 7 2 0-9 2012633D-02 -01917V822-50-02 7 3 O
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
0b6t5d2D-Ue -0t6bT ubJO-Oe -o 8061o -oi
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10 7 01400323310-01 -0461122370-02 10 a 01242763380-01 -0894658037U-02 10 9 01089319100-01 03246507080-0Z to 10 09420318070-02 -01092549030-02
11 1 0250080345U-01 00 12 2 0247295792D-01 -03272497200-02 11 1 0238236040-01 -0465750840-OZ
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13 1 OJ759311363D-0 00 13 z 0369567035D--01 -0 49767b78O-02 13 1 03504739OD--01 -0767970759D-02 13 4 OJ22605456-l -094519b0160--0 13 5 0290473b6W-O -011072o08O0-0 13 6 02078087070-01 -01129502600-01 13 7 02274224610-01 -0107773504U-01 I3 b 02007312000-01 - 70bg2b13U-04 13 9 01787774U60-0 -08277044200-02
13 10 016218710ID-0l -05269839340-U2 3 1 0 b32039b00-01 00
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16 3 0619080329D-01
16 4 05222507440-0 16 5 04318463710-01 16 b 03b61939240-01
16 7 0297027034D-01 16 a 02525984700-01
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17 2 O98189811D-Ul 17 3 074b9917140-01
17 4 0b92371118D-03 17 5 046u2438250-01
17 6 0370412024 -01 17 7 0302335544D-Ot 17 8 02o37106080-01
Ie I 0132606904c 00
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ORIdINAL PAGE ISOF POR QUALIIY
50
28 1 28 2
29 1 29 2 30 1
30 2 31 1 31 2
32 1 32 2
UBK2
-068889924bD-03
04121553900-03
04064117160-03
0187194278D-02
0307810624D-02
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051303787D-02
05869087620-02
00
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0171122738D-02
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O12310962 01
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09$24333140 00 098252S7950 00
09957920860 00 09958142570 00
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-01 i4229893D-02
0549540520D-03
03983631 2D-03
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00
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51
03449630470-01
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
0339770359D-O2 025402769002o Uz43905u001 -O1494910U
01593413760-02 02150bbUD-02 02130231290-01 -00317l81OD-UI
ABSOLUTE VELOCITY C
2 2 2 2
2
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4 4 a
4 4 4
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5 5 5 5
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6 6
6 6
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WITH TE X COO
VV
OS40636473D-03 09672405090-03
03366740540-03 0136210924D-02
0709522853D-03
0t8015897D-02
02019452950-02 0135137459D-02 078708Z3210-03
0774098423-03
0048167061-03 0339348380-02
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02289242963-02 052852b667gt-02
0542b4812D-02 050501b7lb-z
0452947945-02 0392061865002
02796393760-02
O203437049)-02 0748160094j-02
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03085b2bebO-02
039945219ID-02 010014bt090-01
0101077730-Ut
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THETA
002966186
349646847445
3126612
00
3256789
3452464276861
1I20397
498476 00
3375649 34538683544602
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22 0~8d56446 00 005 25 I 092167651) 00 00 25 2 0893J64534) 00 0001 26 1 09040658800 00 00 26 2 0900934770 00 00649
55
ORIGINAL PAGE IS
OF POOR QUALITY
27
27
28
z2
29 29 30 30
31 31 32
32
2 2
3 3 4
4
5 5
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7 7
8 8
9
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12 13
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17
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00 00 01580443260-02 2-41823
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-_
56
27 1 27 2 28 1
28 2 29 1 29 2
30 1 30 2
31 1
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32 1
32 2
3
2
2 2
1 3 2 3
4 2 4 1 5
2 5
b 2 6 1 7 2 7
8
2 8
9 2 9
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VVBY
09244b3373 00 00
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THEIA
0333155355 -03 264571)
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08782424470-03 803728 0I55522518) 00 2874401 0183615469D-02 858777
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04247129490-02 368700 03049732360-01 3231207 0267629951D-02 534605
OZ3b69236u0-01 3341487
57
PBX PBY Vectors containing the values of boundaries
on the scroll
VV Vector containing the absolute values of the velocity vector V at each grid point
VVBX VVBY Vectors containing the-absolute va-l-ues- of the Velocity vector V at the boundary points for each I and J mesh lines respectively
THETA Vector containing the values of the angle between the velocity vector V and the x-axis at each grid point
41
IDATA UfI IlL SCtOILL ClPrS Ct I I(iJ
GA tUNSTAJT 211140000 (ANA 140000 STAG TLMP - 300000UO FIAD EXIT VCL- 200o0000
I 2
)O 1270000
III I9046~ Il = 007447
II Ob9fr4 RL 00 1) 940000 UNIT DIM e-0
Id
In 0 IPAP 0 I r vo= 0 T
ITMAX 2O0
LPSMAX=O0(00000 I to 40 0950100
WP = 0 000
IV 400 1 VIPLtpt= I IPLT 20
IX = 005000 )Y = 005000 IC II JC a MI J2 N3 = A
PtIoSU45 ampuvIlct DIS AT I-AC 01O1 PUINT
0219090J 0I91905 0 2196905 02190005 02I94 9ub ((0 00 00 00 00
I = 3 0l2(9u5 0219091 0 02910105 UoeI9t5 OS OflUt95 U0 OU U0 0093 O191905
I = 4 02(01 0 2140905 0 19tI0So U 0) lO 19 U905 0-1 90905 02 1P4105 00 00O)9t9196190
-a U 1 t 9 0 00
0 I 119 = o0r 0 51100905~lO 0 2 19090b 0bull 211090b U229U90D 0 190U5u 0 191I)95 b 0 1 5 U0
U219090S 02191905 OPI9090 02190905 02198905 0219f905 0 2191905 0-4190905 0419U905 00
I = 02191905
7
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ORIGINAL PAGE IS=549o9270-o-OF POOR QUALITY
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ORIdINAL PAGE ISOF POR QUALIIY
50
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29 1 29 2 30 1
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
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ABSOLUTE VELOCITY C
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ORIGINAL PAGE IS
OF POOR QUALITY
27
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OIGINAL PAGE 18
OF POOR QUkLI Y
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ORIGINAL PAGE IS
OF POOR QUALITY
27
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OIGINAL PAGE 18
OF POOR QUkLI Y
52
0339770359D-O2 025402769002o Uz43905u001 -O1494910U
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