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TITLE:
Fiow Simulation Using the Finite Analytic Method
with Mdti-Block Body-Fitted Gnds
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Date: 9 December 19,1997
TABLE OF CONTENTS
LIST OF FIGURES .................................... .............................. ................................. v
LIST OF SYMBOf AND ANDBREVLAmONS ....,........................................ ......... vii ... ACKNOWLEDGEMENTS ............................. .... ..... ....
ABSTRACT ................... .... .................................................................................. k
1 . INTRODUCTION ................... ... ...... ... ..... .... ..............*................... 1
............................ 2 . THE TWO-DIMENSIONAL FINITE ANALYTIC METHOD 5
2.1 Principles of the FAM ........................ .......................................... .................... 5
2.2 Selection of FAM Ceil Boundary Function ...................................................... 7
.......................................... 2.3 Numericd Problems Evaluating FAM Coefficients 11
............................................................... 3 . BODY-FI'ITED GRID GENERATION 17
3.1 Background ................................................. .......... ......................... 17
3.2 Elliptic Grid Generation - the Laplace System ..................................... ............ 18
3.3 Elliptic Grid Generation - the Poisson System ..................................... .......... 22
4 . THE TRANSPORT EQUATIONS ........................................................................ 26
4.1 The General Scalar ................. .. .......... ............................................................ 26
4.2 The Momentun equations ............................................................................. 3 0
4.3 Continuity and the pressure and pressure correction equations ....................... 34
4.4 Overail Solution Procedure ............................................................................... 37
5 . MULTI-BLOCK CONSIDERATIONS ................................................................. 39
5.1 Background ................... .. ............................................................................ 39
............................ 5.2 Corner Grid Points ..... ............................................................ 40
.................................................. 5.3 CeU-centred Gzid Points ................................... 41
.................... 5.4 Face-centred Grid Points ... ....................................................... 42
. 6 SIMULATIONS ....................................................................................................... 43
6.1 Flow Simulation Software - CFDnet ............. ........... ................................ 43
6.2 Cavity Flow ...............* ......................... ...... ............................................... 43
63.1 Cornparison to the Finite Difference Method ............................................. 44
................................................................................... 6.2.2 Non-oahogonal grid 4 6
...................................................................... 6.3 Flow Around a Circular Cylinder 4 8
6.3.1 Background ................... ......................................................................... 48
6.3.2 Geometry and Boundary Conditions ...................... ............................ 4 9
............................................................................................................ 6.3.3 Grid 51
6.3.4 Tirne Step ........................ ...................................................................... 51
...................................................................................... 6.3.5 Steady flow Results 52
6.3.6 Unsteady Flow Results ................... ........ ......................................... 5 4
7 . CONCLUSIONS ................... ...... .................................................................... 5 7
REFERENCES ............... ......... ................ ........ . 59
LIST OF FIGURES
Figure 2.1 Typical FAM Calculaaon CeU ......................................................................................... 5
Figure 2.2 Comparison of FAM Cell Boundarg Functions Along the Northem CeU ............................................................................ Boundary . CeU Reynolds Number 2AGG 8
Figure 2.3 Nurnber of Surnmation Ternis. Accwacy. and Preusion Used Over a Range of ............................................... Celi Reynolds Nurnbus for FAM Coefficient Cdculations 13
Figure 2.4 Geometry and Boundary Conditions for the Convection of a Step Disconanuity ...................................................................................................................................... Problem 14
Figure 2.5 Variation in 4 dong vertical centre-line of domain. Re=50.000. angle of .......................................................................... convection dpha=45O. N=100 x 100 cells 15
Figure 2.6 Sensitiviry of Numerical Diffusion to convection angle. Re=50.000 ...................... 16
............................................................................. Figure 3.1 Sarnple Structured C d n e a r Grid 17
............................................... Figure 3.2 Arrangement of Neighbours on the Curvilinear Grid 20
Figure 3.3 Effect of P and Q Control Functions Near a Southeni B o u n d q (q=0) F o m p s o n et al 19851 ............................................................................................................. 23
....... Figure 3.4 Sampk Single Block hfesh using Elliptic Grid Generaaon . Poisson System 25
....... Figure 3.5 Sample Single Block Mesh using Elliptic Grid Generation . Poisson Syscem 25
.......................... Figure 4.1 Non-orthogonal FAM Ceii for West Face Velouty Components 31
Figure 4.2 Generic Continuity Control Volume ........................................................................... 35
Figure 5.1 Sample Multi-block Stnicnired Gnd with Six Blocks ............................................... 39
...... Figure 5.2 Arrangement of Cell-Centred Neighboun - Four Neighbour Discretizations 41
..... Figure 5.3 Arrangement of Cd-Centred Neighbours . Eight Neighbour Discretizations 41
Figure 5.4 Arrangement of Face-Cenued Nughbours on an "Active" Block Boundary . Eight Neighbour Discretkations ......................................................................................... 42
Figure 6.1 Geomeûy and Boundary Condiaons for the Lid-Driven Caviry Fiow Problem . -44
Figure 6.2 Calculated u velocicy dong the Caviv Centerline. Re=400 ...................................... 45
Figure 6.3 Cornparison of total computation tkne, Cavity flow problem, Re=1000 .............. 45
Figure 6.4 Comparison of Orthogonal and Non-orthogonal Grids for the Cavity Flow Problem ...................................................................................................................................... 46
Figure 6.5 Comparison of Cdculated Solutions for the Cavicy Flow Problem. Sueamlines. Re=400 ....................................................................................................................................... 47
Figure 6.6 Comparison of Cdculated Solutions for the Cavity Flow Problern. Pressure Contours. Re=400 .................................................................................................................... 47
Figure 6.7 Calcdated u velocity dong the Cavity Centerline. Re=400 .................................. 47
Figure 6.8 Geometry and Boundary Conditions for the Cylinder Flow Problem .................... 49
Figure 6.9 Detail of the 0.25D Grid Near the Cylinder ............................................................... 51
Figure 6.10 Cornparison of Calculated Cylinder Flow to Expenmental Data of Tancda [1979] aac Re=26 . (Strearntraces are used for visualization. streamtrace spacing has no
.............................................................................................................................. significance) 5 3
Figure 6.1 1 Streamline Visualization for the unsteady flow around a cylinder ac 4 time steps .......................................................................................................................... (dt= 1. Re= 100) 54
Figure 6.12 Stceamhrmion value in the wake. 5D behind the cylinder on the axis of ............................................................................................ symmetry. Re=100. 0.1D mesh 55
Figure 6.13 Cornparison of Calculated Strouhal Number to Experimend Resuia Fritton 19711 .......................................................................................................................................... 56
LIST OF SYMBOLS AND ABBREWATIONS
F~~
FDM
Re
Finite halyt ic Method
Fini te Difference Method
Reynolds number
Cartesian coordinate directions
F M convection speed coefficients, evaiuated at point "P"
FAiM neighbour and source coefficients, respectively
generd scalar
diffusion coefficient
fluid density-
nuid dynamic viscosity
Source tm
Cartesian veloaty components in the x and y directions, respectivdy
generalized curvilinear coordinate directions
ve1oQt-y components in the local 5 and c d n e a r coordinate directions
Jacobian of the inverse coordinate transformation
inverse metric coefficients
Contravariant velocity components
pressure
mommnim equation pressure coeffiaent
ACKNOWLEDGEMENTS
1 wish to gratefülly acknowledge the input and support of Dr. Julia ;Milimer, as weil as the
other talented researchers previously under his s u p e ~ s i o n , induding Mohammad Reza
Shariati, Ali Mahallau, and Dr. Yuping Sun. Tiieir substantial body of work provided the
basis for the presenr conaibuaon. Appreciation is due ro Dr. F. Hamddahpur and Dr. G.
Kember for reviewing this thesis and pa r t i üpa~g on die examination comrnitree. Speual
acknowledgrnent is also given to Antonio Bemfica and Dr. Jim Chuang, whose computer
expertise provided die parallei computing environment used for the simulations of d u s
work.
1 also wish to express my deepest gratitude to my wife, Melissa Day, for her unending
support and undersmding.
The principal objective of the present work was to study several of the problems associated
with the traditional Finice ha ly t ic Method ~Ahf), and then propose and test solutions to
these probluns chat improve both the accuracy and applicability of the method for solving
fluid flow problems, pdcu l ady at relatively high c d Reynolds numbers. The identified
problems and cheir proposed solutions indude an investigation of the FAM cd boundary
fimcaon, and development of new FAM coefficient expressions using a cosine boundary
h c t i o n . The c d Reynolds number iirnit, normally evperienced by Finite Analycic
algorithms, is overcome by introduchg coeffiaent calculations based on ubitrary preùsion
arithmetic. Control h c t i o n s for eiiiptic grid generation are proposed diat produce
smctured body-fined grïds in whidi dic gnd iines are orthogonal at the boundaries. h novel
solution procedure based on primitive variables is presented in which the discretizntion
equations for the u and v Carresian vdocity components on a staggered, body-fitted grid are
developed Çrom an aigebraic manipulation of the local curviiinear velouty components. This
method results in strongly convergent and uncomplicated pressure and pressure correction
equations, and avoids the tensor algebra normally assouared with curvdinear velocity
cornponenrs. Fially, to accommodate multi-block grids, a strategy for handiing ghost poincs
around each block and assigning grid neighbours near the block boundaries is proposed.
These proposed modifications to the traditional F A ? are incorponted into a computaaonal
fluid dynamics program c d e d CFDnet, which is used sirnulate nvo fiows and test the
accuracy and stability of the overd solution procedure. The Ekst is the steady Ivninar
recirculating square cavity flow. Solutions are compared widi those produced by a traditional
h i t e difference technique, considering both accuracy and corn putation time. Grid
independence is established by comparing the caviq flow solutions produced on a highly
non-orthogonal grid to those produced on an orthogonal grid, and other numericd data
Eiom the iiterature. In the second simulation, the s tady and unsteady flow around a circuiar
cyiinder is solved for the range of Reynolds nurnbers R e 4 to 100. Using a rdativeiy coarse
gnd, accurate predictions were obtained for flow profles, size and location of artached
vortices, the transitionai Reynolds number from steady to unsteady flow, and the vortex
shedding frequency (Strouhal number).
1. INTRODUCTION
The Finite Difference Method (FDM) is one of the most Mdely applied numerical mediods
to solve fluid flow and heat cransfer problems [Gerald and Vlrhearley 19841. In the FDM, the
convection and diffusion tems in the goveming partial differential equations are
approximated by different order mncated Taylor series. This approximation c m resuit in
undesirable numerical diffusion [Patankar 19801, particularly in convection-dominated flows
where the grid is not aLigned with the flow direction. The Finite h a l y t i c Method (FrûCI),
introduced by C. J. Chen et al. 119801, addresses some of the limitations of the conventionai
FDM. In the FAM, the caiculaaon domain is sunilarly divided into suitably smaii cells to
f o m a sauchued grid. The FAM, however, uses the local analytic solution to the linearized
governing partial differential equaaons to relate the field variable value(s) at a grid node to
the values at ail its surrounding neighbom. The main advantage of the FAiM is the
automatic simulation of the convection-diffusion e f fec~ which minimizes numerical
diffùsion and instabilities, particularly at high Reynolds numbers [Sun and blilitzer 19921.
Despite this strong advantage, severai perceived or acmd shortcomings have prevented the
FAM from reaching its fd potenaal as a viable aitemative to the FDM for solving practical
problems of fluid flow and heat m s f e r .
Perhaps the most common problem associated with the Fm[ is the percepaon that the
computation tirne is significantly longer than the traditional FDM, due to the increased
complexity of the coefficient calculations. Some authors report that this increase is slighdy
compensated by the supenor stability and convergence characteristics of the FAM, resulting
in fewer global iterations to achieve a solution [Sociropouios et al 1994; Shariati et al 19951.
In the present contribution, FAM coeffiaent calculations are developed based on a cosine
boundary function. Unlike traditional irnplementations presented in the Literature [Chen ec al
1980; Shariati et ai 19961, a calculation procedure that attempts to minimize the nurnber of
floacing point operations is proposed. A direct cornparison of solution accuracy and total
solution time is made benveen the resulting implemenation of the FAV and the traditional
FDM.
Another problem îssociated with uaditional implementations of the FAV is the cell
Revnolds number limit, which occurs when the infinite series involved in the F M
coefficient calculations become inaccurate at high c d Reynolds nurnbers due to the tinite
precision of computer arithmetic. Consequently, traditional implemenntions of the FAiM
cannot be applied to problems where the ce11 Reynolds number, defined in terms of the
FAIM cell dimension, exceeds about 60. In the present contribution, an initial calculation is
oetfonned to esumate the numericd predsion required for accurate coefficient calculations, L
and, if the required accuracy exceeds the standard computer accuracy, special summation
routines are used to accurateiy calculate the coefficients. These calculations use the GNCT-
iLP publidy available Lïbrary functions for a r b i q preusion arithmetic [GNU-MF 19961.
Most problems of practical interest involve complex or c w e d geornemes. Such geomemes L
can be handled in general by discretizing and solving the nansforrned governing equanons
on a NNilinear, body-fitted grid F o m p s o n et al 19851. These transformation techniques
have been applied to the solution of fluid flow problems based on the F M where the
choice of dependent variables has either been the u and v Cartesian velocity components
aligned with the global x and y directions [Sharîaa et al 19951, or the curvilliear veloury
cornponents aligned with the local cunrilinw coordinate directions [Sotiropoulos et al 19941.
In the case of the former, die applicability of the scheme may depend on the onentaoon ot
the grid relative to the Cartesian reference frame If(arki and Patankar 19891. In the case ot
the latter, complev source terms resuit fiom the aansfomiation of die momentum equarions
into curvilinea coordinates mki and Patankar 19891. In the presenc contribution, a novei
solution procedure using u and v Cartesian velocity components on a staggered, body-fitted
gxid is presented. The discretkation equations for the Cartesian velocity components are
deveio~ed from an algebraic manipulation of the local cwvihear veloucy components. This
rnerhod results in strongly convergent and uncomplicated pressure and pressure correction
equations, and avoids the tensor dgebra nomialiy associated wich cuMlinear velocity
cornponents.
Some problems
cannot easily be
involvuig multiply-connected domains or
modded with a singie body-fitted gid , as
pdcularly complex boundaries
required by the FAIM. in such
cases, a multi-block grid is necessary, where the domain is subdivided into a number of
smder, ' l o g i d y rectangular" blocks F o m p s o n et al 1985; Hauser and W i a m s 1994. h
strucnired mesh is then generated in each block, and the solution process involves the
exchange of information dong the boundaries beween adjacent blocks. Because
discretizaaon equations based on the F M include the influences of ail grid point
neighbours, including the diagonal neighbows, speùai considerations are required at the
block boundaries and particdarly the block corners. In the present contribution, a strategy
for assembhg grid point neighbours near the boundaries of blocks on a general multi-block
grid is proposed and tested.
These modifications to the traditional F M have led to the development of a general nvo-
dimensional computational fluid dynarnics solver for transient incompressible flows cailed
CFDnet [Ham and il[ilitzer 199q. To demonstrate the accuraq, grid independence, and
robusmess of CFDnet, two recirculating flows were simuiated.
The fist simulation was of the steady laminar recirculating square caviry flow. Due to the
relatively simple geometry and strongly convergent nature of this problem, it is commonly
used as a test problem for flow simulation algoriduns [Shariaa et al 1996; Sun and hLiliaer
1993; hksoy et al 19921. Solutions were compared with those produced by a traditional finite
difference technique based on die power law [Patankar 19801, considering both accwacy and
total cornputaaon time. Grid independence was established by comparing the cavity flow
soluaons produced on a highly non-orthogonal grid to those produced on an orthogonal
grid, and to other numericd data kom die literature plukhopadhyay et al 1993; Peyret and
Taylor 19831.
In the second simulation, the steady and unsteady flow around a circular .Linder was solved
for the range of Reynolds numbers Re=O to 100, comparing results with both experimental
and numerical data from the Literature Fritton 1971; Taneda 19791. Using a relaavdy coarse
grid, accurate predictions were obtained for flow protiles, size and location of attached
vortices, the transitionai Reynolds number from steady to unsteady Elow, and the vortex
shedding frequency (Strouhal number).
The aforementioned work is organized in the chapters of the present contribution as
foliows. Chapter 2 presents the fundamental aspects of the FAV, induding the boundary
function selection and how to overcome coefficient evaluation problems. Chapter 3 presents
the development of the body-fitted structured grid generation as applied to the method
presented here. Chapter 4 presents the disaetkation of the two-dimensional transport
equation for the general scalar on the body-fitted gnd. The method is then extended CO die
m o m e n m equations. A k r this the continuity equation is used to develop the pressure and
pressure correction equauons. Having discretized al1 equaaons, an overaii soluaon procedure
for incompressible flow problems is presented. Chapter 5 presents the proposed strategy for
handling multi-block grids with the FAM. Chapter 6 presents the results for the two flow
simulations and their cornparison with available data. Firndy, chapter 7 presents the major
conclusions and recommendations for future work.
2. THE TWO-DIMENSIONAL FINITE ANALYTIC METHOD
2.1 Principles of the FAM
The numerical solution of problems in fluid flow and heat transfer using the FAM requires
the division of the problem domain into a smctured mesh consisting of small rectangular
cells, as shown in Figure 2.1.
Figure 2.1 Typicd FAM Caiculation Ceii
each ce& the exact analytic solution to the linearized govemlig partial differential
equation(s) is used to relate the field variable value(s) at the centre grid node "P" to its
surroundhg neighbours. Detailed denvation and mathematicai proof of the rnethod is
available in the Literanire [Chen and Li 1980; Sun and Milieer 19921, however, to provide a
basis for discussion, the method is summarized here.
The convection-difhsion equation goverring the nansfer of mass, heat, momentun, and
other scalars for 2-dimensional, transient flow problems can be written in terms of the
general scalar @ as
where p is the £luid density, u and v the components of velociry in the x and y directions
respectively, r the diffusion coefiaenf and S the appropriate source tem. The scalar 4
might represent a component of velodty, die energy, the turbulent kinetic energy, the
turbulent dissipation, or the concentration of a certain species.
Discretization by the FAM requires the Linearization of Eq. (2.1) over die Fm1 calcdation
ceil. Ce1 velodties u and v, densiry p, diffusion coefficient T, the source term S, and the time
derivaave, are aU assurned constant and equal to the value at the center of the FA\[
calculation ceil - node "Pm. The hearized result is most comrnonly expressed
where the convection speeds are dehed
and the forcing term includes both the source rem and the time derivative, expressed as a
backward difference in terms of the ame step, A t , and the previous solution, 0"
Transformation of the dependent variable in Eq. (2.2) using
yields the following homogeneous equation for which an analytic solution exists
B y assurning a hc t iona l variation between neighbourhg 4 ,, values around the entire
boundary of the F M cell, the analyàc solution of Eq. (2.5) cm be found duoughout the
c d by the method of separation of variables. Evaluaaon of the analytic solution at the
center grid node ''Y yields the following algebraic expression
where C, and Cf are the neighbour and source Fm[ coefficients
expressions depend on the choice of boundary b c t i o n specified
celi.
respectively, whose exact
on the FUI calculaaon
2.2 Selection of FAM Cell Boundary Function
The accuracy of the F M , as weU as the mathematical compledty of the coefficient
expressions, depends, to some degree, on the selection of a suitable Frtçl cell boundary
function. The most widely applied funcaonal variation is the exponential plus iinear function
[Chen and Li 1980; Chen et ai 19881. Beween the 3 neighboun on the northem boundq ,
for example, the variation of$, is assumed to be
where the coefficients a, b, and c are d e h e d
This ensures that the ~ c ü o n passes tlilough each of the boundary values. This choice of
boundary hnction results in reiatively simple expressions for the Fmf coefficients, and
reduces the nurnber of infinite summauons thac need to be calculated CO one, thus
consenring computation rime. However, as origindy addressed by Sun [1992], this boundary
function violates the mawimum-minimum p ~ c i p l e required by the homogeneous equation,
Eq. (2.5), and may thus lead to physicdy wealisac results. Consider, for example, the
~otential case when the NW, NC, and NE homogeneous boundary values are 5,10, and 10 L
respectively. Figure 2.2 shows that the Linear plus exponenual hct ional variation of Eq.
(2.7) may generate values outside the range of the actual boundary values, thus violaang the
maximum-minimum principle.
Figure 2 2 Cornparison of FAM CeIl Boundaq Functions Along the Nonbem C d Boundary - CeU Reynoids Number U h = 6
To overcome this problem, a piece-wise hc t iona l variation c m be applied benveen the
neighbours Seiecuon of an appropriate piece-wise €unaion will ensure rhat die maximum-
minimum principle is satisfied.
In the present work, the following piece-wise cosine function is assumed.
By Limiting the range of the cosine function to either -n/2 co O or O to x/2 , the maximum-
minMun p ~ c i p l e is saasfied. The coeffiaenc expressions based on the cosine h a i o n are
now presented, wrinen in a way to minimue the number of floating point operations
required in their calculation.
In NIO dimensions, the infinite summations required for the FAIM coefficient calculations
depend only on the foiiowing 3 non-dimensional parameters, associated with the ceil
Reynolds numbers and cell aspect ratio.
Within the summation loop, 12 unique surnmations must be cdculated. In the present work,
the summation loop was terminated when the change in the highest order summaaons (S,.,
and S ,.,) no longer affected the k t 8 signi fican t digits.
From the above sumrnations, it is convenient to calculate the following ternis that are
cornmon to the coefficients.
From these terrns, the neighbour coefficients c m be compactly written as
-Bk C , = e Tx, + e*Ty, -Bk
C, = e TL4 -Bk C,, = e T,,, +e-*~,. , Ah
C W C = e q . 4
-Ah CEc=e Ty.4
Csw = e B k ~ , , + e * ~ , , , Bk
C, = e TX.4
Cs, = e B k ~ x , + e-Ah~y,
and the source coefficient
2.3 Numerical Problems Evaluating F A . Coefficients
For low cell Reynolds nurnben (2Ah or 2Bk < about 60), the inhi te summations of die
FAM coefficients, Eqs. (2.1 1) and (2.12), converge rapidly, and can be vuncated \kithout loss
of accuracy after about 25 terms. As the cell Reynolds number increases, the nurnber of
terms required to produce converged summations increases, requiring increased
computauon cime. At even higher cell Reynolds numbers, however, a more serious problem
arises: the summations themselves become numencaily inaccurate irrespective of the number
of terms. This is a direct result of the LiMted numerical accuracy of the cornputer, and can
make the FtLM unstable or, in the worst case, highly divergent
Ensuring accurate calculacion of the FAM coeffiaents ar high ceIl Reynolds number requires
arithmetic precision higher than the 64 bit "double" precision usudy available. To overcome
this problem, the sumrnauon calculations required for the FiUV coeffiaents were modified
to incorporate the publidy available GNU-MP library of functions for arbinary precision
arithmetic [GNU-MP 1 9961. To ensure the slower high precision routines are used only
when required, it is necessary to devdop a preusion requirement calcuktion that can be
perforrned at the start of eve y celi's coefficient cakulaaon to determine whecher n o d
"double" precision routines c m be used, or what level of arbitrary preusion is required in
the GNU-MP routines. An expression for the preüsion requirement c m be developed based
on an order analysis of the coefficient expressions as follows.
Through inspection of the infinite sumrnations of the FMf coefficient expressions, Eq.
(2.1 1) and Eq. (2.1 2), it is apparent thac the largest terms in the sumrnations, and thus the
terms that govem the maximum preùsion required, occur at relative. s m d values of the
index n. Recognizing that the largest FA.%[ coefficients ML1 be of order 0(1), because the
sum of FAhd coefficients is always exactly 1 [Sun and hliliaer 19921, the absolute numerical
preusion required CO produce accufate sumrnations must be about the order of the lü-gesr
terni Li the coefficient surnmaaons, plus about 10 decimal digits for the numericai accuracy
of the final coefficients. Thus we c m write the approxirnate ccpressions
and
where N is the bit preusion required, and n' is the value of n at which the summ
is maximum. Because it is only necessq to calculate the preusion accurately when the c d
Reynolds nurnber is relaavely high, and there is a possibilig that normal "double" preusion
might be inadequate, Eq. (2.17) and Eq. (2.18) c m be M e r simplified by appro2ùmathg
the hyperbolic cosine expression.
It rernains to determine the srnall value of the index n* at which the summaaon term is
maximum. A numerical investigation of the surnrnation terms over a broad range of cell
Reynolds numbers and aspect ratios has shown that the value of the surnmation index at
which the maximum term occurs varies with the Reynolds nurnber and ceil aspect ratio.
However, the sirnpli£jring assumpaon no = O (except in the numerator, where n' = 1 is
used) produces adequate resuits for these preusion calcuiations. hlgebraic manipulaaon of
Eq.(2.17) and Eq. (2.1 8) then yieids the h a 1 preusion requirement calculation.
1 bit precision N = - [ a b s ( ~ , h) + abs(~, k) + ln(10'9 - rnin(h, k) JA ,' + B,' ]
ld2)
Figure 2.3 presents the required number of surnrnation tems and the actual precision used
for ca icu ia~g FAM coeÇ6cients over a range of ce11 Reynolds numbers. For this particular
figure, convection is considered at 45O to a square FAiM caiculaaon ceil (h=k), and die FAM
coefficients are based on a piece-wise Linear boundary hc t ion .
n u m b e r of ternis in summations 100 --
O coefficient accuracy
80 -- .- 1 .OE-06
number of coefficient ternis in 60 - - accuracy
summations '- ' 'OEa9 (1 -sum(Cnb))
! 1 .OE-18
O 50 100 150 200 250 300 350 400 Cell Reynolds Number (Re=2Ah=2Bk)
64 bit 96 bit , 1 28 bit 160 bit 192 bit ,
normal ' p recision Precision used for caluclations
Figure 23 Number of Summation Tems, Accuracy, and Precision Uscd Ovct a Range of Ccil Reynolds Numbers for FAM Coefficient Calculationr.
The GNU-MP routines step up the preasion Li Licremenn of 32 bits (4 bytes) as required to
meet the preusion requirement calculation of Eq. (2.20). Ln this example, the transition fxom
regular to higher predsion routines occurs at a c d Reynolds number of about 70. Figure 2.3
also presents the coeffiaent accuracy, which makes use of the property chat the sum of the
FAM neighbour coefficients should be exacdy 1 .O.
With the coefficient calculations now extended over the entire range of Reynolds numbers,
it becomes possible to quanti^ the reduction in numerical difision that the F M can
achieve for skewed, highiy convective flows. To do dis, the standard problem of
convection of a step discontinuity on a 1 x 1 square is considered [Patankar 19801.
Figure 2.4 shows the geometry and boundary conditions of the problem. By solving the
convection difhsusion equaaon, Eq. (2.1), on the square domain at high Reynolds nurnber
and source term S=O, difhsive effects will be minimal, and the step in the boundary
conditions should be convected directly through the centre of the domain. Any srnoothing
observed in the step discontinuity away from the boundary can be atuibured to numerical
difhsion r e s u i ~ g from the inaccuracies of the discretization technique.
boundary condition: r
X
boundary condition: +=
Figure 2.4 Geometiy and Boundary Conditions for the Convection of a Step Discontinuity Problem.
Figure 2.5 shows the calculated variation in + dong the vertical cmtreline of the domain for
two different discretization methods: 1) a discretization of the governing equation based on
the F i t e Difference Method (FDM) using the power law [Patankar 19801, and 2) a Fm[
discretization using the GNU-MP library h c t i o n s for arbicrary precision arithmetic. Resdrs
are presented for convection angle a=4S0, with Reynolds Number Re=S0,000. The FAIM
captures the sharpness of the step disconcinuity much more accurately.
-0.5 -0.4 -0.3 -0.2 -0.1 O 0.1 0.2 0.3 0.4 0.5
y (along vertical centre-line of domain)
Figure 2.5 Variation in (I dong vertical centre-lioe of domain, Re=SO,ûûO, angle of convection alpha=45O, N400 x 100 ce&
Quanti+g the nurnericd diffusion as the average absolute variation between the exact step
and the calculated results dong the vertical cenaeiine, the sensiavity of numerical diffusion
to the angle of convection to the grid, a, can be caicdated. Figure 2.6 presena die sensitiviry
of numencal diffusion to convection angle with the average error calculated in th is way.
Because the FAM indudes the influences of al1 eight neighboun, its numencal diffusion
experiences a minimum at 45 degree intervais. The FDM, which indudes the influence of
only 4 neighbours, experiences xninimum numencal diffusion at 90 degree intervals when the
convection is aligned with the grid.
average error
l
O 1
O 5 1 O 15 20 25 30 35 40 45
angk of convection relative to mesh. alpha (degres)
Figure 2.6 Sensitivity of Numerical Diffusion to convection angle, Re=SO,OUO
It c m be conduded that the F M coefficient calcularions incorporaung the GNC-iL[P
arbinary preusion routines produce an accurare and stable scheme capable of significantiy
reducing the numerical diffusion when the angle of convection is skewed with respect to the
3.1 Background
To extend the application of the FAvI to the more generai case of curved geomemes, i t is
necessary to use body-fitted grids. The generation of body-fined grids is discussed
extensively in the literature F o m p s o n et al 1985; niornpson 19841 and specificdy with
respect to the F M [Shariati et ai 19961. As such, only the basic principals and the spedfic
techniques employed in the present work are now presented.
Thompson [1984] describes grid generation as ". . .a procedure for the orderly distribution of
observers over a physical field in a way that efficient communication among the observers is
possible, and d physical phenornena on the entire continuous field may be represented with
sufficient acniracy by this finite collection of observarions." In the present contribution, the
"orderly distribution of observea" was achieved by arrangmg the observers, or grid points,
at the intersections of cuMlinear grid lines distributed chroughour the domain (Figure 3.1).
The resulting grid is commonly refened to as a smctured, curvilinear grid.
Grid lines are indexed as Lines of constant 5 or q. By convention, a grid h e spacing of
A&Aq=l is used. For relaavely simple domains, such as that shown in Figure 3.1, each
domain b o u n d q corresponds to either the first or last cuMLinear grid Line of constant 5 or
q. More complex domains, induding multiply comected domains, can be sub-divided into a
number of simpfer sub-domains, referred to as blocks, and a separate grid generated in each
F o m p s o n et al 1985; Hauser and Willliams 19921. Complete speuficaaon of the grid, then,
involves the specification of (x,y) for every intersection of 5 and q coordinate iines.
Mathematicaliy, the grid lines describe a c&ear coordinate space, where, just as each
point in the domain can be uniquely idenafied by its @,y) coordinate, each point c m dso be
uniquely idenafied by its (5,q) coordinate. That is, there is a one-to-one mapping or
transformation that exists betareen (x,y) and g,Q. If the grid line positions do not change
with cime, this general transformation of independent coordinates can be written
3.2 Elliptic Gnd Generation - the Laplace System
If the values of 5 and 1 are defined dong the dornain boundaries, the calculaaon of 5 and q throughout the intenor of the domain is a dassic boundary value problem, whch c m be
solved by assumlig the distribution of eïther 5 or q throughout the domain is governed by
Laplace's equaaon F o m p s o n et al 198q.
Because Laplace's equauon is part of the ellipâc f d y of partial differential equaaons, this
technique for g e n e r a ~ g body-fitted grids is reférred to as eiiiptic grid generation. In
addition, the homogeneous nature of the goveming equations ensures that the distribution
of (5,q) throughout the domain wi.U be continuous and obey the maximum-minimum
principle (i.e is Limited to values beween the maximum and minimum value of 5 or q
defined on the boundaries), producing smooth, continuous grids.
Because the specification of the mesh requires an (x,y) for every intersection of 5 and
coordinate lines, it is the inverse problem that is generdy solved. Using the chah d e
relations hips,
die inverse boundary value problem for Eq. (3.2) c m be stated
where the coefficients are d e h e d
ax Tenns like - are referred to as inverse memc coefficients, and can be calculated using ac 6nite difkrences based on the grid point x and y values. Because the x and y values are
changing as the grid is calculated, inverse metric coeffiaents must be recalculated as the grid
caiculations converge.
Figure 3.2 Arrangement of Neighbours on the Cunilineu Grid
Discretking Eq. (3.4) on the curvilliear grid dows us to write algebraic expressions that
relate the x and y values ar any intenor grid point 'Y' to the ne ighbou~g x and y values
shown in Figure 3.2.
where
Based on Eq. (3.6), the overali solution procedure for calculacing the grid can be stated as
follows.
1. Divide the encire domain into one or more blocks in which a cuMlinear grid c m be
envisioned.
In each block, define the 5 and q directions, and set the number of grid iines in bodi
directions 0.e- set Qnax and qmax).
Inside each block, define (x,y) for each grid point &q) along the boundaries. In the
present work, a uniform spacing of @d points along the boundaries was used.
Guess an initial @,y) for each internai grid point by interpolation from the boundary
values. Transfinite interpolation is suitable for this iniaai guess F o m p s o n et ai 19851.
Calculate the inverse metric coefficients using central differences based on the curent
grid point locations, then calculate the coefficients for Eq. (3.6) for each interna1 grid
point.
Solve Eq. (3.6) and calculate a new x and y value for each intemal gnd point
Renun to step 5 and repeat und convergence is achieved.
In the present work, the convergence cntena used were based on the mavimum change in x
and y between w o successive iterations, k and k+l, throughout the encire Field.
J is the Jacobian of the inverse coordinate transformation, defined in terms of the inverse
meuics
To make the convergence aiteria of Eq. (3.8) suitable for arbitrary domain scales, the
absolute x and y changes are non-dimensionalized with f i , a characteristic dimension of the
local control volume.
3.3 Eiliptic Grid Generation - the Poisson System
Although grid generation based on the homogeneous Laplace's equaaon guarantees a
smooth and continuous body-fitted grid, it is ofien desirable to exercise some control over
the distribution of grid h e s , such as dustering Lines near a boundary of interest, or
increasing the density of grid lines in a region of high gradients. For this, Thompson et al
[1985] recornmend the use of Poisson-type equations.
P and Q are called control functions, and can have a unique value at every gnd point. By
adjusting P and Q, it becomes possible to exercise control over the disaibution of grid lines.
The inverse problem corresponding to Eq. (3.10) is
Discretizing Eq. (3.1 1) on the cuMLinear grid results in the same algebraic expressions for x
and y as Eq. (3.6) with the source terrns modified as foiiows.
Figure 3.3 Effect of P and Q Control Functions Nez a Southern Boundary (q=O) pompson et al 19851
Figure 3.3 illustrates the effect that negative values of either P or Q have on the local gnd.
Negative values of control function Q WU tend to move the q-lines in the direction of
decreasing 11. Positive values have the opposite effect Similady, negative values of control
funution P tend to move the c-lines in the direction of decreasing 6. %%en the boundary
points are Bxed, as is the case for the constant 5-Lines shown in Figure 3.3, the effect of
negative P-values is to cause the 6-Lines to lean in the direction of decreasing 5.
Thompson et al [1984] propose iterative expressions for P and Q dong the boundary to
produce orthogonal grid lines at the boundary, and a speufied spacing beween the
boundary and the fïrst grid he. Interpolation of the calculated boundary values diroughout
the interior of the domain is then used to produce grid smoothness.
In the present work, the charactenstics of die conuol h c t i o n s were used CO develop new
iterative correction expressions for P and Q on the boundary that produced both orthogonal
grid lines at the boundaries, and even spacing of the first nvo grid Lines moving away frorn
die boundary. The correction expressions for the boundary P and Q values are as follows.
For southern boundaries (q=O):
For northem boundanes (q=qma.x):
For western boundaries (5~0):
c2 Q = Q * - f - J
For eastem boundaries (S=@ax):
In Eqs. (3.13) to (3.16), P* and Q' represent the m e n t values of the conuol functions, P
and Q the new, corrected values, fis a weïghting factor (in die present worlg f=0.01 CO 0.1
was suitable, and smaller values could be used if convergence diEculties were encountered),
C, and J are defined in Eqs. (3.5) and (3.9) respectively, and were calculated using finite
differences benveen the current grid point positions on or near the boundary (one-
directional second order fGte ciifferences were used when necessary), and d l yid d, are the -
distances beween the boundary point and ht point into the domain and the 6rst point and
second point into the domain respectivdy. Inspection of these correction equations shows
that, as the grid ka become orthogonal at the boundary (cl = O), and as the spa&g
between the h t rwo grid lines becomes the same (dl = d,), the correcuon applied to P and
Q approaches zero.
Figure 3.4 Sample Single Block Mesh using EUiptic Gnd Generation - Poisson System
Figure 3.5 Sample Single Block Mesh using Elliptic Grid Generation - Poisson System
Figure 3.4 and Figure 3.5 illustrate two single block grids generated using eiliptic grid
generaaon with P and Q conuol functions as desabed above. Al1 g i d generaaon in the
present work is based on this technique.
4. THE TRANSPORT EQUATIONS
4.1 The General Scalar
As stated in chapter 2, the convection-ciifhision equation goveming the transfer of mass,
heat, momentum, and other scalars for 2-dimensional, transient flow problems cm be
written in ternis of the general scalar as
where p is the fluid density, u and v the componencs of velocity in the x and y directions
respectivdy, r the diffusion coeffiaent, and S the appropriate source term. The scalar 9 might represent a component of veloùty, the energy, the turbulent kuietic energy, the
turbulent dissipaaon, or the concentmaon of a certain species.
Soluaon of the transport equation on a non-orthogonal, curvilinear grid requires the
transformation of the transport equation Lito general curvilinear coordinates. This is
accomplished by using the chain d e relationships of Eq. (3.3) to transform the x and y
partial derivatives into 6 and q partial derivatives and inverse memc coefficients. M e r some
algebraic manipulation, the transport equation in general curvilinear coordinates c m be
written. It is presented here in non-conservative fonn, because this is most convenient for
discretization based on the Finite Analytic Method.
The connavariant velocity components, U and V, are related to the Cartesian vdoaty
components as
The other equauon coefficients are dehned in ternis of the inverse memc coefficients as
çollows.
a 2 x aZy aZy cv mi2 ~c2fi)-+(c,-+c3-- ayni a< 36:
2 kZg) The linearization of Eq. (4.2) throughout the NNilLiear finite analytic cell is accomplished
by assurning ail parametm constant and e q d to cheir value at die ceIl centre point "P".
This assumption aiiows the transformed transport equauon to be rewritten
where the constant coefficients, evaluated based on the values at centre point "P", are
dehed as foiiows. (Noce that both the time derivative and the cross derivative have been
included in the forcing term.)
Inuoduang the coordinate stretching hc t ions proposed by Shaxiati et al [1996],
Eq. (4.6) c m be aansformed and rewtirten as
where the new convection speeds are
The Çorm of Eq. (4.9) is identical to the form of Eq. (2.2), and to the form presented in most
of the iiterature [Chen and Li 1980; Sun and hLiliaer 19921. For the transformed F M ceil,
with dimensions
The finite analytic solution to Eq. (4.7) can be expressed
where C, and Cf are the neighbour and source FA?% coefficients respectively.
Throughout the present work a unit grid spacing was used ( A c = Aq = 1 ), simplibng the
transforrned FAM ceil dimensions.
Subsututing the general expression for the forcing term FP into Eq. (4.10) and expressing the
time derivative as a backward difference,
results in an irnpliat expression for $ at a given Mie step in ternis of its neighbouring 4 values, and @) vaiue from the prevïous cime step:
If an expression for the source term is available, Eq. (4.15) cm be used as the basis for an
implicit solution technique.
4.2 The Momentum equations
The solution of the momennim equations in genaaiized coordinates normally involves the
use of either the p&dy transfomed equations and solution of u and v Cartesian vdoaty
components or the M y transforrned equations, and soluaon of the c d e a r veloary
components. Although the solution of the Cartesian velocity components is preferable due
to the sigmficandy reduced complexity of the momentum equation source cerms P(arki and
Patankar 19891, the solution may be dependent on the relative orientation of the mesh, and
speüal treament of the resultant 9-point pressure equation is required to ensure
convergence [Sharîaü et al 19961.
Karki and Patankar [1989] proposed a solution technique using the cunrilinear velociaes on a
staggered grid. To avoid the complicated source terni normaliy assoaated with the use of
curvilinear components, they derived the discretization equation for the cunrilinear velocity
components from a linear combination of the discretization equations for the Cartesian
velocity components. This technique resulted in a strongly coupled 5-point pressure and
pressure correction equauon. Because, however, neighbouring cunrilinear components are
not necessarily in the sarne direction (due to grid curvanue), it was necessaq co account for
this difference in direction in the source term, r e s u i ~ g in addiaonal cornputation and
bookkeeping. Another problem assouated widi the use of the curvilinear velocity
components occurs in multi-block problems when boundary velodues and menic
coefficients must be passed from one block to anodier. CurviiLiear components on die
boundary of one block will not necessarily correspond to die sarne cuMLLiear components
on the adjacent block's boundary. Furdiemore, any averaging or interpolaaon of velocities
that is required becomes addiaonally cornplex because the components are relative to the
locate curvilinear coordinace vectors, rather than the global x and y directions.
To overcome these problems, a discretization of the momanim equauons was developed
based on the Cartesian u and v components stored at aLi faces on a staggered grid
configuration. Because the F M coeffiaents are the same for both u and v components at a
single point, the additional computation time required to operate on borh components ac
every face is not sigmficant. In any case, both veloaty components are required at di faces
to caldate the FAiM coefficient convection speeds Ap and Bp.
Figure 4.1 Non-orthogonal FAM Celi for West Face Velocity Components
Consider the non-orthogonal, cunrillliear grid shown in Figure 4.1. \Ve desire an expression
for the u and v Carresian vdocity componens from the disaecized m o m e n m equations. It
is possible ro substiture these Cartesian components directly for 6 in Eq. (4.15) to develop
the discretization equations, however the resulting source terms, which involve pressure
derivatives in the x and y directions respectively, will make the source temi expressed in
temis of the curvilinear denvauves somewhat complicated, and derivation of the pressure
and pressure correction equations aiso complicated.
Because the momentum equauon govems the consemation of momentum along any
direction in the flow, not just the x or y direction, we can choose a more suitable direction to
apply the discretized momennun equation. For the western face veiouty shown in Figure
4.1, the momenturn equation can be applied along the local 5 direction, yielding the
foilowing discretbation equation.
c, J' a2uc., - - '[i 1 + a cDbu;.nb + - SSSw - X f c 2 - + -
nb=I p a g h i + a U'*w 1 O
The prime is applied to the neighbouring velocities because they are in the 5 direction as
d e h e d at the West face, and not necessarily in their own local 6 directions, which may differ
due to grid curvature.
The advantage of this choice of coordùiate direction is that the pressure derivative assodated
with the source term can be expressed in terms of the two neighbowing pressures only
(pressure at point "P" and 'TV"). For example, for incompressible Iaminar flow, the source
term for the momentum equation dong the local 5 direcaon c m be expressed
At the same West face location, the local velocity component is assumed to be interpolated
fiom the 4 nearest neighbouring q veloaq components (see Figure 4.1).
where Wnb are appropriate interpolation factors, or weights. Once again, a prime is applied
to the neighbour velodties because they are in the 11-direction as defmed at the west face,
and not necessarily in their own local q-directions, which may differ due to grïd cwanire.
In die present work, di four interpolation factors were sec equal to 0.25, equivalent to h e a r
înrerpolation benveen velociues on the nansformed gnd.
Using the following relationship benveen the local curviiinear velocity components and the
Cartesian u and v components
and the following inverse relauonships
discretizaaon equanons for the Cartesian vdocity componenn c m be developed based on
Eqs. (4.1 6) to (4.1 8). Csing centrai differences to discreuze the pressure derivatives, die
momentum equations c m be written as
where the double hat notarion is used in the spirit of the hat notation introduced in the
original devdopment of SIMPLE and SMPLER [Patankar 19801. The various terms in Eq.
(4.21) are defïned as Çoliows.
where the hatted velocities are defïned:
and the over-barred velocicy terrns are defked:
Using the sarne process, analogous expressions can be derived for the south face Carresian
velocity components.
The double-hat velocities for the southem face have the fotlowing slighdy different
expression.
In Eq. (4.29, ds has the same form as d, in Eq. (4.22).
4.3 Continuity and the pressure and pressure correction equations
The continuig equaaon can be expressed in p e r d cuMLLiear coordinates as
wvhere U and V are the contravariant velociaes defïned in Eq. (43), and J the Jacobian of the
inverse coordliate transformation, d e h e d in Eq. (4.4). To simplify the developrnent,
consider the density independent of time.
/ both u and v
Figure 4.2 Genenc Continuity Conml Volume
Integrating this Eq. (4.27) over the c o n ~ u i t y control volume shown in Figure 4.2 with
A t = AT = 1 yieids:
An equauon for pressure c m be developed by replacing the contravariant velocities in Eq-
(4.28) with expressions in ternis of the vdociry discretizations for the momentum equations
gwen in Eqs. (4.22) and (4.26), pelding.
where the coefficients are dehned:
The double-hatted contravariant
cornponents as
velocities are defined in terms of the double-hatted velou ty
It is this relatively uncomplicated form of die pressure equation which jusafies our earlier
deasions with regard to discretization of the momennim equations.
A pressure correction equation can be developed by considering the iterative process widun
a given time step. Considering the western u vdocity, for example, at some point before
convergence of the t h e step, the incorrect veloucy (denoted with a "star" superscript) d
be related to the incorrect pressure field by the momennim equation discretization as
Correct (at leasc more correct) veloaaes that saas@ the continuiq equauon are related to
correct pressures as
Subuacting Eq. (4.32) kom Eq. (4.33) produces an expression for the con~uity-satisfying u
velocity in terms of the neighbouring correct and incorrect veloaties and the pressure
corrections
where the pressure correction is related to the correct and incorrect pressures by
Eq. (4.34) is sirnplified by setting the nùghbouing velocity corrections contained in the
difference of double hat velochies to zero, recognizhg diat this simplification will not
introduce any error into the final converged solution.
halogous expressions exist for the o t h a velodty components. Subsuniuon of chese
expressions for conànuity-saas*g velocity into the continuicy equanon yields a pressure
correction equation with coefficients identical to the pressure equaaon, Eqs. (4.29) and
(4.30), but with a the following source term.
where the starred contravariant veloàties are calculated kom die starred v e l o ù ~ field using
the definition of contravariant veloaties given in (4.3). Note thac the source terni, b, is m
expression for chc local m a s source, and should eventually reach zero in the converged
solution.
4.4 Overd Solution Procedure
The details of the solution procedure for a uansient problem, based on the discreazation of
momentum and c o n ~ u i t y equations derived above, is now presented. The scheme is s i d u
to the SIhPLER algorithm of Pa& [1980]. In the present work, the impliac caldation
of the pressure and pressure-correction equauons (steps 6 and 8) ac each ireration used the
m-diagonal matrix algo rithm (T'DM) sweeping altematively wes t-easc and souch-no rth to
enhance convergence patankar 1 9801.
1. Define the grid and calculate and store the face-centred inverse memc ternis (e-g. ci, c2.
c3, J, etc.) using central differences.
Set the initial condition for velocity (if the initial condition is not known, such as in a
periodic solution, an initial time step scaling technique may be used, as discussed in the
cylinder flow simulation presented in chapter 6).
Guess the veloaty dismbution. Normaiiy, the initial condition or value from the
previous srep is a adequate.
Select or modify (if desired) the time step.
Calculate the momentun equation coefficients and use expressions Like Eqs (4.22) and
(4.26) and calculate 6, S and d for all faces.
Solve pressure equation, Eq. (4.29).
* A
Use the new pressure field to calculate the face-cmtred veloaties based on the û , Y and
d values calculated in step 4, and expressions like Eqs. (4.21) and (4.25). Consider the
resulting velocity field the uncorrecred, or starred veiocities.
Solve the pressure correction equation using the sraned velocity field from step 7.
Convergence rates were fastesc when the pressure correcaon was zeroed at the star r of
every iteraaon. This corresponds wirh the original recornrnendaûons of Patankar [1980].
Use the pressure corrections to correct die velouty field using expressions Like Eq.
(4.36).
10. R e m to step 4 and repeat u n d the maximum change in primitive variables (u, v, and p)
in any given iteration drops below a given convergence aiterion.
I l . Move to the next tirne step by setring the previous time srep values (uU, vU) to the curent
converged values (u, v), and r e m to step 3. Repear und either a steady solution is
reached, or a speufied number of steps have been calculated, or a specifred total time
has elapsed.
5. MULTI-BLOCK CONSIDERATIONS
S.1 Background
For many problems of practical engineering interest, a single curviiinear stnicmred mesh,
such as that shown in Figure 3.1, is not suitable or perhaps not even possible. This is the
case when the domain is not simply comected, or the domain boundaries cannot be cleaxly
assigned to the rectangular north, south, east, and West boundaries of the cunrilinear
coordinate system. To overcome this problem, Thompson et al [198q suggest the
subdivision of the domain into a number of smailer "logicdy rectangular" regions [Hauser
and Wïiliiarns 19921, c d e d blocks. In each block, a unique cunnlliear grid is generated, and
correspondence of grid points dong the shared block boundaries is forced. The overd
soluaon is then generated by applying the solution procedure to each block and regdarly
updating any necessq dependent variable information dong the boundaries benveen
neighbouring blocks.
Figure 5.1 Sample Mdti-block Structured Gnd with Six Bloch
Figure 5.1 illustrates a mula-block snuccured M d where the domain has been subdivided
into 6 blocks. Each block has at most four neighborisùig blocks, dthough those at physicai
boundaries will have fewer. It is possible to envision the above grid with nvo or more
adjacent blocks combined into a single block In the general case, however, dus could result
in blocks Mth more than four neighbouring blocks, a condition thar would have to be
accommodated in the sharing of information beween block boundaries. In the present
the
work, to simpliQ the solver software as much as possible, the four neighbour Mt was
enforced when 'blocking" the domain. It should be noted chat diis d e in no way limirs
grid in accommodating certain domains, but simply results in a gceacer nurnber of blocks.
A significant advantage of the multi-block grid generauon and solution techniqua is that
thev naturaiiy lend themselves to parallel cornputauon, where a separate computer or CPU
solves each block, and block boundary information is evchanged becween cornputen in the
form of messages. Hauser and Wiarns il9921 applied this multi-block technique to a ceii-
centred 6nite volume scheme, demonstrating a near-hear speedup urith up to 512
processors. At each block boundary, their method required two addiuonal layers of gnd
points, which they c d e d ghost points, to faditate calculaaon of the third-order accurate
fluxes of the solution scheme.
At present, there are no examples of applying che F M to multi-block smictured grids in the
Literam. Because the influence of all eight neighbours is induded in the FhiiM
discretization, speaai attention musr be paid ro the arrangement of ghost points, particulady
near the block corners. In the present wock, the ghost point requirement and d e s for
assigning neighbours were developed for the FAM. These are oudined in the foilowing
sections for each of the three possible locations diat data can be stored on the swggered
grid, nameJy corner, cell-centred, and face-centered.
5.2 Corner Grid Points
Corner grid points refer to the points that Lie ac die intersecaon of the 5 and q gnd lines. In
the present work, corner grid point data induded the r and y values, and the P and Q
control functions used in grid generation (sec Chaprer 3. BODY-FI'ITED GEUD
GENEUTION). Because boundary grid points were set explicidy based on uniform
spaàng, no exchange of corner boundary information between blocks was requxed.
5.3 Cell-cenued Grid Points
Cd-centred grid points refer to the points located at the geometric centre of the ceils
formed by the 6 and q grid iines. In the present work, cell-centred grid point data induded
the pressure and pressure-correction. Both the pressure and pressure-correction equations
involve ody four neighbouring cell-centred values. Consequently, only a single layer of ghost
points are required dong each boundary. Figure 5.2 illustrates that, men when there are
more than four blocks coming together at one corner, no specid neaunmt of the ghost
points is required, and they can be directly considered as neighbours.
NORTHERN \ / / BLOCK
active point (calculated) 0 ghost point (passed from block neighbour)
CURRENT WESTERN 1 1 - ! 1 BLOC%
BLOCK
Figure 5.2 Arrangement of Cd-Cenued Ncighbours - Four Neighbour Discretizations
For the case of ceU-centred values where the discretized govcming equation indudes the
influences of aii eight surrounding neighboun (e.g. calculacion of temperature based on a
F M discretizauon of the energy equaaon), an additional ghost point must be scored ac the
start and end of each set of boundary ghosc points. To accornmodate the general case where
an arbi trq nurnber of blocks meet at a corner, the caldation of the diagonal cd-centred
neighbours may require interpolation between ghost points, as shown in Figure 5.3.
NORTHERN
O
O
O
WESTERN BLOCK
active point (calculated) ghost point (passed from block neighbour) interpolated point
Figure 5 3 Arrangement of GU-Ccntred Ncighbow - Eight Neighbour Discretizations
5.4 Face-centred Grid Points
Face-centred grïd points refer to the points Iocated along the western and southem cell
faces. In the present work, ceii-cenaed grid point data induded the velocicy components,
hatted velocities, d - d u e s (see Eqs (4.21) and (4.25)), and face-centred inverse metric
coefficients. Because some of the face-centred values lie directiy dong a shared b o u n d q
benveen two blocks, it is unnecessary to calculate these values in both blocks. Instead, for
each boundary chat is shared by two blocks, one block is designated as "active", and includes
the calculation of the face values along the b o u n d q , while the second block is designated
"inactive", and it sirnply uses the calculated face values passed from the active block as a
Dirichlet boundary condition.
NORTHERN \ / / BLOCK
active point (calculated) a ghost point (passed from block neighbour) 0 interpolated point
WESTERN 1 j / CURRENT BLOCK BLOCK
Figure 5.4 -ment of Face-Cenmd Neighbours on an "Active" Block Boundvy - Eight Neighbour Discrethations
To accommodate the g e n e d case where an arbirrary number of blocks meet at a corner, the
calculation of certain face-cenaed neighbours may require interpolation between ghost
points, as illustrated in Figure 5.4.
6. SIMULATIONS
6.1 Flow Simulation Sofbare - CFDnet
The grid generation and flow soiution procedures developed in rhe present work were
incomorated into a software package called CFDnet. CFDnet stands for '~Cornputa~onal
Fluid Dynarnics on the Intemet", and gets this name from its unique, Java-based user
interface which aiiows users to develop and solve dieu Computational Fluid Dynamics
(CFD) problems interacavdy over the Intemet fiom within a Java-compatible browser
[Ham and hIiliaer 1 9971. Implemenration of the mdti-block meshing and solving routines
makes use of the Parallel Vimal Machine (PVPUI) software [Geist et ai 19941 to distribute the
processing over a network of cornputen. At the cime of w r i ~ g , however, a given mdti-
block solution was entirely handled on a single computer, so no padelization speedup c m
To test the convergence characteristics and soluuon accuracy of the CFDnet sofnvare, nvo
separate laminar, incompressible fIow problems were investigated:
The lid-hven cavicy flow was solved on an orthogonal Cartesian mesh to compare total
solution times between the Finite Analytic discreazation of the momennun equaaons to
that of the more tradiaonal Gnite difference discretization based on che power law
pacuikar 19801. The cavit): flow problem was also solved for a highly non-orthogonal
rnesh to demonsaate the robusmess and grid independence of the CFDnet sohvare.
The flow around a c i rdar cylinder was simdated in both the steady and unsteady
regime, and the calculated flaw patterns and Erequency of vortex sheddlig was compared
to experimend data from the lirerature.
6.2 Cavity Flow
Figure 6.1 presents the geometry and boundary conditions for the Lid-dnven caviry Bow
problem. Due to the relaavely simple geomecry and strongly convergent nature of this
problem, it is commonly used as a test problem for fiow simulation algorirhms [Shariati et al
1996; Aksoy et ai 19921.
Figure 6.1 Geometry and Boundary Conditions for the Lid-Driven Cavity Flow Problem.
In the present work, the cavity fIow was uxd to compare cornpucational performance and
solution results of the Finite Andytic Method with the more widely used Finite Difference
discreuzation of the momentum equations based on the power law [Pacankar 19801. The
cavity flow probkm was also used to establish the grid independence of die soluuon
procedure by comparïng solutions produced on a highly non-orthogonal gnd to those
produced on an orthogonal grid.
6.2.1 Cornparison to the Finite Dinerence Method
The most Mdely used mechod to discretize the Navier-Stokes equauons on suuctured
meshes is the Finite Difference Method O M ) . When the convective forces dorninate the
flow, cenrral ditference discretizations can become unstable, and the upwind technique is
required to produce stable algosirhms. Patankac's power iaw pacankar 19801 provides a
smooth transition between the central didierence discretization and the upwind scheme,
based on the local Pedet number. The FAM naturally produces this upwindlig effect and
indudes the influence of the diagonal neighbours, thus reducing the numerical diffusion
when the flow is highly skewed relative to the mesh.
One deterrent to the increaxd use of the FAiM is the complexity of the coeffiaent
caiculations, and the perception of substantiaiiy greater computation h e . To compare the
caiculated solutions, and to quantify the diffaence in total computation tirne, the cavity flow
problem was solved on an identicai 31 x 31 orthogonal uniform mesh using both the FALI
and the FDM. Figure 6.2 compares the calculated u velocity component along the centreline
for a Reynolds Number of 400.
I -t Power Law l
!
4.4 4.2 O 0.2 0.4 0.6 0.8 1
u velocity
Figure 6.2 Caiculated u velocity along the Cavity Centeriine, Re=4ûû
The velocity profiles are nearly identical, differing slightly in the region of ma-uimurn negaave
u velocity. Because both solutions exactly satisfy idenacal c o n ~ u i r y equations, diis slight
variation is a resuit of the difference in momentum equation discre&auons, and the
consequent difference in the way momentum is transporred throughout the domain.
Figure 6.3 compares the computation time for the nuo methods, broken down in cemis of
the total time spent on the major cornpucationai sceps.
- -
O 1 O 20 30 40 50
computation time (seconds on a Pentium-Pro 200 PC)
Figure 6 3 Cornparison of total computation Mie, Cavity fiow pmblem, Re=1000
Nthough the FDM momentum equation calculations were significantly faster per iteration
when compared to the FAiM, the difference in overd solution time was only 20 - 25%. This
relaâvely s m d difference c m be p d y atmbuted to the fact that, for a given level of
convergence, the FDM required G to 25% more global iteraaons, with the higher difference
corresponding to higher Reynolds nurnber problems. It shouid be nored that both the FDbI
and FAM solution procedures used the fully transformed curvilinear form of the governing
equations, and as a consequence, both procedures indude the additional overhead time
associated with calculating and managing the inverse metric coefficients. Although the
governing equations written in their simpler format in ternis of (x,y) coordinates could have
been directly discretized on the rectangular grid used for this problem, this simplification is
not generdy applicable, and thus was not induded in this cornparison.
6.2.2 Non-orthogonal grid
To demonstrate the grid independence and robustness of the CFDnet software appiied to
non-orthogonal grids, the 2-dimensional caviry flow problem was solved using the FAiM on
both the regular, and the highiy non-orthogonal muia-block grids shown in Figure 6.4.
Figure 6.4 Compatison of Orthogonal and Non-octhogod Grids for the Cavirg Row Probfem
Figure 6.5 and Figure 6.6 use the suearnhuiction and isobars (pressure contours) respectively
to compare the calculated solution on both grids. Figure 6.7 compares the calculated u
velocity dong the cavity centreline for the taro grids to some other numerical data kom the
literature.
Figure 6.5 Cornparison of Cdnilnted Solutions for the Caritg Flow Problem, Streilmlines, Re400
Figure 6 6 Cornparison of Caicuiated Solutions for the Cavity Flow Problem, Pressure Contours, Rc=400
Figure 6.7 Cdculated u velociq dong the Cavity Centerline, Re400
In ail cases, the calculated solutions are accurate, correlating well with each other and
numerical data, despite the highly non-orthogonal grid and meeting of three block
neighbours at the centre of the non-orthogonal grid.
6.3 Fiow Around a Circular Cylinder
6.3.1 Background
The £low of a free stream, velocity U, around a circular qlinder, diameter D, changes kom a
steady to an unsteady flow pattern Li the wake of the cyhder as the Reynolds number,
Re = pUD / p , is increased. Based on experimentai observations [Panton 19841, below
Re=4 the flow is steady Mth slight asyrnmeny, dividing at the stagnation point and reuniung
on the Çar side of the cylinder. Between R e 4 and Re=35-40 the flow remains steady,
however flow separation results in nvo symmemcal standuig V O ~ C ~ S on the back face.
Above Re=40, the unsteady flow pattern consisn of the altemate shedding of vorüces From
the upper and lower rear surface of the cylinder. The oscillaaon frequency of the vortex
shedding can be charactenzed non-dunensionally by the Strouhal number.
where fis the shedding frequency for one complete cycle, D the cylînder diameter, and U
the free Stream veloaty. In the range of Reynolds numben from 100 to 105, the Strouhal
number is relatively constant with value approximately 0.2 panton 1984; White 199 11
decreasing notably in the relatively low Reynolds number range below Re=100.
There is a substanual arnount of both experimental and numerical analysis of the cyiinder
flow problem in the Literature, perhaps because of its importance in avil engineering where
the periodic forces produced by vortex shedding can synchronize Mth the resonanr
frequenues of structures with devas t a~g consequences [Panton 19841. The 6rst
experimental work investigating the cylinder flow involved characterization of the vortex
shedding frequency by Strouhd [from Goldstein 19651. More recent expebental work has
induded the low Reynolds number investigation of Trinon [1959,1971] , and the
visualization experimenu of Taneda [1979].
The € k t analytic investigation of the vortex shedding was presented in the famous paper by
Karman [from White 19911, and the alternating pattern of vortices in the wake of the
cylinder has since been referred to as the Kamian vortex Street. More recent numerical work
has investigated both the steady and unsteady solutions in greater detd. Some audion have
forced a steady, symmeaic solution by considering only half of the problem domain,
induding the calculation of the average drag coefficient by Son and Hanratty [1969], or the
inclusion of heat transfer modeling and the average Nusselt number calcularions of Chun
and Boehm [1989]. The unsteady flow has been simulated using the vorticity-stream
h c t i o n formulation by Lin, Pepper & Lee [1976] and Ta Phuoc Loc [1980], and the
primitive variable (veloaty-pressure) formulation by Braza, Chassaing and Ha Minh [198q,
among others. Occasiondy, authon wili use the cylinder flow problem to test a new
numerical solution technique or a l g o d m pfukhopadhyay et al 19931.
For aii the numerical studies sited, a relatively high degree of mesh refinement, particularly
near the cylinder, is required to accuratdy resolve the periodic vortex shedding. This is due,
in part, CO the oscillatory recirculaàng nature of this flow, which may result in significant
numerical diffusion when the local flow direction is highly skewed with respect to the gnd
[Patankar 19801. In the present work, rdatively course grids were used to demonsnate the
capability of the FmI to reduce numerical diffusion significantly.
6.3.2 Geometry and Boundary Conditions
northern boundary u= 1 v=o
I southem boundary
Figure 6.8 Geometry and Boundary Conditions for the Cyiinder Row Problem
Figure 6.8 presents the calculauon domain and boundary conditions used for the majority of
simulations. The cylinder was positioned near the inlet with centre at (x, y) = (0,4.5D) to
maximite the downstrearn region where the Karman vortex street was expected. Unless
stated othemise, the total dimensions used for the domain, in terms of the cyiindec diameter,
were H=GD, L=20D.
The boundary conditions are described as foiiows:
Met Boundary. Constant u-veloaty, magnitude=l .O.
North and south boundaries. Constant u-veloùty, magnitude=l.O. This boundary
condition does not d o w for any normal outflow of fluid before the cyiinder, or influx of
fluid afier the cylinder. Consequently, the accuracy of the solution wiii depend somewhat
on ensuring that these boundaries are located far enough above and below the cylinder.
A cornparison between low Reynolds nurnber, steady solutions with H=5 and H=10
showed no significant difference in the calculated flow profüe.
Cylinder surface. The standard no-slip boundary condition (u=O, v=O) was applied to
the cylinder surface.
Outlet. A constant pressure of p=O was applied at the outlet, and the exit velocity was
set normal to the outlet boundary (v=O). The accurate use of this exit boundary
condition requires that the outlet boundary be located suffiuently far downscream of the
cyiinder that there is no remaining trace of the vortex street (i.e. any vortex motion has
been completely dissipated by the fluid viscosity). Although the resulting unsteady
solution indicated that diis was not a valid assumption, a cornparison between unsteady
solutions with L=20 and L=35 showed no significant difference in caldated flow
profile near the cylinder or the predicted frequency of the vortex shedding.
The fluid density was set to 1, and different Reynolds nurnbers we achieved by a d j u s ~ g the
fluid viscosity, in the governing momentum equations.
Three levels of grid refïnement were investigated corresponding to an average ceii dimension
near the cytinder of 0.25D, 0.15D, and 0. ID. Figure 6.9 shows a detail of the 0.25D mesh
around the cylinder.
ire 6.9 Demil of the 025D Grid Near the Cy
6.3.4 Tirne Step
Because the solution procedure developed in the present work is implicit in any given step,
there is no applicable stability airerion for the Mie step, such as the Courant condition, and
theoreticdy any tirne step value could be used to obrain convergent solutions. This p ~ c i p l e
c m be exploited when the dtimate steady solution of a problem is dl thar is desired. In such
a case, a numerically large t ime step can be used in the discreuzation of the momentum
equaaons, making the weighting parameter, a , effeavely O. The solution procedure can
then be used co irerate direcdy to the steady solution in a single t h e step, and the initial or
previous condition is never accessed, and therefore does not need to be speufied or even
stored in memory. This approach was used for simulation of the steady flow acound che
cylinder below Re=3 5.
When a transient solution is desired, however, accurate resolution of the solution as it
changes over time requires some knowledge of the solution t h e constans or periodiuty.
For the cylindcr flow problem studied in the present work, the non-dimensional frcquency
of vortex shedding as characterized by the Suouhal number is approximately S ~ 0 . 2 over a
broad range of Reynolds numbers. This corresponds to a non-dimensional pesiod of about
5. Consequendy, the choice of h e step dr should be based on how many Çrames, or steps in
tirne we want to resolve in a given period. In the present work, non-dimensional time sreps
of 0.25,0.1, and 0.025 were used, correspondhg to an expected resolution of approxhately
20,50, and 200 "frames" per period, respecavely.
m e n solving the h s t step of an unsteady problem where the eventual soluaon is periodic,
the iniaal condition is not known explicidy. It is itself a step in the soluaon of the problem.
One alternative is CO use an impulsive start, speciwng an initial condition of u=v=p=O
throughout the domain, and cycle und the expected periodidq is observed [Braza et ai
19861.
If only the final penodic solution is of interest, as was the case with the present simulation of
the cylinder flow problem, a fast and stable approach to the evenmal periodic solution was
achieved by scaling the h e step in the first steps of the solution. Consider the simple
scaling formula:
where &* is the scaled tirne step used in the calculauon of a in the rnornennim equations, i
is the curent step number (O, 1.2.. . etc.), and Ns is the number of initial steps that the
h e scaling should be applied to.
Using this technique, it is no longer necessary to explia* know the initial condition for the
£irst step of the solution because the scaled time step is numericdy large for this k t step.
However, when a numericaily large tirne step is applied to a problem thar is nanirdy
unsteady, it becomes impossible to achiwe a converged solution at that step. Consequendy,
calculaüons at each of the initial Ns scaled Bme steps are stopped afier a i i m i ~ g number of
iteraaons. For the present work, h e step scalmg was applied to the fisr 10 steps (N, = 10
in Eq. (6.2)) to srart ail unsteady simulations.
6.3.5 Steady fiow Results
Figure 6.10 compares the cdculated steady flow profile at Reynolds number 26 on the hner
mesh to the experimental data of Taneda [1979]. The vertical dashed h e s in figure are
insened to d o w cornparison of the location of the vortex centres and the ceanachment
point. As illustrated, caiculations are in good agreement with the evperimental data.
Figure 6.10 Cornparison of Calculateci Cyünder Fiow to Experimental Data of Taneda Il9791 at Rc=26. (Streamaces are uscd for visuaiizatio<i, streamtrace
spacing has no sigrScance)
6.3.6 Unsteady Flow Results
In the present work, unsteady periodic flow solutions were obtained for Reynolds nurnbers
of 40 or grata, agreeïng weii with the experimencal value for the onset of the Karman
vortex Street of Re=35-40 [Panton 1984; White 1 99 11. Unlike other numerical investigations
in the iiterature praza et ai 19861, it was unnecessary to mgger the vortex shedding, and,
once stabilized, the periodic vortex shedding pattern could be observed indetiniteiy.
Figure 6.1 1 shows a streamline visualization of the calculated unsteady flow around the
cylinder at Re=100 for the 0.10D grid at four successive snapshots in cime.
Figure 6.11 Strtamlinc Viuaiization for the unsteady fiow -und a cylinder at 4 timc steps (dt=i, Re=100)
Although the solution procedure was based on solving the primitive vaxiables (velocity and
pressure), the Stream function is most effective to visualize the flow. In addition, its
conünuity throughout the domain demonstrates that mass flow has been accurately
conserved at the ceii level, and consequently throughout the entire domain.
The Çrequency of the vortex shedding c m be extracted from the unsteady solutions by
plotûng the strearn h m o n a@st cime for a single downstream point Mdun the Karman
vortex Street. Figure 6.12 shows the sneamhction vs. cime at a locaaon SD from the back
face of the cylinder.
0.4
OP
Streamfunction O Y
Non-dimensional time t
Figure 6.12 Streamfunction value in the wakc, SD behind the cylinder on the axis of symmetry, Rc=IOO,O.lD mesh.
M e r an initiai development cime, of the order of ~ 2 0 , the perïod of the o sdaaons
stabilizes and remains constant. In dus case the period was 6.9, correspondhg to a Strouhal
number of 0.145.
Figure 6.13 compares the calculated Strouhal nurnbers for the Re=50 and Re=100 solutions
to some experimentai data hom the literature Fritton 19711.
Strouhal Nurnber St=fD/U
A present w ork - 0.1 0 0 grid present w ork - 0.150 grid
O present w ork - 0.250 grid -- Tritton - bw speed mode O - - - Berger - bw speed made - Tritton 8 Berger - hgh speed mode - - - Berger - basic mode
Reynolds Nurnber Re=pUD/p
Figure 6.13 Cornparison of Calculated Strouhal Numbet to Experimental Resuits [Tritton 19711
Results for aii three grid densiaes are reported to iliustrate the progression towards the gid-
independent soluûon.
7. CONCLUSIONS
A comprehensive solunon procedure for the equations o f fluid fiow in arbitrary geomeuies
based on the Finite h a l y à c Method was presenced here. It is capable of providing accurate
solutions for steady and unsteady Bows on relaavely coarse grids. Irs main characte~sucs are
The F M coefficients are based on cosine boundary hct ions , and use arbimary
preusion arirhmetic CO stabilize convergence at ail c d Reynolds numbers.
The discretization equations for the u and v Cartesian velouty componenu on a
staggered curvilinear grid are deveioped kom an algebraic manipulation of the local
curvilinear velocity components.
This method produces strongly convergent and uncomplicated pressure and pressure
correction equarions, and avoids the censor algebra normdy assouated with curvilliear
velocity components.
The solution procedure is extended CO multi-block grids, and a scrategy for handllig ghost
points around each block and assigning grid neighbours near the block boundaries 1s
proposed.
To verify the accuracy and stability of the solunon procedure, the square fd-àriven cavitv
and sready/ unsteady fiow around a circular qlinder were simulaced. The results fiom the
square cavity simulations showed that, when compared ro the more traditional finite
difference method on identical grids, the FAM-based solution procedure required between
20-25% more CPU time to reach a given degree of convergence. Additional simulations
comparing the solutions of the cavity flow problem on orthogonal and highly non-
orthogonal g d s showed a strong condation in solutions, d e m o n s ~ a ~ ~ the grid-
independence of the solution procedure.
The simulation of the steady and unsteady fiow around a Urcular cylinder for the range of
Reynolds numbers Re=O to 100 was used to M e r test the solution procedure's abiliy to
mode1 transient recirculating flows. Accurate predictions of flow profiles, size and location
of attached vortices, the uansitional Reynolds nurnber from steady to unsteady flow, and the
vortex shedding frequency (Strouhal number) were obrained for relatively coarse grids.
Although the use of arbiuary preusion aithmetic d o w s stable FrZiCl coefficient calculations
at al ceU Reynolds numbers, the increased calculation urne associated with the arbiuary
preusion arithmeac, as well as the linear increase in the number of summation t e m s
required with cell Reynolds number (see Figure 2.3) results in a practical iimit for the PC-
based computaaon environment of diis work of about Re=200 to 300. When the majority of L
the cells in a given problem are near or above this practical limih computation cime becomes
impractical, even for the wo-dimensional problems investigated in this contribution.
Desoite this Lunitauon, the positive resdts dius far obtained suggest that hrther work should
be done to extend the solution procedure's capabilines to include three dimensional,
turbulent, and mulaphase flows.
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