numerical analysis of pressure fluctuation on ship … the previous paper, we applied sqcm to...
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Second International Symposium on Marine Propulsors smp’11, Hamburg, Germany, June 2011
Numerical Analysis of Pressure Fluctuation on Ship Stern Induced by Cavitating Propeller Using a Simple Surface Panel Method “SQCM”
Takashi Kanemaru1, Jun Ando
1
1Faculty of Engineering, Kyushu University, Fukuoka, Japan
ABSTRACT
This paper presents a calculation method for the pressure
fluctuation on the hull surface induced by cavitating
propeller. This method consists of two steps: the first step
is the calculation of unsteady sheet cavitation, and the
second step is the calculation of pressure fluctuation. The
first step is the same method which had been presented at
smp‟09 (Kanemaru et al 2009). The calculation method is
based on the simple surface panel method “SQCM”. The
boundary conditions on the cavity surface are the constant
pressure condition and the zero normal velocity condition
according to the free streamline theory. We obtained
reasonable results with respect to cavity shape and cavity
volume by considering the cross flow component on the
cavity surface.
In this paper, we present the second step, that is to say, we
describe how to extend the cavitating propeller problem
to the pressure fluctuation problem. Also, we show some
calculated results about amplitudes of pressure fluctuation
on the hull surface. Qualitative agreements are obtained
between the calculated results and the experimental data
regarding low order frequency components.
Keywords
SQCM (Source and QCM), propeller sheet cavitation,
cavity volume, amplitude of pressure fluctuation, inflow
velocity on cavity surface
1 INTRODUCTION
Pressure fluctuation on a hull surface induced by an
operating propeller in the hull wake causes ship hull
vibration. If cavitation occurs, the amplitude of pressure
fluctuation becomes considerably larger. Therefore, it is
important to predict the pressure fluctuation previously
and many researchers, such as Breslin et al (1982), Kehr
et al (1996), Kim et al (1995) etc. studied the pressure
fluctuation theoretically. In the case of recent larger
container ships, the pressure fluctuation will be more
serious because their propellers will operate in heavily
loaded condition.
It is generally said that the amplitude of pressure
fluctuation induced by a sphere varying own volume with
time is proportional to the second derivative of the
volume variation. Pressure fluctuation induced by a
cavitating propeller has a similar characteristic. Therefore,
a prediction method asks high accuracy about the time
variation of cavity shape and cavity volume.
We have presented the calculation method of propeller
cavitation using a simple surface panel method “SQCM”
in Kanemaru et al (2009). The feature of the method is
that the cross flow velocity which affects the cavity
volume especially near the blade tip was taken into
consideration on the boundary condition. By our method,
we obtained the reasonable results about not only cavity
area but also cavity shapes. We shall refer to Kanemaru et
al (2009) as “the previous paper” in this paper.
In this study, hull surface flow and propeller flow are
calculated simultaneously in order to calculate the
pressure fluctuation on the hull surface including the
interaction between hull and cavitating propeller. At that
time, we change the cavity shape on blade section at each
time step using the calculated cavity shape by the
previous paper. The boundary condition is zero normal
velocity condition on not only the wetted surface, but also
the cavity surface. That is to say, we deal with the cavity
surface as the rigid surface and regard the variations of
cavity shape as the variations of blade section. The
calculated results of the fluctuating pressures are shown
about two propellers of Seiun-Maru-I and we compare the
results with the published experimental data.
2 CALCULATION METHOD
2.1 Outline
SQCM (Source and QCM) uses source distributions (Hess
& Smith 1964) on the propeller blade surface and discrete
vortex distributions arranged on the mean camber surface
according to QCM (Quasi-Continuous vortex lattice
Method) (Lan 1974), which is well known as one of
lifting surface methods. The formulation of SQCM was
described in the paper (Ando et al 1998) and other papers.
In the previous paper, we applied SQCM to unsteady
propeller sheet cavitation problem.
In this paper, we calculate the pressure fluctuation on the
hull surface directly by calculating the hull surface flow
and cavitating propeller flow simultaneously. Most of the
readers will probably think that the calculation method
means the calculation of cavitating propeller under the
ship stern expressed by singularities. But we divide the
calculation method into two steps: namely, the calculation
of cavitation and the one of pressure fluctuation,
considering the practicality.
First of all, we obtain the unsteady cavity shapes at each
time step using the method by the previous paper at
<Step1> (see Figure 1). Here it is important that the time
step is small enough to calculate the higher order
frequency components. On the other hand, we must
consider the computation time for practicality. The
modified method to save the computation time is
described in 3.1.
Next, the pressure fluctuation on the hull surface is
calculated at <Step2>. We calculate the propeller flow
and the hull surface flow simultaneously applying SQCM
to the propeller and Hess & Smith Method to the hull
surface respectively. In the calculation, we use the blade
section including cavity shape obtained at <Step1> at
each time step. At that time, the boundary condition on
the cavity surface is zero normal velocity condition as
well as on the wetted surface. This means that the blade
section includes the cavity shape with the variation of
time and we do not distinguish the blade surface and the
cavity surface in the boundary condition. The
computation time is not so large by not performing the
calculation of cavitation at <Step2>.
Figure 1: Outline of calculation method
It seems that not only the zero normal velocity condition
but also the constant pressure condition should be given
on the cavity surface as well as <Step1>. But we claim
that this treatment is reasonable because the cavity shape
should be satisfied with both of two boundary conditions
by <Step1>. Though the pressure distribution on the
cavity surface in <Step2> does not coincide with the one
at <Step1> perfectly, we regard the difference is small
enough to ignore.
We also have to model the free surface in this calculation.
The negative mirror image is appropriate for higher
frequency propeller problem, but we adopt the positive
mirror image in order to compare our calculated results
with published experimental data which were conducted
using cavitation tunnel.
2.2 Calculation method for pressure fluctuation
The calculation method for <Step1> is described in the
previous paper. Therefore, we outline the main equations
for <Step2> in this section.
Consider a bladed propeller rotating with a constant
angular velocity (= , : number of propeller
revolutions) in inviscid, irrotational and incompressible
fluid. We take the propeller coordinate system
pppp zyxo and the axis px corresponds with the
propeller shaft as Figure 2. in Figure 2 represents the
angular position.
Figure 2: Coordinate system of propeller
Next, the space coordinate system xyzo is introduced
by shifting px of pppp zyxo to the draft as Figure 3
and the plane xy correspond to the mirror. The plane xy
expresses the upper wall of cavitation tunnel in the case
of comparison with experimental data.
Figure 3: Coordinate system of hull and propeller
The propeller blade is divided into panels in the
spanwise direction. The face and back surfaces of the
<Step1>
Calculation of cavitation
<Step2>
Calculation of pressure fluctuation
・Calculation of propeller
( Blade sections vary with time )
+・Calculation of hull surface flow
Blade section including cavity shape
Output
Input
K
n2 n
X
Y
Z
px
po
pz
py
AV
Bas
e L
ine
of
Key
Bla
de
po
o
px
x
z
P
Q
Q
Q
Q
AV
・
・
・
・
・pz
Draftor
Depth
BS M
blade section are divided into panels in the chordwise
direction, respectively. The total number of source panels
on the propeller surface becomes and
constant sources Bm in each panel are distributed. Also,
the hull surface hS is divided into hM in the
longitudinal direction and )(xN h in the transverse
direction. According to the Hess & Smith method,
constant sources hm in each panel are distributed on the
hull surface. The velocity vector due to the source
distributions on the blade surfaces and hull surface
including these mirror images is expressed by using
velocity potential mB on the blade surface and mh
on the hull surface.
mhmBmV
(1)
Where
K
kS
BBmB
B
dSQPR
qm
QPR
qm
14
1
),(
)(
),(
)(
hS
hhmh dS
QPR
Qm
QPR
Qm
),(
)(
),(
)(
4
1
K : numbers of propeller blade, P : control point
Q and Q : singularity and one of mirror image
R : the distance between control point and singularity
Next, the mean camber surface in propeller blade surface
is divided into segments in the spanwise direction
corresponding to the division of the source panels and
divided into in the chordwise direction. The induced
velocity V
due to the vortex model of the QCM theory
is given by the following equation.
wk
wkL
K
k
M L
k
kkL
K
k
M N
k
vvt
vvtV
1 1
1
1
1 1 1 (2)
Where
: numbers in the spanwise direction
: numbers in the chordwise direction
Lk t : strength of ring vortex on camber surface at
Lt ( Lt : numbers of time step)
kv
: induced velocity vector due to the bound vortex of
unit strength on mean camber surface
k
v
: mirror image of kv
NN
rc
2
12
2sin
)(,
L
N
kLk tt
1
)( rc : chord length of section
wkv
: induced velocity vector due to the trailing vortex
of unit strength in trailing wake
: numbers of trailing vortex
wkv
: mirror image of wkv
As for Equation (2), see the previous paper and Figure 4.
Figure 4: Arrangement of vortex system
The induced velocity vector due to each line segment of
vortex is calculated by the Biot-Savart law. When the
control points are on the blade surface in the same blade,
the ring vortices are close to the control points especially
for thin blade. In this case, we treat the ring vortices on
the mean camber surface and shed vortex nearest to T.E.
as the vortex surfaces in order to avoid numerical error.
We call this treatment “Thin Wing Treatment”.
Inflow velocity vector IV
is expressed as:
)),,(( 00AAI VVV
(3)
on Hull Surface
TWVI VVV
+= (4)
on Camber Surface
STWVI VVVV
++= (5)
on Cavity Surface or Blade Surface
Here WVV
means the viscous component of wake
velocity vector and TV
means the tangential velocity
vector by propeller revolution. Though we use
experimental wake WV
in this method, the potential
wake WPV
is calculated by the considering hull surface
flow. In order not to duplicate the potential component of
wake velocity vector, we subtract the potential wake
WPV
from experimental wake WV
and we input the
viscous component WVV
to the calculation. This relation
is expressed as:
WPWWV VVV
(6)
SV
in Equation (5) means the inflow velocity vector by
the variation of cavity shape with time. SV
is important
term to consider the dynamic effect of cavity for the
pressure fluctuation. SV
is expressed as the time
derivative of cavity thickness as following equation since
it is very difficult to deal with SV
precisely.
t
hVS
(7)
Where
N
KNM )( 2
mV
M
N
w
F
Spanwise
Shed Vortex
1
AG
H
J
1
BC D
E
I
Bound Vortex Free Vortex
Streamwise
Trailing Vortex
1
h : cavity thickness, t : time
In this way, the velocity vector around a propeller or a
hull is expressed as:
(8)
The boundary conditions at the control points on the blade
surfaces, mean camber surfaces and hull surface are zero
normal velocity condition. Therefore, the equation is
given as follow:
on Hull, Blade and Camber Surface (except T.E.) (9)
at T.E.
Where is the normal vector on the control points.
is the normal velocity at T.E. to satisfy the Kutta
condition (see the previous paper). Thus, we can derive
)()( xNMKMNN hh 2 linear simultaneous
equations which determine Bm , hm and Lk t .
The pressure on the blade surfaces and the hull
surface are calculated by the unsteady Bernoulli equation
expressed as:
(10)
Where
: the static pressure in the undisturbed inflow
: the density of the fluid
: the perturbation potential
Here the time derivative term of in Equation (10) is
obtained numerically by two points upstream difference
scheme with respect to time.
The pressure of the hull surface is expressed as the
following pressure coefficient pK in order to compare
the calculated results with experimental data:
220
Dn
ptpK p
)( (11)
Also the fluctuating pressure is expressed as the
difference from the time mean value pK as follows:
ppp KKK (12)
Furthermore the fluctuating pressure is expressed in terms
of Fourier coefficients as follows:
1i
iKiKpp iKKK ))(cos( (13)
iKpK : amplitude of i-th blade frequency component of
pressure fluctuation
iK : phase angle of i-th blade frequency component of
pressure fluctuation
K : total numbers of blade
3 CALCULATED RESULTS
We select conventional propeller (CP) and highly skewed
propeller (HSP) of Seiun-Maru-I as well as the previous
paper and add the body in this method. Tables 1 and 2
show the principal particulars of these propellers and the
body, and Figure 5 shows the body plan.
Table 1: Principal particulars of propellers (Seiun-Maru-I)
NAME OF PROPELLER CP HSP
DIAMETER (m) 0.22095 0.2200
NUMBER OF BLADE 5 5
PITCH TARIO AT 0.7R 0.95 0.944
EXPANDED AREA RATIO 0.650 0.700
HUB RATIO 0.1972
RAKE ANGLE (DEG.) 6.0 -3.03
BLADE SECTION MAU Modified SRI-B
Table 2: Principal particulars of full scale ship
(Seiun-Maru-I)
Seiun-Maru-I
LPP (m) 105.000
LWL (m) 108.950
B (m) 16.000
D (m) 8.000
d (design) (m) 5.8000
CB 0.576
CP 0.610
CM 0.945
Lcb (%LPP) 0.66
Figure 5: Body plan of Seiun-Maru -I
In this paper, we refer the experimental data by Kurobe et
al (1983) in order to validate the calculated pressure
fluctuation. This reference is not the same to one by Kudo
et al (1989), which was referred in the previous paper.
But both of these experiments were conducted at Ship
Research Institute in Japan (SRI, present National
Maritime Research Institute) and can be regarded as the
same experiments owing to the same conditions of these
experiments.
3.1 Calculation of cavitation
We have already presented the validation that the
calculated unsteady cavity shapes by our method agree
with the experimental data in the previous paper. After
that, when we were developing the calculation method of
pressure fluctuation, we found it important that the time
step t must be small enough to get higher order
frequency components accurately. Therefore, we give
small t as following ( = 1.0 deg.) and recalculate the
unsteady cavity shapes.
nt /./ 0360 (14)
V
mI VVVV
0nV
NVnV
n
NV
)(tp
tVVptp I
22
02
1 )(
0p
The trouble is that the computation time becomes very
long by small t .In order to overcome this problem, we
calculate the cavity shapes about one blade only as key
blade ( 3020NM ) and other blades are calculated
without cavitation and with rough paneling
( 1410NM ) (see Figure 6). Figure 7 shows the
comparison between the previous calculation ( =2.5
deg.) and the present method about cavity volume. The
effects of time step for both propellers are small.
Figure 6: Panel arrangements for cavitation model
(Seiun-Maru –I)
Figure 7: Variations of cavity volume
(CP, TK =0.207, n =3.06 ; HSP, TK =0.201, n =2.99)
3.2 Calculation of pressure fluctuation
In the calculation of pressure fluctuation, all blades‟
cavitations have to be taken into consideration at the same
time. Because of it, we apply the cavity shape of key
blade to all other blades and calculate the pressure
fluctuation with ship stern. Figure 8 shows the panel
arrangements of propellers for pressure fluctuation and
Figure 9 shows the panel arrangement of double body
with propellers. As Figure 9 shows, we take the aft body
only. The body is divided into 80hullmain )(hN
panels in front of the position of cut up on the keel line,
40overhangstern )(hN panels behind the position in
the transvers direction respectively, and 23hM panels
in the longitudinal direction. Here the important point is
that the panels are small near the measuring points. (see
Figure 10).
Figure 8: Panel arrangements for pressure fluctuation
model (Seiun-Maru –I)
Figure 9: Panel arrangements of hull and propeller
Figure 10: Pressure measuring points on stern hull
We must calculate the potential wake component WPV
at
the propeller plane so as to calculate the viscous wake
component WVV
according to Equation (6). Figures 11,
12 and 13 show the distribution of WPV
, WV
and WVV
.
The potential wake component WPV
is small regarding
the axial component but similar to the experimental wake
WV
regarding the component at the propeller plane. We
input the viscous wake component WVV
as inflow
velocity vector in the calculation.
Figures 14 and 15 show the amplitude distributions of
pressure fluctuation about the 1st blade frequency
component on the measuring points by CP and HSP
comparing with the experimental data. In these figures,
we also show the calculated results and experimental data
without cavitation respectively. The calculated results
without cavitation show good agreement with the
experimental data. On the other hand, the results with
cavitation overestimate to the experimental data. But we
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Dyp /
Dxp /
P3 P2 P1 S3S2S1
F3F2
S4
A3
A2
C
Key Blade
CP
CP
HSP
HSP
-60 -30 0 30 60 900
1
2
3
[103]
Angular Position (deg.)
Cav
ity V
olu
me
(mm
3) Pres. Cal.
CP
Prev. Cal.10)
Exp.17)
HSP
Pres. Cal.
Exp.17)
Prev. Cal.10)
.deg.52=
.deg.01=
can see that the calculated amplitudes with cavitation are
much larger than those without cavitation, as well as the
phenomenon in experiment. Also, the calculated results
express that the amplitude of CP is larger than that of
HSP. The difference between CP and HSP is similar to
that of the experimental data qualitatively.
Figures 16 and 17 show the amplitudes of pressure
fluctuation about the 2nd blade frequency component. In
the case of Non-Cav, the amplitudes about CP and HSP
are hardly seen in both the calculated results and the
experimental data. On the other hand, the calculated
amplitudes with cavitation are large as well as the
experiments. By the way, the reason of the disagreements
between the calculated results and experimental data with
cavitation about longitudinal distribution is that the
calculated results slightly shift to portside about
transverse distribution comparing with the experimental
data.
Figure 18 shows the amplitude of pressure fluctuation at
point C including higher order blade frequency
components. The maximum values of calculated results
are added in this figure in order to compare with
experimental data measured as maximum data. The
calculated amplitude about the 1st blade frequency
component can express the difference between CP and
HSP qualitatively. On the other hand, it is very difficult to
get reasonable results by calculation with respect to
higher than the 2nd blade frequency components. It is
said that the pressure fluctuation is a complicated
phenomenon involving many factors other than sheet
cavitation, which we consider in this calculation method.
Especially tip vortex cavitation is very important for the
calculation of pressure fluctuation. This will be the most
important aspect of our future work.
Finally, we discuss how SV
in Equation (5) affects the
calculated results. Figure 19 shows the comparison of
calculated result with SV
, without SV
and experimental
data about the 1st blade frequency component. In the
result without SV
, the propeller loading and blade
thickness including cavity shape are considered. The
dynamic effect by cavity variation is not included in this
case. The result without SV
is closer to the experimental
data than the result with SV
. This result denotes that the
overestimation regarding the 1st blade frequency
component in the present calculation may be caused by
the treatment of SV
. Figure 20 shows the calculated
amplitude distribution about the 2nd blade frequency
component. It is interesting to notice that the calculation
with SV
reproduces the 2nd blade frequency component
in comparison with the calculation without SV
. Thus, the
consideration of SV
is very important to express the
amplitudes of higher order frequency component. There is
room for improvement about the consideration of SV
for
realistic simulation. This will also be an aspect of our
future work as well.
Figure 11: Potential wake distribution at the propeller plane
Figure 12: Measured wake distribution
at the propeller plane
Figure 13: Viscous wake distribution for input data
pz
py
W
Propeller Disk
Azy VVpp
/
0.0 0.2
=0.1
0.08
0.12
0.14
0.160.18
pz
py
W
Propeller Disk
Azy VVpp
/
0.0 0.2
=0.1
0.6
0.2
0.3
0.40.5
0.7
pz
py
W
Propeller Disk
Azy VVpp
/
0.0 0.2
0.1
= 0
0.20.3
0.40.5
Figure 14: Amplitude distribution of pressure fluctuation
by CP (1st blade frequency component)
Figure 15: Amplitude distribution of pressure fluctuation
by HSP (1st blade frequency component)
Figure 16: Amplitude distribution of pressure fluctuation
by CP (2nd blade frequency component)
Figure 17: Amplitude distribution of pressure fluctuation
by HSP (2nd blade frequency component)
Figure 18: Amplitude of fluctuating pressure (Point C)
Figure 19: Comparison of inflow velocity conditions about
amplitude distribution of pressure fluctuation
by CP (1st blade frequency component)
0.02
0.04
0.06
0.08
0
Kp5
C S1 S2 S3 S4P1P2P3
CP
Cavitating Cal. Exp.
16)
Non-Cav Cal. Exp.
16)
0.02
0.04
0.06
0.08
0
Kp5
S1 F2 F3A2A3
0.02
0.04
0
Kp5
C S1 S2 S3 S4P1P2P3
HSP
Cavitating Cal. Exp.
16)
Non-Cav Cal. Exp.
16)0.02
0.04
0
Kp5
S1 F2 F3A2A3
0.02
0.04
0.06
0.08
0
Kp10
C S1 S2 S3 S4P1P2P3
CP
Cavitating Cal. Exp.
16)
Non-Cav Cal. Exp.
16)
0.02
0.04
0.06
0.08
0
Kp10
S1 F2 F3A2A3
0.02
0.04
0
Kp10
C S1 S2 S3 S4P1P2P3
HSP
Cavitating Cal. Exp.
16)
Non-Cav Cal. Exp.
16)0.02
0.04
0
Kp10
S1 F2 F3A2A3
1 2 3 4
0.02
0.04
0.06
0.08
0
Kp i
K
CP Cal. Cal.(Max.)
B.F. (i)
HSP Cal. Cal.(Max.)
Exp.16)
Exp.16)
0.02
0.04
0.06
0.08
0
Kp5
C S1 S2 S3 S4P1P2P3
CP
Cal. Exp.
16)
Cal. w/o
Cavitating
0.02
0.04
0.06
0.08
0
Kp5
S1 F2 F3A2A3
Figure 20: Comparison of inflow velocity conditions about
amplitude distribution of pressure fluctuation
by CP (2nd blade frequency component)
4 CONCLUSION
In this paper, we presented a calculation method for the
pressure fluctuation on the hull surface using a simple
surface panel method “SQCM” to the propeller and Hess
& Smith method to the hull surface. This method is an
extension of our calculation method for unsteady sheet
cavitation. We regard the calculated cavity surface as the
rigid surface in the step of calculation for pressure
fluctuation. And we adopt the positive mirror image as
the free surface condition. Comparison between the
calculated results and experimental data led the following
conclusions:
(1) The present method can express the difference
between CP and HSP in the experiment qualitatively
with respect to the 1st blade frequency component. In
the case of non-cavitation, the calculated results
agree with the experimental data not only
qualitatively but also quantitatively.
(2) The calculated results show the amplitudes of higher
than the 2nd blade frequency components by
cavitation, but it is difficult to get the reasonable
results regarding higher order frequency components.
It seems that the consideration of sheet cavitation
only is not sufficient as the calculation for pressure
fluctuation.
(3) The inflow velocity component by the variation of
cavity shapes on the boundary condition contributes
to the calculation of amplitude about higher order
frequency components. The present treatment of the
inflow velocity component should be improved in
future.
REFERENCES
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0.02
0.04
0.06
0.08
0
Kp10
C S1 S2 S3 S4P1P2P3
CP
Cal. Exp.
16)
Cal. w/o
Cavitating
0.02
0.04
0.06
0.08
0
Kp10
S1 F2 F3A2A3