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

Ando, J., Maita, S. & Nakatake, K. (1998). „A New

Surface Panel Method to Predict Steady and Unsteady

Characteristics of Marine Propeller‟. Proceedings of

22st Synposium on Naval Hydrodynamics,

Washington D.C., United States.

Bleslin, J. P., Van Housten, R. J., Kerwin, J. E. &

Johnsson, C-A. (1982). „Theoretical and Experimental

Propeller-Induced Hull Pressures Arising from

Intermittent Blade Cavitation, Loading and

Thickness‟. SNAME Transactions 90, pp. 111-151.

Hess, J. L. & Smith, A. M. O. (1964). „Calculation of

Nonlifting Potential Flow about Arbitrary Three

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Kanemaru, T. & Ando, J. (2009). „A Newmerical

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Kudo, T., Ukon. Y., Kurobe, U. & Tanibayashi, H.

(1989). „Measurement of Shape of Cavity on a Model

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Kurobe, Y., Ukon, Y., Koyama, K. & Makino, M. (1983).

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Fluctuation on a Model of the Training Ship “SEIUN-

MARU” with Reference to Full Scale Measurement‟.

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Thin Wing Theory‟. Journal of Aircraft 11(9), pp.

518-527.

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