A method to predict fluctuating pressures induced by a cavitating propeller

A Method to Predict Fluctuating Pressures

Induced by a Cavitating Propeller

Tetsuji Hoshino*

In order to calculate the fluctuating pressures induced by a cavitating propel/er, the cavity geometry on the propel/er blades must be known as a function of time. Since purely theoretical calculation thereof needs application of a complicated unsteady lifting surface theory including effect of unsteady cavitation, attempts were made in this study to estimate the unsteady cavitation using available methods (an unsteady propeller lifting surface theory in non-cavitating condition and an approximate method of lift equivalence for determination of cavity extent).

As a result, the geometry of the unsteady cavity was found to be estimated by modified lift equivalence method with correction factors considering the unsteady effect on cavity, and by an empirical equation based on the calculation of two-dimensional foils

and the measurement of cavity thickness on propeller blades.

The fluctuating pressures calculated with thus obtained cavity geometry were found to agree well with the measured values not only on model but also in full scale. Surface forces calculated by integrating the fluctuating pressures on the hull surface were found

to be reasonable compared with existing data.

1. Introduction

Propulsion of ships is ordinarily made by screw

propel-lers (hereinafter called propelpropel-lers) operating in general be-tween a hull and a rudder. Since the flow around a propeller is varied in space because of the presence of hull and rudder,

forces and moments exerted on the propeller itself, on the

hull and on the rudder are fluctuating with time. Those

fluctuating forces and moments are called unsteady

propel-ler forces and cause vibration of a ship especially in the

after-body.

In the recent years, reduction of vibration and noise have been more and more stressed for prevention of damages on

hull and instruments and for betterment of habitability.

Since, as mentioned above, unsteady propeller forces are a

principal source of ship vibration and noise, it is desired to

estimate them in an initial design stage and to prepare

appropriate countermeasures if necessary.

The unsteady propeller forces are classified into two categories according to how they are delivered to the hull. One of them is propeller shaft forces transmitted through the propeller shaft and its bearings as a sum of fluctuating force and moment of each blade, and the other is surface

forces resulting from fluctuating pressures around a propel-ler propagated to the hull surface through the water.

A number of proposals have been made for prediction

of propeller shaft forces either by a rigorous method based on unsteady lifting surface theory(1)_(6)or by an

approxi-mate approach based on a quasi-steady technique17 In a review on the state of the arts on unsteady propeller forces,

Tanibayashi8 states that the fluctuatiön of thrust and side

forces is in general a few percent of the mean thrust of the

propeller, while surface forces exceed sometimes 20% of the mean thrust, thus being more important as a source of ship vibration than propeller shaft forces. In this paper,

therefore, the author deals with surface forces of a propeller especially with unsteady cavitation on its blades.

Surface forces are, as mentioned above, excitation due

to distribution of fluctuating pressures over a hull and a rudder (or rudders). Therefore, calculation of the surface

Dr. Eng.. Nagasaki Technical Institute, Technical Headquarters

forces is usually made of calculation of free-space pressure

field, consideration of solid boundary factors and

integra-tion of the pressures over the hull. An alternative approach

has been proposed by Vorus9 (10) to calculate surface forces directly. This method may be more suitable if solely

the total surface forces are concerned, but main difficulty is in calculation of velocity potential due to unit vibratory

motion of a hull, and further it is not easy to verify the

method by comparison with experiments. Therefore, the

present study has been performed along the line of a con-ventional method of obtaining surface forces by integrating fluctuating pressures.

Theoretical study on fluctuating pressure field generated by a propeller seems to have been made first by Gutin(h1) who dealt with noise due to a propeller of airplanes. And in

the field of marine propellers, theoretical studies on the excitation were started by Taniguchi (1958)h12), Breslin (1958)113) and Pohl (1959)(14) independently at about the

same time. Shortly after, Breslin and Tsakonas

(1959)'>

derived a formula including the effect of blade thickness,

and showed fairly good correlation of the numerical results

with experimental values. Later, with the development of lifting surface theory of marine propellers, it has become possible to predict fluctuating pressures around a

non-cavitating propeller with good accuracy as a sum of loading effect obtained a lifting surface theory and blade thickness

effect represented by source distributions along the chord

of blades116 l7)

Increase in fluctuating pressures due to cavitation was

reported first by Takahashi and Ueda (1969)118) as a result

from measurements in a cavitation tunnel. Johnsson and Søntvedt (1972)119) performed both full scale and model

measurements on a 230000 DWT tanker, and showed (Fig. 1) that full scale values become much larger than those of

the model with increasing number of revolutiôns of the

propeller. With further measurements of fluctuating pres-sures (20)(21) it was shown that effect of unsteady

cavita-tion must be taken into consideracavita-tion when dealing by

theory with fluctuating pressures around a propeller.

113 0.2

j

0.1 Maximum Minimum on-cavitatñg o 30 50 70 90

Number of propeller revolutions (rpm) Fig. i Amplitude of fluctuating pressure on a 230000 DWT tanker19 (pressure transducer above a propeller)

For this, Huse (1972) formulated the effect of

un-steady cavitation on fluctuating pressures around a propel-ler and showed that the component due to volume change of a cavity is most impôrtant. Hoshino (1979)(23) extended

the theory of Takahashi124> to calculate the fluctuating

pressures around a spherical bubble advancing with varying radius, and showed that the main contribution comes from the second derivative with respect to time of the volume of the spherical bubble.

For practical application of the above theories,

behavi-our of unsteady cavity must be known as a base. However,

since a purely analytical approach is too difficult to be

tacked with, alternative methods have been proposed.

Takahashi (1976)(25> used an empirically obtained

amplifi-cation factor for the effect of cavitation. Johnsson and

Søntvedt (1972)(19) applied Knapp's dynamic similarity

law261to pressure distribution over a non-cavitating propel-ler blade on a quasisteady base. Noordzij (1976)127 applied

the two.dimensiónal theory of Geurst (28)129) _{on cavitating}

foil stripwise at each radius. Many other proposals01341

have been made for theoretical prediction of unsteady

cavi-tation and fluctuating pressures, but they are represented

by either of the two above.

Of these methods fòr prediction of unsteady cavitation on a propeller blade, very few numerical examples have

been provided and no attempts seem to have been made to

verify the theory by measuring the cavity shape on the

propeller blades. Therefore, eventual côincidence of

fluc-tuating pressures between theory and experiments does not

necessarily mean the correctness of the estimated cavity shape.

In view of the situation described above, the present study was intended to establish a practical method of

pre-dicting fluctuating pressures and surface forces of a propel-ler. In the course of the study, particular attention was paid

to the practicality of the method; an available unsteady

Loading point

Field point P

cavity

Fig. 2 Coordinate systems of propeller

lifting surface theory on non-cavitating propellers is cou-pled with semi-empirical liftequivalance method for deter-mination of cavity extent on propeller blades, Further the

results of the calculation are còrreòted incorporating experi-mental data on cavity extent, cavity thickness and the

fluc-tuating pressures measured in the cavitation tunnel. The

whole work is detailed in the author's doctoral dissertation, of which the principal part is described below.

2. Fundamental theory of a cavitating propeller

2.1 Coordinate systems and geometrical representations of propeller and cavity

Let us consider a propeller rotating with a constant an-guiar velocity in a non-uniform flow of incompressible and inviscid flow extending to infinity. The propeller consists

of N blades of identical shape attached to the boss

axisym-metrically and cavitation covers the back side of blades only. Face cavitation is ignored because the area and the

thickness of it is very small.

In representing geometrical shape of the propeller and

the cavity, we adopt a rectangular coordinate system O-xy. fixed in space. Now, we choose the xaxis so as to coincide

with thé axis of revolution of the propeller with the origin O at an appropriate point on the axis. The z-axis points vertically upward from the origin O and the y-axis com-pletes the right handed coordinate system with x and z. The máin flOw is directed toward positive x-axis with the velocity VA.. Next, we introduce for convenience a cylin.

drical coordinate. system O-xrçt, fixed in,space, where the

radial coordinate r is meàsured from the axis of the

pro-peller and the angular coordinate p is measured clockwise

from the z-axis around the positive x-axis. Then, the rec-tangular coordinate system O-xyz is transformed intothe

cylindrical coordinate system O-xrçc by the relations

x=x, yr-sinp, z=r.cosp

(1)

as shown in Fig. 2.

The propeller radius and the boss radius are denoted by

r0 and rj,, respectively. The propeller is rotating with the angular velocity 2 in the negative p-directiOn, and the generating line of the first blade is assumed to pass

thez-axis ät time t=0. If we define another angular coordinate O

fixed to one of the blades, measuring clockwise from the generating line of the blade, the relation between O and p

t)=

k2fr, t) - ski(r, t)

Fig. 3 Section of k-th propeller blade at radius r (7) Trailing edge Generating line

Fig. 4 Blade outline of k-th blade projected to the plane

perpendicular to x-axis

Further, the mean camber line of the blade section

n0(r, s) and the thickness of the blade section tb(r, s) are written as

n0(r,s)= [n2(r, s) +n1(r,

s)]/2

tb(r, s)

= n2(r, s) - n1(r, s)

If we denote the thickness of the cavity on the back side of the k-th blade at time

t

by tCI((r, s, t), the height of the mean camber line of the section of the blade including the

cavity

n(r,

s, t) and the thickness of the blade including the cavity tS(r, s, t) are expressed by

nfr, s, t)

n0(r, s) + tCk(r, s, t)/2 s, t)

=

tb (r, s) + tek (r, s, t) where tekfr, S,

t)r

I sL(r)<s<sl,l(rt)

O for

s;2fr,t)<ssr(r)

(1O) k3fr, s, t) -n2(r, s) for ski fr,t) s Sk2fr,t)J

Then, the face and the back of the blade including the

cavity n(r, s, t)is given by

n=n(r,s, t)n'(r,s,

t)

+(-1)'t(r,s, t)/2

where

n(r, s, t) =

n1(r,s), s, t) = _k2('

r)< s

si-(r) 12- (r, s) for 5L (r) S< Ski(r, t). 'k3fr' s, t) for

_ki',

t)

Now, let us introduce a parameter y so that y

= 1 at

the leading edge and y = i at the trailing edge. If we denote 0-coordinates of the leading and trailing edges by 0L (r) and Or respectively as shown in Fig. 4, then the s-coordinate of the blade section can be expressed as

s= [0,w(r)(r)v]

Ja(r)2+r2 ₍₁₂₎

where,

1v i

OM(r)= [Or(r) +OL(r)}/2,(r)= [OT(r)OL(r)]/2

(13) Using the parameter y instead of s, the variables- n(r, s, t),

t(r,

s,

t) and n(r, s,

t) shall be rewritten byn(r,

y, t),

t(r, y,

t) and n(r, y,

t) respectively. Further, the v-co-ordinates of the leading and trailing edges of the cavity on

MTB1 50 May 1982

t)

3

for the k-th blade at any instant t is represented by

û=O

°k ìt

(2)

where

0k271(k-1)/N,

k1,2,

N (3)

If we cut each blade by a cylinder with radius r, the base line of this blade section is considered to be a helix with

the pitch 2ira(r).

In order to represent the shape of the

blade sectiòn and the cavity on the back side of the k-th propeller blade, this cylindrical surface is expanded into a plane. Then, we introduce a new rectangular coordinate

system (s, n) fixed to the plane, where the s-axis- is chosen so as to coincide with the base fine of the blade section and

we take its positive direction from the leading edge to the

trailing edge as shown in Fig. 3. The origin is located in the

plane including the x-axis and the generating line. The n-axis is chosen normal to the s-n-axis directing from the face

to the back of the blade section. Further, let the rake angle of the propeller blade be 6, being positive backward. Then, we have the following relations between the two rectangular

coordinate systems (x, rO) and (s, n) for the section at the

radius r:

x= [a(r)srn]/-s,/a(r)2+r2 +(rrb).tan6

_i

O=

[s+a(r)n/r]/.Ja(r)2+r2

r

(4)

In representing this blade section using the coordinate

(s, n), we distinguish the face and the back of the blade by

= 1 and 2 respectively. Let the s-coordinates of the lead-ing and traillead-ing edges be SL(r) and s

r(r)

respectively, then we have

n=nK(r,$)

(5)

where

sL(r)ssT(r), K=land2

In the same manner, in representing the shape of the cavity

on the back side of the k-th blade at any time t using the

coordinate (s, n), the s-coordinates of the leading and

trail-¡ng edges shall be written by Skl(r, t) and sk2(r, t) respec-tively, then we have

n₌

'k3fr' s, t)

(6)

where Ski (r, t) < s < Sk2fr, t)

The chord length of the blade section C(r) and the length of the cavity¡Ck(r, t) are expressed by

0(r)=

sT(r) - sL(r)

ick (r,

the back side of the r-section shall be vkl(r, t) and Vk2(F t) respectively. Then, any point (x, r, 0) on the surface of the

kth blade including the cavity is represented by

X=Xbfr,V,t), r=r, O=eb;fr,v,t)

(14) where, Xbfr, V, t) =a(r)[OM(r) +

(r)v] + (r rb)tan

* rng(r, y, t)

-(15)

x=xg, r=r,

0=0

(18) where,

x=h(00+L'),

Of(J=Ob+J1,

0<'L'<°°

Then, the velocity potential Ø induced by the çavitating

propeller operating in non-uniform flow can be represented as follows: 0 = øi +Or N

10 f (r)\/h2+r2dv

4k'=1 r,

-i 7 (r' , 0k'

_a'

(-)V

r'

47rk=1frbfi' v' t_-s--)

+ ,,2 dv' 1 N r0

9'

i

= - 41T

k'l'1b

dr'f a(rÇ

v'

t---)---+ ,,2 dV where, Rf = 's,/(X - x'flJ)2 + r2 +r'2 2rr'.cos(p 9'fO °k'

Rb = V'(x hO'ci)2 + r2 + r'2-2rr'.cos(p

O' + lt),

a 1 a

ha

./h2+r'

'ax'r'a0,0

x'm= h(O'0+ 0'fO=

9= 8M(r') +

7*(r y, t) =f"7(r, vÇ t+ (r)(v' v)/2)(r)..Jh2+r2 dvÇ

y(r,v,t)=0 forv1,

f'a(r, y, t)(r)..Jh2 + r2 dv = O

Let the pressure at any point and at infinity in the fluid be p and p. respectively, and let the density of the fluid be

p. Then, from the Bernoulli's theOrem for the unsteady

motiOn, the following linearized pressure equation is obtain-ed:

p

aø ao

(22)

where, pp PVVA ',4P(v2+v,+v,2)

Next, let the pressure on the propeller blades be [p]

and the following approximations are introduced:

VA +[vx}(bk,VAW*

h

Jh2

+ r2

v'h2+r

where V2, and L41* are taken to be the average valués

with respect to s and t. Then, we obtain the following

equation:

Fig. 5

Velocity diagram for

blade section at radius r

ObKfr,V, t}=9M(r)+(r)v+

*

a(r)n,(r y t)

rJafr)2 + r2 2.2 Velocity potential and boundary conditions

2.2.1 Velocity potential due to cavitating propeller and

pressure equation

A propeller is assumed to operate in a non-uniform flow

consisting of a main uniform flow with the velocity VA in the positive x-direction and a small disturbance varying in

space. Let the components of this small disturbance in the

x, r and p-directions be v, r and v, respectively.

Let us ôonsider the flow field around the cavitating

pro-peller operating in this non-uniform flow. In formulating the flow fiéld, the propeller blades and the cavity on the blades are replaced by a vortex system, which consists of

the bound vortices 7(r,

y, t) distributed Over the mean camber surface of the blades including the cavity and the

free vortices shed from the bound vortices, and the sources

o(r, V, t) distributed over the above mentioned surface.

Then, the. vortex distribution represents the load acting on

the blade and the source distributiOn the thickness of the

blade and the cavity.

However, it is too complicated to treat-the mean camber surface over which the singularities are distributed and the

locus of the free vortices precisely. Therefore, we assume that the mean camber surface and the locus of the free vortices can be replaced approximately by the helical sur-faces with a constant pitch of 2irh independent of r. Then,

by using an average axial velocity VA' and an average

an-gular velocity 2* during one revolution of the propeller, h

is expressed as (see Fig. 5)

h = VA*/Z*

(16)

where,

vA*=vA+v;, ç*_Ç*/,.

average value of the axial disturbance and in-duced velocities during one revolution of the

propeller

v*/r

: average value of the angular velocity due to the

tangential disturbance and induced velocities

during one revolution of the propeller

The mean camber surface over which the bound vortices

and the sources are distributed can be approximated by the following surf àce:

x=h90, rr, 6=00

(17)

where

0O=0M(r)+(r)v, 1v1

In the same manner, by using a parameter i, the surface of the free vortices can be represented as

[p*}

8[0]bko

*

-

at

W

[Ws](bko)

(-1)

O 2

W7fr,L,t_»

(23) h

_íaø

r aø where, [ws](bko) /h2 +r2

and [ ] denotes the value on the face or the back of

the blade including the cavity, and E

_{] (b) denotes the}

value at the point (hO0, r, O) on the surface defined by

Eq. (17).

2.2.2 Boundary conditions on propeller blade iñcluding

cavity

For a cavitating propeller, the following two boundary

conditions are necessary due to the presence of the cavity: [11 there is no flow across the blade surface and the cavity

surface (kinematic boundary condition), and

[2] the pressure on the cavity surface is equal to a vapour pressure Pv of the fluid (dynamic boundary condition).

At first, let us consider the kinematic boundary condi-tion [1]. The kinematic boundary condicondi-tion on the surface of the blade including the cavity is given by neglecting the

radial velocity components as follows:

raobKI axbK av 1. at + VA + [v ax%,,I _raebK ao

rak°

öv 1, at

+ÇLr+[v

+-where K = i and 2

When we add the above two equations corresponding toK =

i and 2, and multiply the result by 1/2 we have the

follow-ing equation by neglectfollow-ing small quantities of higher order:

t)

where (25)

r

lao

_{h ao}

[wfl](bko) =

/h2+

r2 1ax'

(bkO)7

}bko)f

(24) =

k'=l

dr'f o

,-,y', t

-(r)Jh2

+r'2 dv'

Here, Ø represents the induced velocity potential in

non-cavitating condition and O represents the induced velocity

potential due to the presence of the cavity. Then, Eq. (25)

can be divided into two equations as follows: * *an0(r s)

[wfloJ(bko)=Wa+W

_as'

(35)

[wflc](b,,=-f-(-I--+ W*_)tck(r, s, t)

₍₃₆₎ where,

r

J ao0 h ao0 W,0 _(bkO) -./h2 + r2 ax hlbko)7 ](bkOJ

r

ao h _ao [wnc](b)=

- /h2+?

In the same way, Eq. (26) can be divided into two

equa-tions as follows:

at(r,v,t_ûkS

=w*tfr,5)

as

,fr, y, t

= a

T at tck(r,s, t)

Now, substituting Eq. (37) into Eqs. (33) and (34), we

have consequently as the velocity potential due to the

source distribution i NSr0 i

dr'

watb(rs')

i

-r, _as' Rb

O(r)./h2+r2dv

N r0 a =

f

_dr'f

( +

_{)t'(r s}

_t) 4k=1 rj '-1 at as'

_L.

(r)\/h2 + ,,2_c-j1' Rb MTB1 50 May 1982 (29) Rb (37) 5

the boundary conditions, the bound vortices 'y(r, y, t) and the sources c(r, V, t) can be divided into two terms, one treating the blades and the other treating the cavity respec-tively as follows:

'y(r, y, t)= _0fr,y, t) +7c(r,y, t) c(r,v, t)=00(r,V, t)+Uc(r, y, t)

where, 'yo(r,v,t)=7(r,v,t)=Q forv1

Therefore, the induced velocity potential due to the cavitat-ing propeller can be expressed as follows:

0=00 + Oc (30) where,

0= 010+ 0,

Oc =Oic + 010= _{47Tk. $,.'°}dr'f'

(r)Jh2

+r'2dv'

fyOk'

a i r, y, t--b-) .__(_).,V/h2+r2dO (31) 1 N

°dr'f'

W(r)Jh2

_+r'2dv'

0/C 4J

(r', v' t _') an'f

(I)..

/h2+ r2dO oto ₌

4nk'=lLf-' o(r

v t--)

.;-(r')-sJh2

+r'2dv'

W*sJ(V»2+ (r)2,

a = tan'

(f_))

tan_1(4

+ [vs]_(bkO)' r _{2r+ [v](bko)J}

In the same way, when we subtract the Eq. (24)

corre-sponding to = i and 2, we have the following equation

by neglecting small quantitiés of higher order:

0k

,atb(r,$)

a a

a(r, V,

t--)=W

as

+(+w*)tCk(rs

t) (26)

Next, let us consider the dynamic boundary condition

[21. Let the pressure over the cavity surface on the back of

the blades be [Pl ck anew. Then Eq. (23) can be rewritten

approximately as

[P](ck) [p*} a[01(bko)

p

Ws} (bko}y' V,

t_)}

(27)

With the vapour pressure of the fluid

P, the dynamic

boundary condition can be expressed as

[P] (ck) Pv, where Vklfr, t)

V k2fr t)

(28)

Next, let the pressure on the back side of the blades in non-cavitating condition be [o]wk2). Then, we have the following equation from Eq. (27).

[Po](2) []k

ò[øcIbko

P

-

at

+

W*{[wX](b)+

(40) where, *

[o](k2)- [P ](bk2)

_at 'y0(r, y, _pW* [wso}(bko)+ 2 h ô00

r

- Jh2 +

2

[--]bko--F] (bko)'

[W],kO)= /h2+r2{0)+1(0)}

Here, the dynamic boundary condition is expressed by Eq. (28).

The bound vortices y0(r, y, t) in non-cavitating condition

can be obtained by solving Eq. (35). Further, the bound vortices 7c(r, y, t) due to the effect of the cavity and the

thickness of the cavity tck (r, y, t) can be obtained by solving the simultaneous equations which consist of Eqs. (36) and

(28).

2.3 Fluctuating pressures induced by a cavitating pro-pelle r

A fluctuating pressure L.PP P induced by a

cavitat-ing propeller consists of the followcavitat-ing four components

such as

AP10 : component due to the loading of the propeller

in non-cavitating condition,

component due to the loading induced by the

presence of the cavity,

to component due to the thickness of the propel-ler blades, and

AP : component due to the thickness of the cavity.

Then we have from the linearized pressure equatiOn (22)

and the equation of the velocity potential (30).

AP=_P_P*__Apio+APic+LPto+1,.Ptc

(41) where, N , _i

e,

L,,, =-

_{10 4lr),}

('

WO(r')dr'f 'y,,lr' u' t__k..)

'V,-,-r'(x hO'o)rh.sin(pOO'+Zt)

- AI

f'O(fI

'vJr' V'

4 k'=1b

dv'

Rb3 -i '

''-'

} Rb3 V =

--e--T°

W*r)drf' atb (r s')

4irk'=l Tb -i

VZ (x - h9) + rr'2.sin(ip -

_{- 9k' + lt)}

"b

/h2+r2dv

N

r

''2' t)

_{2 a2} k=lfSTb

(r')dr'f

(r' t)

(

+ at as

tk'fr' sÇ

Jh2

_{+ r'2 dv}

Rb Sro_ t) _a a

Tb°" Svk,1(;

t)

w*-)tCk

(r s'. t)

V (x - h9) + rr'.sin(ip -

- °k' + 2t)

3

"b

(45)

.h2+2dV}

The components APq, àP». and P«j in the fluctuating pressure are of the order of 1/Rg, while the component L\P contains the term of the order of 1/Rb. Therefore, the main contributions to the fluctuating pressures cOme from the component due to the thickness of the cavity at

rela-tively large distances from the propeller.

Further, let us simplify the Eq. (45) of the fluctuating pressure L.Pt due to the thickness of the cavity. Assuming

that the distance Rb in the integrand of Eq. (45) is

approxi-mately independent of the variable y' for a field point rela-tively far from the cavity as compared with the dimensions of the cavity, the sources representing the cavity can be regarded as concentrated on the mid-chord of the cavity

y

v. Then, integrating Eq. (45) with respect to y', we

have ÇTO(ò SCk'(r, t))+

W*atEk(r t)

1

dr'

C

4 k'=l irb

at2 at $T

[asCk(r. t)

_{+ W*tEk,(rÇ t)]} Tb at

V2 (x - h90) + rr'Lsin(o -

O - °k' + 'ìt)

_.dr(46)

where,

R,,c = '.J(x - hO0) + r2 + r'2 - 2rr'cos(p r- 8 Ok'+t),

0c= OM(r') + r')vC, VC [Vkl (r

t) + (k2 (r

t)] /2, Vk2(r, t)

SCk(r, t) f

vkl(r t)tCIfr s, t)ds,

tEk fr, t) = tCk[r, vk2(r, t), t] (47) Here SCk (r, t) represents the sectional area of the cavity at

each section of the blades, and tEk(r, t) represents the

thickness of the cavity at its trailing edge.

3. Practical method for estimation of fluctuating pressures

induced by a propeller

3.1 Currently proposed methods

With the strength of the bound vortices 70(r,

y, t),

'YC(r, y, t) and the thickness of the cavity tCk(r, y, t)

detér-mined by the boundàry conditions mentioned in Chapter 2,

we can calculate the pressure fluctuation induced by 'a

cavitating propeller. Of the above three unknowns, the

strength of the bound vortices 'y0(r, y, t) in non-çavitating condition is relatively easy to be solved with the present techniques fOr numerical calculation, whereas it takes too

much computation time at present, to determine the

Table i Various methods for predicting fluctuating pressures

of the cavity tCk(r, y, t) by solving a mixed boundary value

problem with respect to the cavity. Lee

solved this

mixed boundary value problem directly in the time domain with a Vortex Lattice Method. In spite of time consuming computation, the result did not always agree with the ex-periments and further improvements to the method are

considered to be necessary for practical use.

Most of the current methods of predicting fluctuating

pressures around a cavitating propeller are therefore based

on simplifying assumptions and empirical corrections for

them in estimating behaviour of unsteady cavities. Namely, they estimate the instantaneous cavity geometry by quasi-steady approach from pressure distribution over the blades

of a non-cavitating propeller. Tanibayashi18 classified the various methods proposed until now to calculate the

fluc-tuating pressures induced by a cavitating propeller as shown in Table 1. According to this table, the pressure distribution over a blade is estimated either by unsteady or quasi-steady

theory of a non-cavitating propeller. Then for each instan-taneous pressure distribution, the cavity extent and

thick-ness are estimated stripwise at each blade section. This is further classifièd into two methods.

One of them (19 30 36either applies Knapp's

dy-namic similarity law on a spherical bubble26 or assumes,

as supported by many experimental resu Its(37H39,,that the lift coefficient is unchanged by the appearance of cavita-tion, so called "Lift Equivalence Method." Since this meth:

od provides no information on cavity thickness, cavity

thickness must be determined by the experiments or the

simplified theory.

The other method27 3 may be represented

by that of Noordzij27, who employed Geurst's free stream

line theory (28H29) for steady flow around a

two-dimension-al hydrofoil with the same lift coefficients and cavitation

number as those of each blade section of the propeller. The

results, however, underestimated the thickness of cavity

and accordingly fluctuating pressures, because this method does not take into account unsteady behaviour of the cavity on a blade passing a wake peak. In this method, an iterative calculation or the graphical calculation is necessary, because

MTB15O May 1982

the cavity extent and thickness can not be given explicitly. Further, physical

me-aning of the results is doubtful when the

cavity covers 70-100 % of the chord of

the blade.

Besides the above methods, Kato and

Ukon37 proposed the use of either the free stream line theory or the lift equiva-lence method depending on the angle of

attack and the thickness-chord

length ratio of the blade section. This method is,

however, too complicated for practical

application.

Next, in

calculating the fluctuating

pressures induced by the cavity, Huse221

replaced the displacement effect and the volume variation of the cavity by a simple point source, while Noordzij27> represented the cavity by the source distribution like the present method. Theoretically the latter method is more rigorous, but it should be kept in mind that the accuracy of

prediction of fluctuating pressure is more dependent on the

accuracy of estimation of cavity geometry, which is

de-scribed in the next section.

3.2 Method to estimate cavity geometry

There are various methods to estimate the cavity

geome-try on the propeller blades as mentioned in previous sec-tion, but none of them are complete in expressing com-plicated unsteady cavitation. Therefore they are

supple-mented in most cases by experimental corrections corn par-ing the calculated values with the experiments.

In this paper, 'lift equivalence method' is adopted from the practical point of view because of its simplicity, and experimental correction factors are introduced to consider

the unsteady effect of the cavity.

In order to calculate the cavity extent by lift equivalence

method, the pressure distribution over the propeller blade

in non-cavitating condition must be known as a function of

time. At first, the bound vortex distribution y0(r, y,

t--k-)

in non-cavitating condition is

obtained by solving the

boundary condition Eq. (35) and we have an equivalent

two-dimensional blade section corresponding to the vortex

- _- M

distribution y, (r, y, t ----) at each radius. Here the effect of the viscosity of the fluid is considered with a correction factor Ck to the slope of the lift coefficient as follows:

y, t----) = Ck.7o(r, y, t_----)

(48)

where Ck = 0.882

Next, we calculate the pressure distribution over the equivalent two-dimensional blade section (40) by Moriya's conformal mapping method41 Then, we consider this to

be the same as the pressure distribution over the propeller

bi ade.

Let the pressure distribution [p0] over the back of the blade section in non-cavitating condition be Pb(r, s, t).

7

Authors _{non-cavitating condition}Pressure distribution in Cavity extent Cavity thickness Johnson (19)

Søntvedt Quasi-steady theory

Dynamic similarity law

on spherical bubble Diameter of tipvortex Noordzij27 Quasi-steady theory 2-dimensional theory on cavitating foil Szantyr (30) Unsteady lifting surface

theory on spherical bubbleDynamic similarity law Bubble dynamics (31)

Fitzsimmons Quasi-steady theory 2-dimensional theory on cavitating foil Kaplan 132)

Benston Breslin

Unsteady lifting surface

theory 2-dimensional theory on cavitating foil

Vuasa Ishi,

Unsteady lifting surface

theory 2-dimensional theory on cavitating foil

Chiba

Sasajima

Hoshino

Unsteady lifting surface

theory

Modified lift equivalence

method

2-dimensional theory & experimental data

the condition in which cavitation occurs can be expressed with vapour pressure.P as

Pb(r,s,t)P

(49)

With dimensionless pressure coefficient C(r, s, t) and local cavitation number 0L(r,t) at radiusr defined by

Cp(r,s,

t)

Pbfr,S,t)P00

CLfr, t) (50)

Eq. (49) becomes

Cp(r,s,t)oLfr,t)

(51)

With atmospheric pressure Pa, acceleration of gravity g,

and immersion of the propeller shaft I, we have at the

representative position, namely, the mid-chord of the blade

Parn, +pg{Jr.cos(OM(r)

0k 2t))

(52)

Then, the local cavitation number oL(r, t) can be

repre-sented as

ÛLfr, t)

Pv +pg{I -

_y2pW*2r.cos(eM(r)+Ok

2t»

(53)

The starting point of cávity skl(r, t)

is approximately

given by the s-coordinate at which both sides of Eq. (51)

becomes equal first when approaching the blade from

up-stream. We have known from experience that cavitation inception tends to be delayed by the effect of viscosity of

the fluid (36, but we neglect this at present.

Now, from the experimental results of the

two-dimen-sional blades, it is well known that the lift coefficient is

unchanged by some extent of cavitation139. Further, for

ordinary merchant ships the reduction of propeller thrust is

not accompanied by the occurence of cavitation. Such is

the background for adopting the lift equivalénce method in

the present study. The method is schematically explained by Fig. 6, in which the condition of lift equivalence is

ex-pressed by .s (r, t) SjkO

[CP(r,s,t)ÛL(r,t)]ds

Skl(r, t) k2 t) SkO(r,t) [Cfr, s, t) + L (r, t)] ds where,

sA.cj(r, t):

chordwise position at which Cp(r, s,

t) be-comes equal toUL(r, t) again.

For lower aL(r,t) at whichsk2(r, t) can not be found within thechord of the blade, sk2(r, t) is determined assuming for convenience C(r, s, t) aft of the trailing edge to be zero.

When the s-coordinate sk2(r, t) of the trailing edge of the cavity is obtained by Eq. (54), the length of the cavity is

given by Eq. (7), namely,

1ckfr, t) t) 5k1(r, t) (55)

However, cavity length estimated by this simple method döes not agree with experimental observation especially in the case of unsteady cavitation. Therefore, let us attempt a modification with experimental correction factors

consider-ing the unsteady effect of the cavity. At first, we introduce

a correction factor KL to the cavity length defined as

S (r t) EC,,(r, s, t)+crL (r, t)]ds KL = t)

s,(r, t)

S[C,,(r,

s,

t) - aL

(r,

t)]ds

Skie; t (54) (56) Q L) 1.5 1.0 Q. 0.5 0

0.5

B=KL A Pressure distribution of non.cavitating propeller 'O (-,i) Sk2(T,t)

skl(r,t) Mean camber line

Fig. 6 Explanation of present method

for prediöting cavity length

By suitable choice of KL,the length of the unsteady cavity

can be determined with reasonable accuracy. KL = 1 corre-sponds to the ordinary lift equivalence method for the case

of the steady cavitation. However, in the case of unsteady cavitation KL 5 generally smaller than unity because a cavity can not be fully developed within the limited

dura-tion of low pressure around the cavity in the unsteady flow.

Because of the same reason, there is a time lag in the

growth of the cavity. When we solve the equation of motion

of a spherical bubble considering the unsteady effect, the bubble radius reaches maximum shortly after the peak of the negative pressure, as shown in Appendix. Therefôre, this time. lag in the growth of the unsteady cavity must be considered in the theory and shall be represented by

Since

it

is

difficult to obtain these factors KL and ¿t

theoretically, attempts are made to determine them by

experiments.

Next, let us consider the thickness distribution of the cavity. In theoretical treatment, the shape of cavity is

con-sidered to be of closed, semi-closed or open type. Since the unsteady cavity observed in cavitation tunnel appears to. be

like an open, type model, the thickness distribution of the

cavity is represented by an empirical equation based on the

calculation of a cavitating foil according to lsay42 At the downstream end of cavity, however, a finite thickness ¡s

assumed in accordance with observed cavitation patternson

propeller blades in non-uniform flow. Namély, with an an-gle of atack to the equivalent two-dimensional blade

sec-tion a(r, t) and the thickness at the cavity end tk (r,

t),

the thickness of the cavity at any chordwise position. is

given by

tkfr,s, t) =

tkfr, t).["

t)1iíq (57)

ick (r, t)

where, tk

(r, t)= 2a°(/ t).lCk (r, t)

q : parameter to represent the cavity shape

Up to the above, a cavity has been thought to be filled with vapour, but the actual cavity is not always filled with vapour only. Izumida et al.43 measured actual void ratio (or true cavity volume ratio) in the cavity on

sional wing by using a Gamma-ray void meter and reported that void ratio was 30-50% in the case of partiäl cavitation. Considering this, effective thickness of the cavity tck(r, s, t) ¡s assumed to be expressed with a parameteri as

S -Sklfr, t) 1/n

tck (r, s, t) = tEk (r, t) [ ] (58) 1ck

t

where, tEk (r, t) = g. tEk* fr, t)

parameter to represent the effect of void ratio in

the cavity

Here, this parameter is experimentally determined by comparing the calculated value of the cavity induced

fluc-tuating pressure with the measured one. Then, the sectiònal area of the effective cavity SCk(r, t) becomes from Eq. (58)

2 t°1r t)

SCk(r, t) = Kc [ci,fr, t)]2 (59)

-3.-3 Method to predict fluctuating pressures

3.3.1 Method of prediction for non-cavitating condition

For a non-cavitating propeller, the strength of the bound vortex distribution 'y0(r, y,

t-k-)

is obtained by solving the

boundary condition (35), and considering viscous effect of

the fluid by Eq. (48), we have 'y(r, y, t ---) Substituting

this bound vortices into Eq. (42), the fluctuating pressure

zo due to the effect of propeller loading can be obtained.

Next, substituting the blade thickness distribution tb(r, s) into Eq. (44), the fluctuating pressure .Pdue to the effect of blade thickness can be obtained. Then, the fluctuating

pressure in an infinite fluid is given by the sum of Pj0 and In practical application we are most concerned with

the fluctuating pressures over a hull surface of a ship near

the propeller. The fluctuating pressure on the hull surface

LPH can be expressed with- solid boundary factor Sb as follows:

'H Sb.(lO+LPO)

(60)

Here, Sb representing the image effect of the hull surface

can be taken to be 2.0 approximately as reported by

Huse44

Since the fluctuating pressure PH is a periodic func-tion with blade frequency /2ir, it can be expanded into a Fourier series as follows:

= 1/2Po+

_{E [P,,N.cos(mNZt) + ZP,,N.sin(mftJSt)]}

m=1

= '/2z.P0+ E °mNc05(mt ømN)

- (61) m=1 where, mN=

/(iP,N)2 +

)2 mN = tan

N/'PN)

=

PnN1 + PNt, 'nN

= nN1 + flç2irIíz L1P,nNI = nN1= Nt= Sb.--J

10.cos(mNt)dt

ITO Ç 2ir/IZ

Sb--S

P10.sinmN2t)dt 2irffZ

Sb..Ç

¿XPtO .cos (mNíZt)dt Input MTB1 50 May 1982

I

Fluctuating pressure due to blade thickness: _P0 Propeller geometry Wake distribution Operating condition

I

Unsteady lifting surface theory in non cavitating condition: y0(r,s,t)

f

t

Fluctuating pressure due to propeller loading: P10 $

Blade pressure distribution ¡n

non-cavitating condition: Pô(r.s,)

t

I

Geometry of unsteady cavity at

(t+t) (extent and thickness)

I

Fluctuating pressure due to cavity

thickness: _"rC

Fluctuating pressure induced

by a cavitating propeller

Output: _sPHSb.

Fig. 7 Block diagram of fluctuating pressure calculation procedure by present method

2ir/

= Sb.f

P1ysin(mNt)dt

Further, if we define the following notations:

AQ /IADC 12 lADS- 12

mNIV

mNIl +1

mN/I

-1 S C

WmNI -

tan t'mNlI'mN1

mNt=

/(zP,Nt)2 +

ÇbmNt

= tan'

(PNt/zPNt)

-the non-dimensional amplitudes of -the fluctuating pressure are expressed as follows:

mN/ &°mNt ₍₆₄₎

KpmN KPm_{NI =}

Pn2D2 = pn2D2 where,

n : number of revolutiòns of a propeller u/2ir D: diameter of a propeller 2,-e

3.3.2 Method of prediction for cavitating condition The fluctuating pressure around a cavitating propeller

consists of z.Po and as mentioned in 2.3.

Of them, and P10 have been already obtained in

3.3.1. Since the reduction of propeller thrust is not accom-panied by the oàcurrence of cavitation for ordinary mer-chant ships, the change of bound vortices 'y(r, y, t due to the cavity can be neglected, and accordingly we

neglect Lastly, let us consider the component When the sectional area of a cavity SCk(r, t) and the

thick-ness at the cavity end tEk (r, t) are calculated by the method

explained in 3.2, zP can be obtained by substituting them

into Eq. (46). Thus using ¿Pic, z.Pj and

the

fluc-tuating pressure LPH on a hull surface induced by the

cavitating propeller is represented as

'-H

Sb.[iPio+IPfo+PtC]

(65)

Flow chart of the calculation mentioned above is shown in Fig. 7.

i o

____jpl$

Min. cavitation number 0.2 Max. velocity 12m/s

Propeller dynamometer I7HP

Torque (max.) 5kgm

Thrust (max.) 120kg

Impeller motor 7OHP

Fig. 8 GeneraI arrangement of cavitation tunnel in Nagasaki Experimental Tank

According to the calculation of the solid boundary

fac-tor Sb for a spherical bubble moving with varyingradius(23

can be taken to be 2.0 approximately if the distance of the sphere from the wall is larger than twice the diameter of the sphere. Considering that the cavity on the propeller

blade is small enough compared with the distance from the hull surface, Sb is taken to be 2.0 in the following

calcula-tion.

The fluctuating pressure £.PH

obtained by Eq. (65) is

expanded into a Fourier series and its amplitude is dimensionalized in the same manner as the case in non-cavitating condition. Then, L..PH can be expressed by Eqs.

(61) and (62) with the following relations instead of Eq.

(63):

ADC - ,tDC ADC + ADC

mN - mNl + mNt mNc

AS

- AnS AflS AflS

'mN -

mN1 + 1-mNt + 'mNc where,

ç

2/c2

tJ'nNc = 5b-$0

1iPt.cos(mNt)dt

2 2ir/2

= Sb. TSo

A.Pt.sin(mNt)dt

3.4 Estimation of surface förces

When the fluctuating pressures LPH induced by a

pro-peller in cavitating and non-cavitating conditions are cal-culated, we can obtain surfáce forces by integrating them over the hull surface SH of a ship. Let us treat the vertical

surface force F5 only, because the vertical vibration is most

important when considering the ship vibration by surface

forces. Then F is expressed by integration of the fluctuat-ing pressures over the hull surface as follows:

F.. ffsH °HflzdSH

_=ff

H'x dy

upward direction cosine normal to the hull surface projected hull surface to the xv-plane

Here, 1H is a function of the distance from the propeller tip and given by Eq. (61). Substituting Eq. (61) into Eq.

(67), the vertical surface force can be written as

= 1/2F0 +

_{FmN.cos(mN2tmN)}

₍₆₈₎

m1

where,

LFmN=

sJ(ffLPmN.cosømNdxdv)2+ j'fr9mNsin ømN"'Y)2

_sp

(69) (67) Wir me

tflflfl.ffl/. tfl.1

4 mf low Stern frame Model propeller

Shaft center line

Wake

screen

Observàtion window

Fig. 9 GeneraI setup for cavitatiofl test of

propeller in non-uniform flow

mN

tan'[,ffLPmN.sin ømN dx dy)/

(ffmNc0SømNdxdV)]

sp (70)

Usually, the blade frequency component of the surface

forces ¡.FN is concerned most in studying ship vibration. 4. Comparison of cavity geometry with experiments. in cav

itation tunnel

4.1 Test procedure in cavitation tunnel

Most of cavitation tests mentióned in this paper have

been conducted in the cavitation tunnel of Nagasaki Experi-mentál Tank, Mitsubishi Heavy Industries, Ltd. The

cavita-tion tunnel has a closed typè measuring seccavita-tion whose dimensions are 500mm x 500mm as shown in Fig. 8. In

order to minimize the correction for tunnel wall effects,

the static pressure and the speed are measured separately in the plane of propeller by a static pressure tap on a side wall

and a total pressure tube just outside the boundary layer along the side wall145. Non-uniform flow to the propeller ¡s simulated by a wake screen and a wire mesh around a

stern frame in front of the propeller as shown in Fig. 9.

For measuring fluctuating pressures a flat plate is set above. the propeller parallel to the propeller shaft.

Advance coefficient J, thrust coefficient K and cavita-tion number a are defined by

V _T

J=;;--' KT-204

_°fly212

(71)

In uniform flow tests, water speeds are varied to obtain the designed advance coefficients at a constant number of

revolutions of the propeller. In non-uniform flow tests, wa-ter speeds are adjusted by thrust identity method so that the propeller produces thrust corresponding to the design

thrust coefficient at a constant number of revolutions.

Further, in order to establish the designed cavitation

num-ber, the static pressure in the tunnel is adjusted.

The cavitation phenomena on the propeller blades are sketched and photographed through the observation

win-dows by using stroboscopic lighting.

The angular position of the generating line of the first

blade is defined by p, where Pc is an angle measured clock

wise from top. Then, p can be expressed with the angular velocity of the propeller as

1.2 1.0 0.8 ç 0.6 0.4 0.2 Section-A (t/C= 0.05) Experiments Calculations Oû0=4 Lift equivalence "-.

method (K,=l.0) Nishiyama's method46 1.0 2.0 3.0 _Poo-Pv oL_ --w'2

-Fig. lo Comparison of cavity length of

blade section at nr0 = 0.9

The cavitation patterns at po = 00 are observed in the uni-form flow tests, and the cavitation patterns at various an-gular blade positions from inception to desinence are

ob-served in the non-uniform flow tests.

Wake fraction w representing the wake distribution simulated by the wire mesh is defined with an axial

dis-turbance velocity v as

(73) Principal particulars of Propeller A-1 used in the tests are

shown in Table 2. Propeller I is referred to in Chapter 5

only.

4.2 Comparison of cavity extent

4.2.1 Cavitation on two-dimensional wing

In order to examine the applicability of the lift

equiva-lence method for estimatihg cavity extent, reference is

made first to cavity lengths on two-dimensional wings. The

experiments were conducted in the cavitation tunnel on the two plane wings with typical blade sections at 0.9r0

radius and at 07r0 radius of a propeller. These two-dimen-sional wings are called Section-A and Section-B respectively,

and chord length of them is 200mm and both side ends of

them are fixed to the tunnel wall.

Fig. 10 and Fig. 11 show comparisons between thus

: Tested in uniform flow

4.0

a=Z/

s .0

Fig. 11 Comparison of cavity length of blade section at nr,, = 0.7

measured and calculated cavity lengths by lift equivalence

method (KL = 1), where W is the inflow velocity to the

wings and ct0 is the angle of attack of face to the inflow velocity. Results of Nishiyama and Ito's two-dimensional

cavity flow theory1461 are also shown in those figures. For the Section-A with smaller thickness-chord length ratiò, the

calculated results by lift equivalence method and cavity

flow theory coincide with the experiments to the same

degree. However, for the Section-B with relatively large

thickness-chord length ratio, lift equivalence method tends

to give somewhat shorter cavity lengths than the experi-men ts.

Anyhow, the cavity lengths of a two-dimensional wing can be estimated by lift equivalence method with

reason-able accuracy.

4.2.2 Cavitation on propeller operating in uniform flow

Next, let us compare the cavitation patterns observed on

a propeller operating in uniform flow with the results

cal-culated by lift equivalence method (KL = 1). The observa-tion of the cavitaobserva-tion patterns was made on the Propeller F

for a container ship (e), and for the conditions of 2.0 at various J's. Fig. 12 shows the comparison of the cavita-tion patterns.

Table 2 Principal particulars of model propellers and wake distribution

Experiments Calculations

a0=2- Lift equivalence

o a0='

method (KL=1.0)

o=8' Nishiyama's_method6

c0= 8' 1.0 2.0 3.0 P - Pv MTB15O May 1982 4.0 11 Model propeller A B C D E F' G H V Diameter (mm) 250.0 250.0 250.0 250.0 250.0 268.0 268.0 202.28 250.0 Pitch ratio 0.872 1.050 1.000 0.661 0.661 1.000 1.032 1.031 0.852 Expandedarearatio 0.702 0.750 0.850 0.567 0.606 0.652 0.652 0.745 0.654 Bossratio 0.185 0.180 0.180 0.172 0.169 0.191 0.191 0.200 0.180 Number of blades 5 5 6 4 5 5 5 5 5

Rake angle (deg.) 3 8 8 8 5 8 -15 6 6

Skew angle (deg,) 13 16 15 15 15 14 72 12 15

Designed for Container ship (al

Container

ship (b)

Container

ship (b) Tanker (c) Tanker (dl

Container ship (e) Container ship (e) Container ship (f) Liner ()

Wake distribution Wire mesh

Ship (a) Wire mesh Ship (b) Wire mesh Ship (b) Wire mesh Ship (c) Wire mesh Ship (d) Wire mesh Model (e) Wire mesh Model (e) Behind ship Model (f) 1.2 Section-B (t/C=0.07) 1.0 0.8 0.6 0.4 0.2

=0-(p= 0.

J=0.3

J=0.5

i=0.7

lt is evident from this figure that the steady cavitation on a propeller operating in uniform flow can be estimated

by lift equivalance method (KL = 1) for practical purposes.

4.2.3 CavitatiOn on propeller operating in non-u niform

flow

Lastly, let us compare the unsteady cavitation patterns on a propeller operating in non-uniform flow. The model propellers tested were chosen to cover a wide variety of

principal particulars as shown in Table2, differing in pitch ratio, expanded area ratio, number of the blades, skew of the blades and so on. Propellers A-G were tested in non-uniform flow simulated by wire mesh in the cavitation tun-nel of Nagasaki Experimental Tank and Propéller H was

tested behiñd a ship modél (f) in the large cavitation tunnel of Ship Research lnstitute47

The unsteady cavitation patterns on the propeller blades

are estimated by the method mentioned in 3.2, but the

factors KL and Lt representing the unsteady effect on the

cavity must be determined experimentally

in advance. Therefore, these. factors KL and t were obtained by

com-paring the experiments with the calculatiòns for Propeller A. Propeller A was designed for a container ship (a) and

tested in non-uniform flow as shown in Fig. 13.

In order to determine KL. comparison was made on the maximum cavity extent of the unsteady cavitation as shown

in Fig. 14, between the results from the observatiòn and from lift equivalence method. It is evident from this figure that with KL = i the lift equivalence method has a tenden-cy to give wider cavity extent than the experiments. Then,

the calculätion was made for various values of KL with

modified lift equiva lance method, Eq. (56) and it was found

Fig. 13 Simulated wake distribution of a container ship (a)

Propeller A KT=O.20, o= 1.54

Observed Calculated KL=l.O

(Lift équivalence method)

+

9=10

Fig. 14 Ier disk + + o=350 Comparison of maximum extent of

cavity ih non-uniform flow

that KL = 0.25 gives better agreement with the observed cavity extent as shown in Fig. 14. Therefore, in the

follOw-Ing calculations, KL is taken as

KL = 0.25 (74)

Next, let us determine the time lag Lt in growth ofthe unsteady cavity experimentally. To this end, the obseréd

cavitation patterns at variOus angular blade positions were

compared with those calculated in a quasi-steady manner

by the present method with KL

0.25. The results are

shown in Fig. 15, in which about 20 degrees phase lag is

observed between the experiments and the calculations. For

convenience, in the following calculations, the time lag ¿t

of the unsteady cavity is represented by the phase lag I.spò defined as

'Po

180it/ir= 20 in degree

(75)

Then, the calculated cavitation patterns with this phase lag Po are found to give better agreement with the observed ones as shown in Fig. 15.

The values of the factors KL and Po given by Eqs. (74) and (75) are tentatively used in the following calculations.

At first, comparison is made for the five- and six-bladed

propellers (Propellers B and C) with higher pitch ratio for a container ship (b). Fig. 16 shows that agreement of cavita-Propeller F

Calculated Observed Lift equivalence method

(KL = 10)

=0 P0=0W

Fig. 12 Comparison of cavitation patterns

in uniform flow (a = 2.0)

Propeller A Kr0.20, 0,=1.54

Photographs of cavitation

340

Predicted by present method with phase correction (KL=O.2S, Ç0=20)

Ç? =340'

0'

Predicted by present method without phase correction (KL=0.25)

o-0'

Observed in cavitation tunnel

20

Fig. 15 Comparison of cavitation patterns with

and without phase correction Propeller B (N=5) Kr=0.187, 7= 1.40

Observed in cavitation tunnel

+ + + ± +

Ç?0=340 O' 20' 40 60

Predicted by present method (KL=O.25 Ç?0=20')

+ q?0 _340' Fig. 16 ± 20' 40 60' 20 40' + Propeller C (N=6) KTO.183 Cn=l.34

Observed in cavitation tunnel 40

+ -i- + +

340' 0' 20 40

Predicted by present method (K=0.254Ç?o=2O') 60'

60'

60'

+ + 1+

O' 20' 40' 60'

Comparison between observed and predicted cavitation patterns of model propellers for a container ship (b)

Kr=0.157, c= 1.75 (Full load)

Observed in cavitation tunnel

+

1+\

+ +

Ç'0=34O' 0' 20 40 60'

Predicted by present method (KL=O.25, 9=2O')

+ 340' Fig. 17 + Ç?o=340' Fig. 18

Comparison between observed and predicted cavitation patterns of model propeller D for

tanker (c) in full load condition

KTO.177, =1.59 (Ballast condition)

Observed in cavitation tunnel

0'

MTB1SO May 1982

+'

1+\

+ + +

Ç?0= 340' 0' 20 40' 60

Predicted by present method (KL =0.25, Ç?0=20')

60' Comparison between observed and predicted cavitation patterns of model propeller E for tankers (d) in ballast condition

tion patterns between the observation and the prediction is generally good except at angular blade positions where

cav-ity leaves the leading edge of the blade.

Nèxt, comparison is made for the four- and five-bladed

tanker propellers (Propellers D and E) with lower pitch

ratio. Propeller D was tested ¡n full load condition, and Pro-peller E in ballast condition. Figs. 17 and 18 show the com-parison between the observed and the predicted cavitation

patterns for both propellers. In this case, agreement is not so good as the previous case. The cavitation observed in tunnél is limited to tip region of the blades, while the cavi-tation predicted by the calculation extends to inner radius

along the leading edge of the propeller.

In order to investigate the effect of skew of the blades,

comparison is made for a conventional type Propeller F and a highly skewed Propeller G designed for the same container ship (e)(48U. Comparisons between the observed and the calculated are shown in Fig. 19. Good agreement of cavita-tion patterns for the convencavita-tional type is seen as expected

from Fig. 16, but it is to be noted that similar degree of

agreement can be seen on the highly skewed propeller.

Finally, comparison is made for the Propeller H operat-ing behind a model of ship (f)l47) Wake distribution meas-13

Propeller F (C.P.) KT=O.172, =1.57

K=0.l98,

o=

1.87

Observed ¡n cavitation tunnel Observed in cavitation tunnel

+ + + +-

-9=340'

0' 20' 40 60

Predicted by present method (KL =0.25, 9=20)

Propeller G (H.S.P.) KT=0.1S3, o1= 1.67

Observed in cavitation tunnel

,+..

r

20' 50 80' 100'

Predicted by present method(KL = 0.25, 9=20)

20' 50' 80'

Comparison between observed and predicted cavitation patterns of conventional and highly skewed propellers1491

Propeller disk

100'

05 Fig. 20 Wake distribution of a container ship model (f)

ured in the propeller plane is shown in Fig. 20. In this case,

effect of the tangential disturbance velocity v is included in both experiments and calculation, and a good agreement can be seen between them as shown in Fig. 21.

As meñtioned ábove, it is found that the unsteady

cavi-tation on the propeller blades can be predicted with

reason-+

Fig. 21

+ + + + +

9=

340' 0' 20' 40' 60'

Predicted by present method (KL=O.25, _Q0=20)

+

-I-0' 20' 40' 60'

Comparison between observed and predicted cavitation patterns of a model propeller H

operating behind a ship model (f)147

Fig. 22 Arrangement of pin gauges for measuring cavity thickness distribution

able accuracy by the present method with the correction factors KL and Po given by Eqs. (74) and (75) for various propellers and operating conditions.

4.3 Comparison of cavity thickness

In Section 3.2, it was proposed to estimate the cavity

thickness on the propeller blades by Eq. (57). Let us

exam-ine the applicability of this equation and further determexam-ine

the coefficient q by comparison with experiments.

Measure-ment of the thickness distribution of the unsteady cavity

on the propeller blades is, however, very difficult to be

made. Only a few reports123501 are available which

de-scribe the measurement results.

In this study a fairly simple method was adopted; the

thickness distribution on the propeller blades was measured

by pin gauges mounted on the back side of the blades of the Propeller A as shown in Fig. 22. The pin guages were planted at the chordwise positions s/C 0.2, 0.4, 0.6 and 0.8 for the 2nd, 3rd, 4th and 5th blades respectively, and

at the radius r/r0 = 0.6, 0.7, 0.8, 0.9 and 0.95 for each

blade. The pins were shaped with stream-lined sectión arid

were painted with stripwise differing colour to detect the

height of the cavity surface. In a strict sense, the cavitation

on the blades is affected by the presence of the pins, but when they were compared With the cavitation on the first blade without pin, the shape of the cavity upstream of the'

Fig. 23 Close-up of cavitating propeller blades with pin gauges

piñ was not seen to be affected so much.

Measurement was made on Propeller A operating in a

non-uniform flow as shown in Fig. 13. Photographs in meas-urement are shown in Fig. 23, and typical results are shown

in Fig. 24. Looking over the shape of thickness distribu-tions, the parameter q in Eq. (57) was chosen as 2. The

results of calculatiOn with q = 2 is entered with broken line in that figure, which shows generally good agreement with

the measured results except at the angular blade position

where cavity leaves the leading edge of the blade. Therefore,

the cavity thickness tk(r, s, t) is predicted by Eq. (57) with

q=2

(76)

in the following calculations.

The volume of the cavity Vc*(t) is obtained by radial and

chordwise integration of the measured cavity thickness tk(T,S, t) as

r0 s

(rt)

Vc*(t) $ dT'S k2

t

(r s t) ds' (77)

r/, s,1(r, t

The volume variation of the unsteady cavity on a blade is

shown in Fig. 25, comparing with the calculated values. The

calculated cavity volume was found, however, to be fairly

larger than the measured one except in the case of Kr=020 and a, 2.33. The volume of the calculated cavity reaches maximum earlier than the measurement, though the blade position angle had already been retarded by 20 degrees. 5. Comparison of estimated and measured values of fluctu.

ating pressures

5.1 Measurements of fluctuating pressures in cavitation tunnel

The shape of unsteady cavity on the blades of a

propel-ler operating in non-unifOrm flow can be estimated with

pretty good accuracy by the method mentioned in 3.2.

With thus obtained shape of unsteady cavity, the

fluctuat-ing pressures induced by a cavitatfluctuat-ing propeller can be

câl-culated by the method mentioned in 3.3. In this chapter,

let us examine the method of calculatiOn by comparing the estimated fluctuating pressures with those measured in cavi-tatiOn tunnel and those measured in full scale ships.

In a large cavitation tunnel or a depressurized towing

tank, propeller-induced fluctuating pressures are measured

0.20. = 1.54 L.E. 10 320 . Measured at r/r0=O.9 Calculated at r/r00.88 (q=2) po=0-10 o Measured - Calculated (KL=0.25, o=20, =2) KT=O.2O, o=2.33

000000

2-.---r o i 340 0 20 40 60 80 ci [degree] 20 40 60 80 90 [degree] KT=0.20, ,=1.11 MTB15O May 1982 tEK(r,t) I.E. tEK (r,t) 320 340 0 20 40 60 80 90 [degree]

Fig. 25 Comparison between measured and calculated cavity volumes for a propeller A

on the hull surface of ship model set in it. On the other hand, in the case of smaller cavitation tunnel, ship model

cannot be accommodated in it, so a dummy model is used

to measure fluctuating pressures on it. The cavitation

tun-nel in Nagasaki Experimental Tank is of a medium size so

that we replace the hull surface above the propeller by a

flat plate located parallel to thè propeller axis and we

meas-ure the fluctuating pressmeas-ures induced on this flat plate as shown in Fig. 26. In this case the tip clearance ratio 9t/0.

whereg is the distance between the flat plate and the tip of propeller, is kept the same as full scale ship and propeller.

In the measurement of propeller-induced fluctuating

pressures, the measured value may include the component due to the vibration of measuring system. Especiálly when

the measuring system is resonant with unsteady propeller

forces, the influences become very large. Unsteady propeller

forces are periodic function with a fundamentäl frequency 15

L.E. I.E.

Ç'0=30

IEK(r.t)

Fig. 24 Comparison between measured and calculated cavity thickness distributions

Fig. 26 Arrangement for measuring fluctuating pressures and data-acquisition system in cavitation tunnel

P(t256)--íP(t1)2 rP(t1)3 -P(t2), 2S6)2

r

Flexure Struts Tip clearance 9

Fig. 27 Periodic mpling technique used for averaging pressure signals

equal to the blade frequency tiN. Therefore to avoid reso-nance, the natural frequency of the measuring system

should be kept low enough below the blade frequency. For

this purpose we made the mass of the flat plate for meas-urement of fluctuating pressures as large as possible and supported it by flextures as shown in Fig. 26. Fluctuating

pressures induced on the flat plate are measured by pressure transducers mounted on it. The pressure transducers used

in the measurement are of a strain gauge type, 6mm in diameter, having maximum capacity 2 kg/cm2 and their

natural frequency is 20 kHz in air.

A block diagram of data acquisition from the pressure transducers in the cavitation tunnel is shown in Fig. 26. In

order to eliminate non-periodic random component of pres-sure signals, ensemble averaging is applied by using periodic sampling technique of the pressure signals. With 256 pulses per revolution the pressure signals from all transducers are

simultaneously acquired and digitized by A/D converter.

The samples are taken in general of the pressure during 50

revolutions of a propeller. For each pressure signal the

samples are averaged in the following way (see Fig. 27): P(i2)50 Propeller I K= 017 =atm, One revolution

K

2D2 0.05j 0.051 O'-C (O'-Center) g/D= 0.10

- KpmN22

0.05 0.025 -360'

_m=1234

0.05 0.025 r 360'

Averaged pressure signal AmØlitudè spectrum Fig. 28 Example of averaged pressure signals-and

amplitude spectrum in non.cavitating coñdition

1 50

LP(tj)me .P(t1)1, ¡ 1,2,

256

- (78)

50

Thus for each pressure transducer, we obtain the averaged pressure signals during one revolution of the propeller, con-sisting of 256 discrete pressures. The averaged pressure sig-nals P(t)meas. are expanded into a Fourier series as

fol-lows:

M

¿P(tj)meas =h/2O+

[Ncos(mNZtI)

m1 + LP,N.sin(mNZtj)1 M = 1/a +

PmN.c05(mti - mN) (79)

m=1 where,

= './(AP,,,N)2 + (LìP,,N)2 : amplitude of m-th blade

frequency component

ømN = tan'(P,,N/.P,Ç,-N) : phase angle of m-th blade frequency component L.P(CnN ¿.P(ti)meas.cos(mNÇZti) -256, 2 256 2 256

'mN

P(t,)measin(mNti)

256 j

M : maximum orders of blade frequency

Then, a non-dimensional amplitude of fluctuating pressure is expressed as

mN ₍₈₀₎

KpmN

- pn2D2

- 5.2 Comparison of cálculated and measùred fluctuating pressures induced by non-cavitating propeller 5.2.1 Comparison with the measurements in the

cavita-tion tunnel of Nagasaki Experimental Tänk At first we compare fluctuating pressures in non-cavitat-ing condition measured by the method in the previous sèc-tion with those calculated by Eq. (60). Propeller I was used in the measurements, whose principal particulars are shown

in Table 2. Measurements were done in uniform flow for various tip clearance ratiös g/D. An example of the avert aged signals of measured fluctuating pressures and their Fourièr coefficients are shown in Fig. 28. lt can be seen

=

m1 2 3 4

S (Starbord side) 0.05

O.05I

0.025

360' m=1 2 3 4 Pressure Signal Simultaneous

signals conditioners sampling & holding

i pulse/rev.

Pulse

generater 256 pulses/rev. A/O converter

Mass storage/ Mini-computer Data bank Output Pressure signal l.\ 256 pulses/rev. ist 2nd 3rd -50th. t-i pulse/rev.

Propeller I KT=Ol7, c=atm.

--300 0.08 0.06 o-0.02

-NSRDC 4118 360 sa 180 0.06 o 0.04 002

Fig. 30 Comparison of amplitudes and phases of fluctuating pressures in uniform flow (non-cavitating condition)

NSRDC 4118 360

0_1,.0, à

10 Fore '-x/r. 'Aft OExperiment (16) 360 ab - a) 180 0.04 0.02 9/D=0.15,J=0.833, 360 ab -m .', . 180 91/D0.05,J=0.833, o,=atm. o o

o:10

O 1.0 Fore x/r. -. Aft Present method MTB15O May 1982 360 ab 180 0.06 0.04 0.02 -1.0 0 1.0 Fore - x/r0 -*Aft Kerwin's method'6

0,-..:-.--

o

0iO

0 1.0 -1.0 0 1.0 -1.0 0

io

Fore x/r0 -'Aft Fore -X/r0 -.Aft Fore x/r0-.Aft oExperiment'61 -Present method _{----Kerwin's method'6} Fig. 31 Comparison of amplitudes and phases of fluctuating

pressures in uniform flow (non-cavitating condition)

Next we compare the measured fluctuating pressures with those calculated by this method with respect to the

5-bladed propellers NSRDC 4381 without skew and NSRDC 4383 with 100% skew operating in uniform flow. Tip clear-ance ratios are g/D = 0.05 and 0.15. The non-dimensional

amplitudes K5 and phases 0 of fluctuating pressures are

compared with the cálculated values in Figs. 32 and 33. We can see that the calculated values and measured values agree

well with respect to highly skewed propeller NSRDC 4383

as well as skewless propeller NSRDC 4381.

Finally we compare propeller-induced fluctuating

pres-sures in non-unifôrm flow measured by Ne1ka1521 with the

calculation by the present method, for the propellers

NSRDC 4381 and NSRDC 4383 mentioned above. The

non-uniform flow is produced by the 4-cycle screen shown

in

Fig. 34 and therefore 4-cycle velocity component is

dominant. The flat plate for the measurement of fluctuating

pressures is set at the two different positions as shown in Fig. 34 with the tip clearance ratio g/D = 0.05. The

meas-9,/B Experiment calculation 0.06 o

-0.10 0.20 0 0.32 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Port side - y/ro -.Starboard side

Fig. 29 Comparison of transverse distribution of

amplitudes and phases of fluctuating pressures

in uniform flow (non-cavitating condition)

that the signals of fluctuating pressures are very regular and similar to sine curve.

Transverse distributions of non-dimensional pressure

am-plitude K,,5 and phase Ø at the designed operating condi-tions are shown in Fig. 29, comparing with the calculated

values. The measured and calculated values agree very well with respect to both amplitude and phase.

As ¡s shown in this measurement, the phase difference of

fluctuating pressure between the port and the starboard

sides is very large in non-cavitating condition. 5.2.2 Comparison with the measurements at

NSRDC6XS1 )(52)

Next, we compare the results of model tests carried out at NSRDC (David W. Taylor Naval Ship Research and

Devel-opment Center) with the results of calculation by the

pres-ent method.

Denny1161 measured the fluctuating pressures induced

on flat plate in the 24-inch water tunnel of NSRDC. This propeller NSRDC 4118 was 3-bladed with expanded area ratio of 0.60.

The non-dimensional pressure amplitudes K3 and phase 03 measured at the designed operating condition in uniform flow are compared with the calculations by Eq. (60) in Figs.

30 and 31. The results calculated with the lifting surface theory by Kerwin and Leopoldt16 are also shown in these figures. Tip clearance ratios gt/D are 0.05 and 0.15. K3is composed of two parts; the one is the component K3 due

to blade thickness and the other is the component K,,3 due

to propeller loading. As for both calculations agree

very well with experiment, but with respect to K,,3, the

calculation by the present method agrees better than that

by Kerwin and Leopold's method. -0.6 -0.4 -0.2 0 yIr0 -100

0.206

sa a) 180 o 0.03 0.02 0.01 o 1< 0.02 0.01