Pressure fluctuations on the hull induced by cavitating propellers

PRESSURE FLUCTUATIONS ON THE

HULL INDUCED BY

AVIÎATING

PROPELLERS.

by ERLING HUSE Teehiie Hoqeschoo DOCUMENTATIE

I: K 2.t

III

DATUM: ZU APR. 1912

NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION NO.111 MARCH 1972

by

Erling Huse

Contents.

Paae

Abstract.

List of symbols.

i

Introduction.

5

Theoretical investigations.

6

2.1.

Generai consïdèrations.

6

2.. 2.

Theory based on source layer representation of

8

cav1tes!

2.3.

Prèssuré due to cav±ty motion represented by

10

source-sink couple.

2.4.

Prèssure due tó vóluflie varition of cavity.

14

2.5.

Prèssure due to tip vortex cavity.

16

2.6.

Description of computer programs.

18

2.7.

Numerical resilts.

20

Experiments with afterbody model in cavitation tunnel.

26

3..l.

lnstru.méntation.

26

3.2.

Mèaurements on afterbody mödei.

27

3.3.

influencé of test section vibrations and tqail

31

effects

3.4.

Comparison between theory and experiments..

-33

Pressure f luctuatiön

thiè to propeller-hull Vortex

34

cavitation.

5..

Conclusions.

36

Aòknowledqeflìent1

37

References.

38

ABSTRACT.

Numerical methods for calculation of the blade-frequency pressure fluctuations induced on the hull by cavities of given

geo-metry have been developed and programmed. Numerical examples show

that cavities which are constant in time give rise to relatively small pressure amplitudes compared to the amplitudes induced by cavities whose volumes are varying with time.

Experimental investigations of the pressure fluctuations induced on an afterbody model by a cavitating propeller are de-scribed. Pressure amplitudes have been measured which are up to

more than 20 times larger than in the non-cavitating case. The

effect of cavitation tunnel walls in connection with such

mea-surements is discussed on a theoretical basis. The experimental

pressure amplitudes are compared with the results from the

com-puter programs and the discrepancy is found to be within experi-mental error.

Extremely high, local pressure peaks due to propeller-hull vortex cavitation have been measured on a horizontal plate above the propeller in a cavitation tunnel.

LIST OF SYMBOLS.

C(t) integration constant in Bernoullit s equation,

D propeller diameter,

D2f distance between source point and field point,

D D D for elements of line sources m1+ m1

mi+, mi-, sf

m2 and in3, respectively, advance coefficient, constants,

non-dimensional thrust and torque,

K

,K

to qo

M

c/ci

P propeller pitch,

,

Rs integration limit applied in Appendix,

D D

m2, m3 J

C m3 C pc pn Z' tot,Z see (36), page 22,

experimental single amplitude of blade-frequency pressure from cavitating and non-cavitating pro-peller, respectively.,

non-dimensional, blade-frequency, pressure ampli-tude,

ñon-dimensional

propeller blade chord length at radius r, diameter of tip vortex cavity1

e cylindrical coordinates of source point,.

$

i. l or -1 (see Appendix),

T maximum thickness of propeller blade profile,

U undisturbed flow velocity at field point,

relative inflow Velocity to propeller blade section,

U0. at bladé tipi

UT total Velocity at field point.,

Va mean inflow velocity to propeller,

W wall effect coefficient,

Z nuxnbe± of pròpeller brades,

see Fig. 4,

àmplitude. of vertical acceieratidn,

bh height of test section,

b width of test section,

c. tip clearance,

c07

propeller blade chord at 0.7 radius,

cf non-dimensional amplitude of excitatiön force pr.

unit length on hull section,

CZ, C

_m2,Z'

C pa C r d C x, r,

e

h

h0

m1 m3 Po

Pi

imi

im2'

ir3'

_{itot j}

mi ,h'm2,h'

PIn3 ,h'tot,h]

Pv

Pd

r

base of natural logarithms (in Appendix only),

blade frequency (=n.Z),

integer indicating order of harmonic component,

distance from field point to propeller center

(see Appendix),

1:1,

integer indicating line nuirber in Appendix,

line source strength,

point source strength,

propeller shaft .revs.pr.sec.,

tunnel pressure,

blade-frequency pressure amplitude due to tunnel

vibrations,

instantaneous pressure,

instantaneous pressure due to cavity motion, cavity

volume variation, tip vortex cavity and their sum,

respectively,

amplitude of h'th order component of

_{ri '}

'imz '

and

respectively,

saturated vapour pressure,

amplitude of "direct pressure wave",

amplitude of reflected pressure wavê,

image row nuitiber in Appendix, elsewhere radial.

cOordinate

f sotircé point,

inner and outer radius of cavitatirigpart of

pro-peller blade,

blade tip radius,

chordwise coordinate,

chordwise extension of cavity,

width and height of idealized test section,

time,

s

C

t -t. i 12 V n , p, no e o mi-' rni+' °rn2 °rn3 K rake angle,

X _{ware length of pressure wave,}

i.i(r,ê,t) soùrce density,

y see Appendix, page 39,

p1 distance from image sinaularity to field' point,

characteristic attenuation distance, densit-y of water,

cavitation number,

t(r,0,t) cavity thickness distribution,

T maximum cavity thickness at one section,.

Tmax maximum cavity thickness,

velocity potential,

h'm2,h

m3 ,h' tot..h

veJocity poteñtial due to in, rn2 and m3, respectivel..y,

phase angles

°1,h' rn2,h'

rna,h and tot,h' respectively,

see Figs. 33, 34 and 35,

velocity induced by source layer (Fig. 4),

tangential position of propeller blade defined on

page 28, .

.

pitch angle,

-,

for definition see Fig. 3,

defined.by.(7), p.ge 11,

field point cylindrical coordinàtes, propeller open water efficiency,

angular coordinate of source point, measured from 'biade centerline,

The pressure fluctuat±ons induced on the hull by nor

cavitating .prope-llers.have been extsVely studied.. Methods. for

theoretical calculation of such pressure fluctuations, first

published in

[11*,

showgood, correlation with experimental

mea-surements in model scalè, e.g. [2,[3j,[9].Fig. 1 shows the res.i1ts

of an i.nvestigation carried out by the author in connectiot with a

220 000 TDW tanker. Pressure fluctuations at Section 1-1 of the

hull ( 10 percent of propeller diameter. ahead of the propeller

plane) were calculated by means of the computer programs

de-scribed in [3], applying a "solid boundary factor" of 2.0. By

means of pressure transducers fitted flush with thehull surface

the blade-frequency pressure. amplitudes were measured on a l/.Ø

scale model during elf-prop'4lion tests in the towing tank. Later

on corresponding measurements were also carried out on the full

scale ship. As can be seen from Fig. 1, the correlation between

theorétical, model scale, and full salé. pressure. amplitudes is.

very good. In this case both the pressure amplitudes or the hull

and the general vibrator.y level of the afterbody of the ship were unusually low.

In many other cases, however, and especially for ships suffering from severe vibrations, the pressure amplitudes measured

in full scale exceed those calculated by a large factor. Fiq. 2

shows such a comparison for a number of sh-ips where pressure ampli

tudes on the hull above the propeller have been mesured in full

scale. Ship number one LS the tanJer whose results are shown in

Fig. 1. Ship number three is the German research vessel "Meteor"Ê

whose results are. publishe6 in [4.]. The identities of the re-maining ships shown. i.n Fig. 2 cannot be pithlished, bcause the

Numbers in brackets refer th reference list on page 38.

,Z' tot, Z

see Fig. 36.

measürements. have been carried out as commercial jobs.

It is interesting to notice the correlation between in-tensity of afterbody vibrations and pressure amplitudes measured

in full scale. For ships number 2, 4, 5, 8, 9 and 12 in Fig. 2

the vertical, blade-frequency, single amplitude vibrations Of the afterbody exceed one öf the following limits,

acceleration > 0.3 m/s. if i

_{< b<}

]0 ôps, or

velocity > 0.005 rn/s if 10

< b < 150 cps

where is the blade frequency (blade ni er multiplied by shaft

revs.pr.sec.). För the remaininc ships the general vibration

level is below these limits. By general vibration leVel is here

meant the vertical vibration of the rigid afterbody, excluding

possible resonant vibrations in löcal structures.

Possible reasons for the poor correlation between theo-retical calculations and full-scale measurements shown for some of the ships in Fig. 2 may be

1.. pressure fluctuations due to excessive vibratory

motion of the hull plates in the vicinity of the pressure transducers, or

the occurrence Of cvitatión in full scale not

accounted for in the theoretical calculations.

The f i'rst item above has been treated in [5]; The secOnd item is treatedin the subsequent sections of the present report.

2. THEORETICAL INVEST1GÄTÏONS.

2.1. General considerations.

The free-space pressure field around a cavitating pro-peller, i.e. thè pressure f ied that would be present if there were

to originae from the following four effects:

1.,

blade thickress,

average or steady lift on blade sections,

fluctuating lift on blade sections,

motion and vOlume variation of cavities.

Assuming linearity the contributions from these four effects may

.

be superposed to give the total free-space pressure f teld.

.. .

The linearizéd treatment of the first three effects is

well kndwn, [l],[3],[6jand [9]. in [6]

a theOretical treatment of

item 4 abOve is also included, but without any numerical results.

In [7] ,

howeVer, sorne numerical results are presented, but still

without comparison with experiments.

Calculation of the contribution of cavitation to the

pressure field can be done by first determining a hydrodynamic

singularity distribution that satisfies the boundary conditions of

flow around the cavities.

Then the velocity potential due to this

singularity distribution is calculated at the field point (where

the pressure fluctuation is to be calculated).

Bernoulli's

equa-tion is then ápplied to obtian the pressure fluctuaequa-tion.

The main problem in this procedurè is the determination

of thé correct singularity distribution.

Cavitation affects the

pressure field to some extent by its influence on the average and

fiuctuating lift on thê blade sections. This effect is relatively

small, however, bedause è.ven in the case of non-cavitating

pro-pellers, the main contribution to the pressure fluctuation is

usually from, blade thickness and not from lift.

In principle, by

modifying, the input data for thrust arid thrüst fluctuations the

computer programs descrIbed- 'in

[3] can be applied to calculate the

pressure fluctuations due to the cavities' influence upon average

and fluctuating lift.

Therefore we shall here confine ourselves

to study the pressure fluctuation due to the cavities' motion and

volume variation.

This simplifies the determination of, the singu=

larity distribution because, by iieglecting 'lift, a sink-source

distribution alöne will be sufficient to satisfy the bOundary

conditions.

Applying a linearized expression fOr the pressure (see

pressure originating from blade loading, blade thickness and cavity

may be calculated independently.. _{The presence of the blade itself}

may thus be neglected when calculating the contribution from the cavity.

2.2. _{Theory based on source layer representation of cavities.}

As mentioned above., by neg1ectng lift _{a source}

dis-trbution will be sufficient to, represent the _{cav4ties. Denoting}

by _{the velocity potential at the field potht s?t up by this}

source distribution, the instantaneous pressure p at the field

point can be obtained from Bernoulli's equation

- + +

UT2

= C(t),

where

t time,

= density of water,

UT = total velocity at the field point.

The integration' constant C(t) may be a function of. time. At

in-finity, however, _{, p1 and UT will be constants unaffected by the}

propeller, and therefore C(t) must also be a constant. Since we

are only interested in.the fluctuating part of the pressure, we

may thus negiectC(t).

By linearization one obtains from B?rnoulli's equation (see for instancé Appéndix 1. of [3])

w + u

H

(1)

where U is the undisturbed flow velocity at the field point, and ±s the axial còordinate of the field point, Fig. 3.

The velocity potential due to a source layer of strength.

where

s Iïtiear chordwise coordinate,

U5

section inflòwvelocty (Fi. 4),

T(r,O,t) cavity thickness

(Fiai 4).

Physically the first term o.the right-hand side of (3) may be

interpreted as due to the instantaneous flo.ï around a cavity of temporally invariant thickness, while the second term is due to the actual temporal variation of thê cavity thickness.

Accörding to the rélátion betweèn flux and source strength we have V jn.. = - (û r(r",O,t) T (r, O

4rrs

s

Knowino the, ceometry. of the cavities as a function f time thè

source distribution can be determined numerically according to (4). From (2) and (1) the instantaneous pressure at the field point can

= s T (r, O ,t) as

T (r,O,t)

(3) (4)

1jT

d

i (2)

where F is the surface with source distrïbütion, and. Df. is the

distance between ource point and field point. Our main problem

here will therefore be to determine the correct source

distri-bution to represent the cavities. One posibie way to solve this

problem is to use a source layer of strength M(r,Ó,t) ovèr the

cavit.tingpart

of

the blade ánd eventúally outside the blade,

see Fig. 4) . Fig. 3 shows the definition of radial coordiìate r

and chòrdwise angular coOrdinate O of the source point. As

in-dicated in Fig. 4 the source strength p(r,O,t) induces a velocity

Vn át right angles to

the blade section. Inodr to satisfy the

then be calculated. A disadvantage of this method is that (3)

and (4) are valid only on condition that the cavity thickness T

is small compared to its chordwise and radial extensions, or

.4.

« 1.

3s

Especially at the "leading" and "trailing" edges of the cavity

numerical problems may be encountered due to an infinite . This

problem may be àvoided by applying line sources at these edqes as

explained for the leading edge of the blade profile in Appendix i of {3].

As a matter of fact a computer program has been written

at Th Norwegian Ship Model Experiment Tank, based on the procedure

outlined above. The c&iity thicktiess at a large number of points

over the cavïtating part of the blade and for a large number of tangential: positions of the propeller is usd as input data tó

cal-culate p(r,0,t) according to (4). Thus, in order to derive the

full advantage of the accúracy of the program the geometry of the

cavity must be very accurately known. In practice this will

usually not be the case. We shall therefore concentrate ön a

simpler representation of the cavity which will also facilitate the data input procedure.

2.3. Pressure due to cavity motion represented by source-sink

couple.

The simplified cavity representation is shown in Fig.

4. The effect of the motIon of the cavity (corresponding to the

first term on the right-hand .side of (4)) is calculated by divi-ding the cavity into a large number of. tangential strips of

radiai width r. The flow around each strip is assumed to be

pro-duced by a twodimensional line, source of strenqthm1 at a distance

am (measured along the arc) f röm the "leading edge" of the cavity

and a line source of strength -m1 (sink) at a distance amfro its

"trailing edge!'.

The valùésöf»á

and m1 are to be detérmined by

and s m1 ... m V ' S am +

Ç

{l

+

(c)2]

where ó = arctan 5c_2am ' , V V, and U =n[4rr2r2+p2]½ ' (8)

where.n is the propeller shaft revs.pr.sec. and P is the propeller

pitch., V V

These formulae are based on the definition oÎ

two-dimensional source strength given in [8], i.e. the volume of f.uid

emitted pr. unit time and pr. unit length is 2irm1. At the field

1. the distance s

between

the stagnation points

(measured, along the -arc.) shall be

equal.to

the

V chordwisé extension of the real cavity .t, the

same radiuS, and . V

.2. the maximum .thickness _T of' the, simplified

V V ' imax

,

V ,

cavity shall be equal to the maximum thickess of

the real cavity at the s'me radius.

If the radial extension of thé cavity is noti large compared to its thickness, this strip theory will still give incorrect re

suits due to "end effects". If this is the case, the pressure

fluctuation due t the cavity can probably be better predicted by

applyïng the calculatIon technique descrthed in Section 2.5 of this report.

V V

Applying the ieli-known

techniqVue

of first setting up

the expressioñ for the complex potential due to a two-dimensional source-sink couple in homogeneous flow, then calculating the posi-tion of the stagnaposi-tion points, and finally deriving the formula

for the dividing Stream line from the stream function (see for in-stance [81, p. 202), we obtain in our case

U T

'S imax m1

2(ir- 2e,)

point (on the hull abovetheöé1le±) the

eiocity potential

due to the positive liri

s6urcee1thnent in Ar is the same

mi'+

as if it were replaced by a three-dimensional point söuròe.

Since the volume off luid emitted pr. unit

time from a

three-dimensional point source of unit strenath is 4

(definition öf

point söurce strength) the strehgth of this quivalent point source

becomes rn1r/2, giving

¡

2 D

_{n, i+}

where D

_mi+

is the distance between the field point and the source

element.

Addition of the potential due to the negative line

source element and integration in radial direction yield the

in-stantaneous potential due to thernotion öf the cavity,

.2

m m 1;

dr

D. D

'

mi+

mi-

(9)

D

- D+

in1

dr

(10)

D+

Dm

where r1 and r2 are the inner and outer radii of the cavitating

part of the propeller blade.

For the numerical calculations it

is convenient to express D

and D

by. D

which is the

dis-mi+

mi-tance between "cavity mid-chord" and field point.

The distance

between the field point (,p,c) and a source point (x,r,e) on the

blade Is generalLy (see Fig. 3)

Dsf = {(-x)

+ r

+ p

- 2rp

cos(c-e)}2

= {(

-

r2+ p2 - 2rpcos(E-'-O)}

(il)

Thus (10) becomes

y = tangential position of blade,

K rake angle,

O = anqular coordinate of source point measured f ron centerline of blade (see Fig. 3).

Denoting by the value of O at "cavity mid-chord." we obtain

O

D= {(

Kr)2 + r2 + p2 -

2rpcos(E-y-Introducing the relation

a y a at at a.y a

= 2Trn

-a y,, e óbtain from (13) r2 a

r

TIfl m1 (sc'_2arn) at ¡:;; j [4.j.2r2+p2]½ D m2 r D

-D

' (O

-O

mi

mi+

mi

mi+ (sc2am)27tr r Dm2 [4.rr2r2+p2]½

Kr)(-

.) -

rs'in(c__O2)}.

aD m2 ae ma )}½ rn2 (12) (cont.) (14)

For small values of Dm - D. we have

-

-

Kr)(

.!.)

_{- rpsin(c-y-02)}dr}

_(1,3)

7 _{'mi(sc_2arn)}

mi

J

D3[4'Tr +p21½

r1

D

Lm2

' 2n

o

- m2 _{Kr)P + 27rrp.sin(c_y_Om)} dr. (15)}

Derivation of. (13) with respect to yields

mi, Tr

m1(s-2a)

e.

3(-p

- P

Kr)(-

)

- rp.sin(c-ye)}]dr

(16)

Byusing (.5), (6'), (12), (15) and (16,) wemay now calculate the

instantaneous pressure

jm due to the motion of the cavity from

p = mi mi

mii 'w ' t ''.

The way this is done numerically is explained in Section, 2.6.

2.4. Pressure due to volume variation of cavity.

The effect of the Volume variation of the cävity

(corre-sponding to the seond term on the right-hand side of (4)) is

represented by a' line source of strength m2 positioned at

"mid-chord" of the cayity (Fig. 4). ¿2he nstant'aneöus Value of m2 is

determinedin such away that the volume of fluid emitted pr. unit time from m2r His equal to the rate of volume increase of the

strip i.e.

2' =

where s is the sectional. area of the strip. Thé velOcity

poten-tial set up' at the field point by m2'r becomes.

,m r 2 m2 - 2 fl m2 (17) [471.2r2+p21½. _D -1fl2

m2 i 4TrD mz i 4 'rrD m2 a (S.tr) DS

(Lr

+ $

Keeping the number of strips constant so far, we may write

.3 (tr)

Am2

_{- 4rrD} 1fl2 D DS = 4iTD

(Êr

S.

(r2-r1)

tir). (18)

Integration of (18) in radial directiOn yìélds the instantanéous

potential due to the volume variation of the cavity,

m2

m2

Usinq the relation (14.) we obtain

r2 r1 r n D(r2-r1) 2 2(r2-r1) -1 DB D Dt m2 dr. dr, (19) r2 D: = 71fl2 . ₎₊ (cant.) Dt.

r)

m2

-+7m2

1 ayL(r2r1) +

Ud 2

oc

16 111. a(r2-r1) 2(r2-i1) _a-r 'a s ay o ,. . m2 2r D.. .m 2 ay ) ar r1 Taking

12 .rom (12) we obtain by derivation öf (19)

D m2 D rn2 s o

S(-P

- -Kr) dr

Since the form of the approximated cavity section closely resembles that of an ellipse, we may put

- Ir

S = T S

4 imax C

By using (.12), (20), (21).and (22) we may no.w:calculate the

in-stantaneous pressure p due to the vOlume varkation of the

1m2

cavity from .

H

n

=.n

_m2

+

,_Ifl

im2 t

2.5. _{Pressure due to tip vortex cavity.}

A three-dimensional point source in a unifOrm stream

will give rise to a streamline pattern of the. type shown in Fig. 5.

Far downstream the dividing streamlines will form a cylinder. The

relation between cylinder diameter dc? point source strength m3

and free stream, velocity U0 is (see for instance _LB])

(20)

(21)

where

m3 16 D

m3

The point source may thus be. ued to represent the flow azóuida

body of this form. If the point source in Fig. 5 were rotatig

in a circle ma plane at right anglesto the inflow direction, the

dividing streamlines will describe a helicoidal body of a form corresponding tö the form of the tip vortex cavity fröm a

pro-peller blade. We may thus represent the flow due to the tip

vortex cavity by a point sourde at th blade tip. The velpdity

of the blade tip hs to be. substituted for U0 in (24) so that the strength of this point source becomes

n d2

(47r2r2+P2)½ ₍₂₅₎

where d is now the diameter of the tip vortex cavity, and r

the propeller radius.

It should be noted that thé point source adcording to (25) only describes the flow due to the cavity of the tip vortex and not to the. ti vortex i,n general. (This general vortex flow is included in the calculation of pressure fluctuations due to lift described in [3].). It should also be noted that the single

point source representatiön (2S) i ased on the assumption that

thé cavitatïng tip vortex is extending tö infinity downstream with constant diameter.. The finite extensión of the caVity may be

taken into account. by a snk at its downstream end. Even général

variations of ôavity diameter may be taken into account by a

sourcesink line distribution along the. (helicoidal) centérline of

the, cavity.- Since we are mainly intérestéd' in th& pressure f

luc-tuations on the hull directly above the propeller, hall here

negledt thé sourcé-sink distrIbutions further downstream, and we shall .aléò assume dc to be constant during the revolution.

The velocity potential ¿t the fiéld pOint due to the

point source in3 at the blade tip is

(42r2+P2)

m3'27r

Krt)2+rt2+p2_2rtp.cos(C_y_O)}½ (27)

and

O-value at position of point source.

Derivátion of (26) yields (by using (14) and (27))

m3 at n 8.D 3 m3

16D

3 1fl3

Using (28) and (2:9) we may now calculate the instantaneous pres-sure p. due to the tïp vortex from

1m3-- r.. .. -.- . , (28) (29)

d2(42rt2+P2)½.

(_p

Krj

- im3 . 'w' at + U p n 1fl3 . (30)

2.6. Description of computer, programs

As mentioned previously a computer program has been written based on the. source layer representation of the cavities described in SectIon: 2.2.. The numerical results given in.. the sub-sequent Sections 2.7 and 3.4, however, have been obtained by a computer program based on the cavity. representatioh described in

Section 2.3,2.4 and 2.5. This' proram requires the following

in-put data:- .

.

a. propéller .eometry, : field point coordinates, field point velocity, and shaft revs./sec.',

'b. inner and outer radius of cavity at altogether 48 equidistant tanqent-ial positions of the blade (7.5 degrèes between each)',

C. chordwise extension s and maximum thickness r

C imax

at 7radial stations of the cavity with the, blade at the' same 48. tanèntial positiäns,

For each radial station of the cavity and for each tan-gential position of the blade, the quantities m1 and am are first

calculated from (5), (6), (7) and (8) by an iterative process.

Dm2 is calculated from (12). The integrals in (15) and (16) are

evaluated by 6 intervals Simpson integration over the radial

ex-tension of the cavity. Differentiation with respect to y in (15)

is done numerically by stepwise approximations with second-order

polynomials. The instantaneous pressure

irni due to the motion of

the cavity on one blade is calculated from (17) for each of the 48

tangential positions of the blade. Harmonic analysis of this

signal is carried out numerically. The instantaneous pressure due

to the motion of the cavity on one blade may then be expressed by

(neglecting constant terms),

00

p. (a) = p .sin(ha+4, ,)

imi

h=l mi,h where the amplitudes

h and the phase angles h are

deter-mi, mi,

mined by the harmonic analysis.

Numerical calculation of the instantaneous pressure p. 1m2 due to volume variation of the cavities is carried out according to (12), (20), (21), (22) and (23), where the integrals are

evalu-ated by 6 intervals Simpson integration in radial direction. All

necessary differentiations with respect to y are carried out by

stepwise approximations with second-order polynomials. Harmonic

analysis of (a) is then performed to determine the amplitudes

h and phase angles h in the formula

m2, m2,

im2

=

pm2hsin(ha+4)m2h).

(32)

The instantaneous pressure signal due to the tip vortex

cavity is calculated in a straightforward manner from (27), (28),

(29) and (30). Amplitudes and phase angles in the formula.

00

p. (a) =

Pm3 h14)

i.,)

m3 ,.

h=l

are determined by harmonic analysis.

Finally the computer program calculates the sum

p. (a) = p. (a) + p. (a) + p. (a).

itot imi 11P2 1m3

Amplitüdes and phàse ancries in the formula

Pt0t(a) =

hl

are determined by harmonic analysis.

The present program is intendedtö be capable of

calcu-lating up to the sixth harmonic component;o the prèssure

fluctu-ation (cörresponding to the blade frequency component of a

6-bladed propeller). If higher order coiponentS are to be

calcu-lated, the program input data will have to1nclude cavity geometry. at more than the 48 tangential positions mentioned above.

It should be notiöed that tJie pressure signals that we have treated so far, represent the pressure field due to

cavita-tiOn on one blade. It can be shown (see fOr instance [i] for the

analougous non-cavitating case) that in the total pressure induced on the hull by a propeller of Z blades only those components of the

"one-blade signal" which are of order equal t an integer r!u1tiple

of Z will contribute. And their contribution is Z times the

one-blade amplitudes.

Furthermore, the pressure sianals that we have treated so far, represent the "frèe-spacè pressure field", i.e. the pressure

that oul be present if there were no solid boundaries in the

vicinity of the propeller. When the pressure signal on a solid

boundary, e.g. the hull ove the propeller, is wanted, the

free-space .pressure must be multiplied by a "solid boundary factor" Sb

to account for the image efféct of the hull. For a plane surface

of infinite extent in both directions the theoretical value of _Sb

is Sb=2.O. The experimental and theoretical investigations de-scribed in [3]. indicate valués of the solid boundary factor (as defined in [3]) ranging ftom about 1.8 to about 3 on the hull directly above, the propeller.

2.7. Numerical .resuJ.t.s.

In orde± to show the relative magnitudes of the various components of the pressure fluctuations, and for various cavitation patterns, we shall here show some numerical results for the

pat-tern is defined independent of propeller, blade number,. so that it may be applied to propellers of any number of blades.

Cavitation attern :No.- 101.

This pattern 'is,arbitrarily chosen' to consist of a cavity

extending over part of a rope1ler blade of the same oût'line,

dIameter and pitòh as the blades of P-48 shown in Fig. 18. The

cavIty is shown on an axial projection of thé blade in Fig,. 'G. It

extends from r1=O. 3 D to r2O.5 b in radial direction. (D is

pro-peller diameter). Its chordwise extension Sc and its maximum

thicknéss r at various radial sections are aiven in

Fia.

6. In

imax .

addition to the cavity on the bláde shown in

Fig.

6 there is

as-sumed to be a tip vrtex cavity of diameter 'arbitrar±ly chosen

d=oo12 D.

Thé main .fèature of cavitation pattern Nö. 101 is

that d and the thickness and extension o the cavity in both

'±a-dial, and chórdwise diréctïöns are cönstant thiring the téVolution of

the propeller . '' .

Cavitatïoñ 2attern No. 10.2.

In this case the thickness and extension of thé cavity

ontie blade are exactly equal to those described for càvitatïon

pattern No. 101 when a=0 (blade pointing vertically iipwa±ds).

However, the cavity is no no longer invariant with respéct to

time, but changing during, 'the revolution of the propeller in such way that' the third order cotnonent of its völume variation

be-comes the largest possible. This is done in the following way:

When increases frotn zerö to60 degrees va'lües of r -. ,

t

2 1 _irnax

and Sc shöwn in

Figo

6 are multiplied by a factor

(½ ('i+cos(3.))

'Then from =6o degrees. 'to OE=300 degrees. there is no càvitatio....on the blade at all, 'and then from =3'OÒ 'degrees to» =3.60' degrees 'the

th-icknes and extensions increàse again accOrding to théäböve

Cavi.tation pattern. No. 103.

In this case the variation of the cavity is arbitrarily

chosen so as to òbtain a large sixth order omponent in. the volume

variation of the cavity. This is done by applying cavitation

pattern No.. .102 -with the. modification that cos (3) is now substi-tuted by co.s(.6a). in .the;expressions for the cavity's variation with .. The .non-cavitating part of the. .revolution is now from

a=30 degrees.to =33.0 degrees..

Results.

Figs. 7 through 15. show numerical results obtained when applying Cavitation patterns No. 101,. 102.. and 103 as input data

for the program described in section 2.6. Fias. 7, 8 and 9 also

show the cavitation patterns in question on an axial projection of

the propeller blade. T1e calc.ulat-ions. have been carid out for

nl9 revs, pr. sec. and U=1.53 rn/s (Í!lodel scale).

The results are presented in the form, of non-dimensional single

aplitudes cmi c

ZICmZ and c0i and the phase angles

mi,Z' m2,Z' m3,Z a'nd tot,Z' By definition

= Z Sb

_{pmi.z/(pwrl2D2),} C m2,Z' . = z ' Sb

m,3,zwflD)

CtÒ:tZ . Sb .

where the free space pressure amplitudes p , p

'

and

mi, m2, m3,

defined by (31), (32), (33) and (35) have leen multiplied

by Sb to accountfor the image effect of the hull, and by Z to

obtain the. signal from. a propeller with Z.blades. Division by

pn2D2, is.introduced to make the amplitudes non-dimensional.

Thus Cmi

Z m2izi.crn3:,z.:and c0represent the theoretically

calculated non-dimensional single amplitude of the blade-frequency component of the pressure fluctuation induced on the hull by the

caviti

on a Z=bladéd propeller.

They represent the cöntrIbutions

from cavity motion, cavity volume variation, tïp vor tex cavity,

id

the vectorial sum of these three contributions, respectively.

All

the results shozn in Figs. 7 through 15 have been obtaIned by

assuring a solid boundary

ctQr Sb=2.O.

Fig. 7 shows that for Cavitation pattern No. 101 the tip

vortex gives only avery small contributjon to the resulting

pres-sure field.

Since the cavitiesofthis pattern are constant during

the revolution there is no contribution from cavity volunte

vari-ation.

Fig. 7 also showsthat in thiscase of constant cavities

the resulting pressure signal on the hull is relatively independent

of the blade number of the propellerf.

Fig. 8 shows that for: Cavitation pattern No. 102 the

contributïon from. cavity volume väriation is the predominant one in

the case of 3-, 4-, and. 5-bladed propellers.

Only for a 6-bladed

propeller will the contribution from cavity motion be the larcTer.

For Cavitation pattern No! 102 the contribution from the tip VOrtex

cavity is .too small to be shown in Fig. 8., beeing of the order

c

=310.

.

m3,Z

Fig. 9 shöws that in. the case of CavitatiOn patterti. No.

103 thè contribution from cavity volume variation is considerably

higher thañ the contribution from motion for any blade number.

It is alio

interesting to notice the pronounced increase in

pres-sure atpiitude with increasing blade nuniber.

Direct comparison between Figs. 7, 8 and 9 shows that

the constant cavities (Cavitation pattern No. 101,) give much lower

pressure amplitúdes on the hull than the other twO patterns.

(Notice that the scale o.f the Ordinate axis in Fig. 7 is different

from Figs:.

8 and 9).

Figs. 10, 11 and 12 show the influence of the radial

distance of the feld point upoi the amplitude and phase ánglé of

the theoretical blade-frequency component for the vziou

caVi-tation patterns.

An important conclusion to be drawn from these

diagrams. is that the. pressure contribution from cavity motion

de-creases far more rapidly with increasing distance from the'

pro-peller than döes' the contribution from cavity volume variation.

Taking c

_m2,6

(volume variátion) and c

_flti,6

(motion) from the

ampli-tude diagram in Fig

12 we find for instance that for p/D=0 6 the

ratio betweentherti is about 2.6.,. Increasing p/fl to 1.1 this.ratio

becomès i6. '...

.

::

. . .

Figs. i3and 14 showexamples òf how the pressure

air.pli-tudes and.,phase angles are influenced by the field point

coordi-nates

and c.

The phase angle diagramofFig. 14 shows a very

important difference between the various

ótributions to the total

pressure field.

The phase 'angle

of the component due to

volume variation is. independent of c.,

and, according to Figs. 12

and 13, alsoindependent.of.p and

.

This means that.the pressure

signal due to cavity vo1uthe variation occurs with equal phase all

over the afterbody of the ship (neglecting finite propagation

velocity of the pressure wavè).

The. phase of the pressure

compo-rient due to cavity motion ;is, höwever, as shown in Figs. 10

through.14,; especially 14, varying with varying field point

coor-dinates..

For cavitation pattern No.101 the influence of. c upon

phase angle would be even more pronounced, than shöwn for Cavitation

pattern No. 103 in Fig. 1.4.

.

The total excitation force amplitude on the hul.i is

ob-tained

by integrating the pressure amplitude over the. hull,

täking the phase angle at teach position into proper account..

The

phase relations mentioned above thus. makes the excitation ft'rce

amplitude due. tò völume.variation considerably larger than the

force amplitude, due to cavity motion, even if. the corresponding

pressure amplitudes were equal.

To illustrate this phenomenon the pressure signals for

Cavitation patterns Nös. 101, 102 and 103 have been calculated

and integrated over a.hull section òf the arbitrarily chosen form

shown in Fig..

15..

it is of circular form, its radius of curvature

being twice the propeller diametér.

Its axial position is in the

plane of the propeller.. The coordinates of fièld point A are

=O, p=0.8 D,.

i.,e.,identical' to the field point for which

Figs. 7, 8 and. 9' have been obtained.

The diagram of Fig. 15 shows

the results of: the integration.

Cf is the singleaxnplitude of the

blade-frequency component: of, the vertical excitatthn force pr.

unit length (axial direction) òn the séction.

This force pr... 'unit

length has been made non-dimensional by divisïon' by pn2D3.

Re-'turning to Figé 9 we see thtforCavitation.pattern No. 103,

6-bladed propeller, the .pressure amplitude due to volume variation

motion. Turning from the pressure amplitude at field point A to the vertical excitation forcé amplitudes on the section, we see

from Fig. 15 for the sam cavitation pattern and blade number that

the force amplitude due to volume variation is, not about 7, but

about 22 times as high as the force amplitude due to cavity motion. Thé effect of the phase relations becomes still more evident when comparing Cavitation patterns Nos. 101 and.l03.' From Figs. 7 arid

9 we see that for a 6-bladed propeller the total pressure

tp]iude

dué to Cavitation pattern No. 103 is about 12 times as high as that

due to Cavitation pattern No. 101. From Fig. 15 we see, however,

that for a 6-bladéd propeller the resulting force amplitude on the

section due to Cavitatòn pattern No. 103 is, not about 12, but

about 135 times as high as that of Cavitation pattern No. 101. If

integration of the pressuré signals had been carried out nçt only

over the section but also in longitudinal direction to obtan the

total vertical excitation force amplitude on the afterbody, the

effect of the phase relations would have been stili more pronounced. The numerical éxamples presented above are based on

arbitrarily chOsen and quite idealized cavitation patterns and hull

section form. Nevertheless, some practical conclusions may be

drawn:

-The. pressure contribution dué to the tip vortex cavity is relatively, small.

Comparison with the measurements deScribed in Sec-tion 3 of this report shows that the blade-frequençy pressure amplitude due -to Cavitation pattern No. 101

(cavities constant during the revolution) is of,

abòut the same maanitude as the amplitude of the

non-cavitating propeller. The phase angles of the signal

due to Cavitation pattern No. 101 is also about the

same as that of thé non-cavitating propeller. This

means that if the propeller is cavitating as shown by Cavitation pattern No. 101, the resulting

pres-sure amplitude on th hull will be about twice.as

high as if the propeller were not cavitating.

con-ciuded that, from the point of view of excitatión foröes on the hull, cavitation which occurs only during.part of the propeller revolution is far more

dangerous than constant cavitation. Comparison

be-'tween Cavitation patterns No»101 arid 103 for a

6-bladed propeller is illustrating this phenomenon. Their pressure amplitudes differ by a fäctor of about 12 at a certain field point, while the force

pr. unit length amplitudes differ by a factor of

about 135. .

3. EXPERIMENTS WITH AFERBODY MODEL IN CAVITATION. TUNNEL.

3.1. Instrumentation.

At The Norwegian Ship. Model Experiment Tank, equipment has been, developed for measuring the pressure fluctuations in

duced by cavitating propellers ön the afterbody model in. the

cavi-tation tunnel. This equipment has now been in use for some years.

The pressure transducers are described in [3]. These transducers

are fitted in the afterbody model with their diaphragms flush with

the surface of the models. The interior of each transducer is

connected to a vacuum tank outside the tunnel by. an air-tight tube

through the tunnel wall. In this way the low s.tatic pressure in

the tunnel test section is canceled by an equally low static

pres-sure inside the transducer diaphragms. This arrangement makes it

possible tp use very sensitive diaphragms so that pressure &

pli-tdes down tO a few tenths of a millimeter water column can be

measured. The résonant frequency of the diaphragms is about 400 cps.

in water. FIg. 17 shows schematically the instrumentation set-up.

ISAC. is a special instrument for automatic' spectral analysis. The

pressure signals are aldo recorded by an ultra-violet recorder.

It should b noted that the instrumentation includes an

accelerometer fitted ii the model tO measure the vertical

vibra-tions of the model. 'By analysis of thé accelerometer signal the

can to some extent be estimated.

With this instrumentation investigations of pressure

f luctuations induced by various types of propeller cavitation have been carried out.

3.2. Measurements on afterbody modei!

In this Section the results from a series of systematIc experiments carried out by the author in the smaller cavitation

tunnel' of The Swedi$h State Shipbuilding. Experiment Tank are de-scribed. These results were first published in [il]. The tunnel

has quadratic test section of 70 x 7Ó cm2 'area. Fig. 16 shows

its outline.' The instrumeitation described above was used.

The measurements were carried out with propeller P-548.

This propeller is shown in Fig. 18. ts open water performance

curves are shown in Fig. 19. The afterbody model is shown in Fig.

'20, which also showS the poSitions of the four pressure

trans-ducers Ti, T2, T3 and T4. Ti is positioned 'in the centerpl'ane,

directly above the propeller blade tips1 T2, T3 and T4 are all at

a section 0.15 b ahead Of Ti (D=propeller diameter). 'T3 is in the

centerpiane while T2 and T4 are positioned 0.3 D from the

center-planS, on the starboard and port side respectively. The

aOceler-ometer mentioned in Section 3.1 is fitted inside the model directly

above the propeller.

At warline W

24 (Fig. 20) a horizontal

plate extending far outside the model is fitted. A sectional view

of the test section is shown in Fig. 16.

Tests were carried out with the propeller shaft at four different vertical positions, corresponding'tö tip clearance c

equal to 0.07 D, 022 D, 037 D and 0.62 b (c measure4 vertically

froth the blade tip as shown inFig. 20). For éach tip clearance

tests were 'carred out for the following four cornbinatiòns öf

ad-vance coefficient J and cavitation number

e0:

1) J=0.42 o=0. 388,

2) J=0.59 c0=0.538,

3) 1J=0.84 a =0 . 76 5

.po-pv where

(V2+(O.77rflD) 2)

= mean inflow velocity to propeller (based on thrust identity),

static tu.rnel pres.sure.in undisturbed flow,

= pressure of saturated vapour, at the temperäture of ie tunnel water.

Ail measurements on the afterbody model were carried out

with n=i9 revs pr. sec.

Simultaneously with the recording of pressure fluctua-. .tions by oscillograph and tape recorder, the cavita'don pattern was

sketched and piotographed for various tangential positions of the

propeller. The thikness of the cavities was estimated by direct

visual observation.

In additon to the values given above the

propeller-indticed pressure fluctuations were also recorded for non-cavitating propeller at the. same J values and the same tip clearances.

During, all the tests described, here thern axial wake

dis-tribution was as shown in Fig. 21 (measured by pitot tube). This

distribution was ob..taed by fitting wire mesh to the surface of

the afterbody model. ' . ..

, Results of the tests are presented in Figs. 22 through

29., rigs. 2:2 through 25 show the observed cavitation patterns on

the' suctior side ç,f, the propeller blades at the various. o-J

combinations and tip clearances.. Pressure sIde cavitation did not

occur. The tangential position angle a is defined in such a way that a=O. when one propeller blade is pointing vertically upwards. Increasing a is in the direçtion of propeller rotation

(right-handed for propeller P-548). In addition to the regular variation

varying in a random way. The dotted and solid lines of the

cavi-tat-ion patterns in FigS. 22 through 25 indicate the limits of

this random variation.

c07 is the propeller blade chord length at the 0.7

radius. In Figs. 22 through 2.5 the observed maximum cavity

thick-ness

t

is given in percent of c . It should be noticed that

max 0.7

it is based on direct visual ôbservation and. thus not very

accu-rate. The standard deviation of the observed values of

t-belived to be of the order ±40 percent of_- T..

. max

Figs. 26 and 27 show the non-dimensional experimental

single amplitude c of t-he blade frequency component of the

pres-sure sianal. These amplitudes have been recorded with the

pro-peller cavitating as shown in Figs. 22 through 25. It should be

noticed that the experimental amplitudes presented in this section represent the sum of the contributions from both propeller blades

and cavities. (In Section. 2 of this report only the contrIbutions

from the cavities are considered). In connection with Figs. 26 and

27 it should also be noticed that different a values have been

used at the variOus J values. The vertical line beside each point

plotted, in the diagrams indicate a rough estimate of the standard devïation (defined in such a way that there is about 66 -percent probability that the correct value lies between the indicated limits).

From Figs. 26 and 27 it can be seen how the pressure

amplitude decreases with increasing tip clearahce. It shoild be

noticed, however, that t-his decrease is also partly due to

de-creasing cavitatiOn. For all tip clearances the pressure amplitude

reaches a maximum at a J-value of about 0.6. The most probable

explanation of this phenomenon is that the volume variation of the cavity on each blade contains an extra high fouth order component at this J value, thus leading to a high blade-frequency component in the- pressure signal. it is also interesting tO notice that transducer T2 (on -the starboard side) gives considerably higher amplitudes than .T4 (on the pdrt side). This may be'explained as a consequènce of the cavities' thickness and extension being max-imuln when the blades have just passed through the high wake

region and are pointing to the starboard side. (Compare with the

ainpltitude diagram in Fig. 14 which shows a definite maximum at

the cavity volume Is maximüm).

Figs. 28 and 29 show an "amplification factor" N de-fined by

M=c

_¡C.

PC Pn

where is the non-dimensional single amplitude of the

blade-f requency component measured blade-for non-cavitating propeller.

Since cr is relatively independent of J, the curves in Figs. 28

and 29 have the same general appearance as in Figs. 26 nd 27.

Comparisonof' the four diagrams in Figs. 28 and 29 shows a clear

tendency of increasing M. with increasing tip clearance. FOr c/D=.

0.62 the.pressure. amplitude for cavitating propeller exceeds the non-cavitating case by a factor of up to about 35.

From the oscillograph records the difference in phase angle of the signals from the various transducers were obtained

directly. It is interesting to notice that for e.g. c/D=O.22 and

J=0.59 the difference in phase angle in the signals from T2: and T4

is in the non-cavitating case 160 degrees. (360 degrees

corre-sponds to one period of the blade-frequency component).. With

cavitating propeller, Cavitation pattern No. 6 in Fig. 23, this

difference in phase angle is reduced from 160 to 40 degrees. As

explained in Section 2.7 this. means that the ratio between the vertical. excitatïon force amplitudes with and without cavitation is considerably higher than the ratio M shown in Fig.' 28.

The spectral analysis carried out by means of ISAC shows that the amplitude of the second order component (frequency equal to twice the blade frequency) is considerabl.y smaller than the

blade-frequency component. The ratio between the two was found

to be of the order of 1,/.1.0 for non-cavitating propeller and up to

about 1/4 for cértain cavitating conditions. The expérimental

in-accuracy in this determination of the second order component is

relatively higher than 'for the biade-frequency component. No

3.3. Influenôe of test section vibratiöns and wall effects.

If thé afterbody model is vibrating the pressure

transduáers' will sènsè a pressure signal due to the vibration in

addition t the signal. induced by the propeller. Experimental

determinatioxT of the latter may become inaccúrate or even

im-possible when, the vibratory amplitude reaches a certain level. In

[5] methods for numerical évaivatioh of this effect have been

de-.veloped for ships vibrating in open water. In the present case of

an afterbody model in thé cavitation tunneÏ, however, these 'methods

are inappl±cáble due to the wall effects. in .the 'test section. We shall therefore have to develop a different procedure in this'case.

The afterbody model is very' rigidly connedted to the tunnel.test sect±ôn, which is in itself a very rigid structure. It is therefore reasonable to asSume that the accelerometer mea-sures the vertical acceleration of the whole tést section,

in-clúding the'afterbody model, asa rigid unit.

Let us cons'ider one unit (metr) length of the test

section; d let us assume that the water contained in- this volume osçillateS,.rertically like. a solid body.' Applying the relation, force=inass multiplied by acceleration, fOr thi's body we obtain

= p . b

.b

oa

r w w ,h y

where

a = single amplitude of' verticalacceleration,

= single amplitude of pressur.e difference between lowér and upper hörizontal boundary of test section due to vibration,

= width of test section,

bh = heïght f test section. .

Due 'to: symmetry the single amplitude o-f the distürbance pressure at

the upper horizontal boundary of the test section is ½ _a' or,

made 'non-dimensional -by division by pn2D2, b 'a

_hv

During. the pressure measurements the blade-frequency component of 'thê 'adc'elerometér iaÏ' was meásured1 and it was found tobe less than 0.02 rn/s2 during the measurements with

at-mospheric pressure on the tunnel (non-cavitating propeller). With

the most severe cavitation on the propeller it reached valués up to O i rn/s2 Application of the formula or Cpa given above re-sults in Opa values of never more than 5' percent of the measured pressure amplitude for any tip clearance or cavitation condition.

This way of estimating the effect of 'tunnel vibrations is

very rough, indeed. The formulaj reglects possible' horizontal

vibrations, other modes Of vertïcal vibrations, and possible pres-sure waves in the water generated in other parts of the tunnel. Furthermore the formula is, accörding to its derivation, valid only close to the upper and lower horizontal boundaries of the

test section. Nevertheless, the conclusion must be that it seems

unlikely that afterbody vibrations hae influenced our pressure

measurements to a significant extent.

The experimental results presented in Figs. 22 through 29' show the effect of various cavitation patterns upon the

pres-sure amplitude on the hull. It should be noticéd, however, that

before these results can be applied to full scale ships, one must first consider possible scale effects due to reflections of the

pressure waves from the walls of the tuniel test section. In the

Appendix öf this' report a method for theoretical estimation òf this effect has been developed.

Conclusions tobe drawn from the calculations in the Appendix are:

i. In' the case Qf non-cavitating propeller, and also for

a cavitation pattern which is constant during the revolutthn, the magnitude of the pressure wave re-flected from the walls is expected to be of minor importance.

2. Thé pressure wave due to cavity volume variation is

reflected from the walls in such a way .that the magnitude of the reflected wave may posSibly become of the same magnitude or even larger than the pres-sure wave induced directly by the propeller at the prèssure transducers.

Instrumentation is at present being developed at The

Norwegian Ship Model Experiment Tafik fo experimental calibration

of the cavitation tunnels with respect to wall effectupon the

measured pressuie fluctuatiöns.

3.4. Comparison between theory and experiments.

By the theoretic calculation methods developed in

Section 2 of this rèport It is possible to calculate the pressure

fluctuations induced by cavities of known geometry. These methods

can to some extent be verified by the experimental results b!

Section 3.2. Besides of possible wall effects the main problem in

connection with this verification is the relatively poor accuracy obtained in the visual determination of cavity thickness.

Results of the comparison between theory and experithent

are shown in Figs. 30 and 31. The experimental curves of Fig. 30

are the same as those presented in the lowerdiagamof Fig. 26.

The theoretical urvês have been obtained by vectorial suinxntion

of the pressure amplitudes measured for non-cavitating propeller, and the pressure amplitudes due to the respective cavitation

patterns in Fig. 23, calculated by the computer prögrain described

in Section 26. In these calculations the radial distribution of

T between r and r was assumed to be the same as the

distri-imax 1 2

buton shown in Fig. 6, the instantaneous r1, r2 and Tmax being

taiçen from Fig. 23. Fi 31 shows a corresponding comparison., the results being plotted on a basis of OID and for a constant advance coefficient J=0.59.

The difference between theoretical and experimental

results Siownin Figs. 30 and 31 is quite considerable. However,

it. must be remèirthered that the standard dion of the observed

cavity thickness, which is the basis of the theoretical results,

is of the order of ±40 percent. Furthermore, taking into account

the possibility of wall effects, it must be concluded that the experimental pressure amplitudes are not in direct contradiction

to the theoretical calculations.

As mentioned before, 'a procedure for experimental

verification of the theoretical calculation methods, a technique for accurate experimental determination of cavity geometry,

espe-cially. thickness, iiustbe developed too. Such.development is also

under way at The Norwegian Ship Model Experiment Tank. At the

time of writing of this report, a technique based on .photogrannetry has just shown very promising results.,

4. PRESSURE FLUCTUATIONS DUE TO PROPELLER-HULL VORTEX CAVITATION.

In [io] aphenomenon denoted.by "propeller-hull vortex

cavitation", abbreviated PHV cavitation., is described. It is the

author's belief that this phenomenon may give r.ise to extremely

high, local pressure fluctuations on the hull. The experimental

results presented on the subsequent pages are tobe. considered in close connection.with the investigations described in [10].

When the investigations desdribed in [io] were carried

out, the; only pressure .tranducers available to the author were of

the type described in [3g. The relatively large diaphragm diameter

of these transducers made them unsuitable for measuring the local

pressures of PHV cavities in model scale. Since that time, a new

type of miniature pressure transducers, Kulite Semiconductor

Pro-ducts,. type MQL-125-5, have been purchased. These.transducers have

a diaphragm diameter of only 2 mm. With these transducérs further

experimental investigations have now been carried out in the Num-ber one cavitation tunnel at The Norwegian Ship Model Experiment Tank. This tunrel has an open jet test section of 360 mm diameter.

Fig. 32 shows the arrangement in the test section. A flat

rec-tangular plate 295 min wide and 250 n'in long (in flow direction) was fitted above the propeller. in such a way that the tip clearance c

could be readily adjusted. A vertical strut with streamline

section.was positioned.ahead of the propeller. By.fitting wire

mesh to this strut a region of high wake was produced in the upper

part of the propeller disc. The.resulting wake distribution was

the same as the one shown in Fig. 5 of [loi. Three miniature

pres-sure transducers MT1, MT2 and MT3, were fitted in the plate with

their diaphragms flush with the lower surface. of the plate. Fig.

Withcertain combinations of J, a0.and c the typical

PHV cavitation could be observed. . A vortex with vitating core

was formed, exlnding from the propeller totheplate. A iiore

corn-plete descrIption of this phenomenon is given in

ÏO3.

The signals from the miniature pressure transducers were recorded by Hottinger KWS/6T-5 amplifiers, 130 cps low-pass filters

and a CEC oscillograph. The propeller P-700, whose data are given

in [10], was used för these investigations. Figs. 33, 34 and 35

show some examples of the recorded pressure fluct.iations. They

have ali been obtained with a tip clearance ratio c/D=O.25 and a

propeller shaft speed of 18 revs.pr. sec. Fig. 33 shows a sudden

decrease in préssure at transducer MT3 t times t1 and t2 while

there is no simultaneous change in pressure at MTl and MT2. The

explanation of this is probably that the PHV cavity is sweeping over or in the close vicinity of the djaphraqln of MT3, which then

records the lower pressure in the vortex. Since the effect of the

vortex: is very local there is nô corresponding pressure decrease at

the other two transducers. .

.

The records of Fig. 34 show that at timés t6:andt there

has been a vortex, in the vicinity of MT3. More interesting,.,

how-ever, are the over-ptessure peaks öccuring at transducer MTl. At

time t8, for instance, a vortex is approaching MT1, leading to a

pressure decrease. Then suddenly, at time t9, the vortex cavity

implodes directly on or.very close to the diaphragm of the MTl. This gives rise to a large peak in the: direction of positive

pres-sure,, immediatelyollowed by a neáative pressure peak due to the

-oscillatory behavioür of the collapsing cavity.

Fig. 35also shows a negative pressure peak for

trans-ducer MT3 at time t10 . At times t11 and t12.positivè pressure

peaks occur again at transducer MTl.

As can be seen from.Figs. 33, 34 and 35 the pressure

amplitudes are vaying considerably also between the extremely

large pressÙre peaks. This is due .to a relatively unstable

cavi-tation. During the measurements descr.ibed in Section 3. 2 the

pressure signais were in most cases considerably more stable. It is interesting to compare the form of .the pressure signals of MT3 in Fig. 33 with the full scale, records shown: in

Fig. 24 of lO (transducer position a). it has notbeen proved

However, theirresèmblance with. the model scale records of Figs.

.33, 34 and 35 in.this. report indicate that.they are. The main

difference between model and full scale records is that in the

latter case the. irregular peaks occur with blade frequency. This

is generally not the case. in Figs. 33, 34 apd35. The reason may

be scaleeffects due to differences in Reynolds number and Weber's

number. . After all, öne cannot expect tha.t PHV cavitation on a

horizontal plate.àbove the propeller in a cavitation tunnel with an artificiel wake field will give rise to exactly -the same ir-regular pressurepeaks. as- on a full scale ship.

5.

CONCLUSIONS.

Numerical met,,hods for calculation of the fluctuating pressure.f-ield.indüced on the hull by propeller cavities of given

geometry have been developed and programmed for a computer. The

pressure fields due to cavity mbtion, cavi'ty 'volume, variation, and tip vortex öavity are,daiculated separately, and added to obtain the total pressure.: field due to cavitation. Numerical examples show that as long as the 'cavities are of constant volume during 'the revolution of the propeller, quite severe cavitation must occur in order to make the' blade-frequency pressure amplitude duet to the cavities as large as the amplitude due to the tion-cavitat.ing pro-peller. However', if the volumes of the cavities are no longer

constant, the pressure amplitude may increase onsiderably more.

The pressure amplitude due to cavity volume variation may become. of the order of 10 -times as high as the amplitude due' to constant cavities, even if the maximum volume of the time-varying cavity on each blade is no larger than the volume of the constant cavity on each blade., If thepressure is integrated over the hull to obtain

the-vertical excitation force, this difference between constant and varying cavities may -increase to a factor of the order of 100,

beöause of a fuñdamental difference in phase angle dependence upon

field point coordinates. '

-Pressure fluctuations induced on an afterbody model in a

cavitatiòn'tunnèl have been measured for various propeller tip clearances and' cavitation, patterns. Blade-frequency pressure

amplitudes more than 20 times as high as the non-cavitating con-dition have, been measured. The results of such measurements

carried out in acavitation tunnel may beinflùencedby

reflec-tions from the tunnél walls Instrumentation is at present being

designed for calibration of the tunnels with respect to such re-flections.

PreSüre signals due to propeller-hull vortex cavitation

have been measured by miniature pressure transducers in a

hori-zontal platé above the propeller. The records show extremely high,

loa1 pressure peaks.

The pressure fluctuations measured on the afterbody model have been compared with theoretical results obtained with the

corn-puter programs. The discrepancy iC within experimental error,

whjch is in thïs case relatively high dué tö the fact that the cavity thickness has been determined only by visual observation in

the cavitation tunnel. Improved expeìmenta1 methods for such

determinatï'on is at present being eveloped.. A method based on

photograxrtmetry has just shdwn promisin results.

ACKNOWLEDGEMENT

The investigations preseñted in this report. have been financially supported by The Norwegian Council foe. Scientific and Industrial Research.

Thé author wishes to express his thanks to the Director and staff of. The. Norwegian Ship Model Experiment Tank .for their

cöoperation and support. Special appreciation is extended to

Mr. Svein Eggen for his .assistance in carrying out änd analysing

the experiments.. . . .

The author s also indebted to those of the staf.f at The

Swedish State Shipbuilding Experiment Tank who have contribüted to the experimental investigation described in Section 3, and to Det norske Ventas for providing sorne. of the éxperimental data

REFERENCES e

[i] Breslin, J.P. and Tsakonas, S., "Marine Propeller

Pres-sure Field Due to Loading and Thickness Effects", Transactions SNAME, Vol. 67, 1959.

[21

Denny, S.B., "Comparison of Experimentally determined

and Theoretically predicted Pressures in the Vicinity

of a Marine Propeller", Naval Ship Research and

Deve-lopment Center, Report 2349, 1967.

[3] Huse, E., "The Magnitude and Distribution of

Propeller-Induced Surface Forces on a Single-Screw Ship Modeitt, Norwegian Ship Model Experiment Tank1

Publication no. 100, 1968.

[1

Keil, H., Messung der Druckschwankungen an der

Aussen-haut über dem Propeller", Schiff und Hafen,

Heft 12, 17. Jahrgang, 1965.

Huse, E., "Hull Vibration and Measurements of

Prop-eller-Induced Pressure Fluctuations", International

Shipbuilding Progress, Vol. 17, No. 187, 1970.

Isay, W.H., "Theoretische Grundlagen der Hydroakustik des Schraubenpropellers",

Ingenieur-Archiv, Bd. 35, Heft 6, 1967.

Armonat, R., "Das Hydroakustische Druckfeld eines

Pro-pellers", Institut für Schiffbau der Universität Hamburg,

Bericht Nr. 209, 1968.

Mime-Thomson, L.M., "Theoretical Hydrodynamics",

MacMillan 1955.

Johnsson, C.-A., "Pressure Fluctuations around a Marine

Propeller. Results of Calculations and Comparison with

Experiment", Swedish State Shipbuilding Experiment Tank,

Husè, E., "Propeller-Hull Vortex Cavitation",

Noreqian

Ship Mödei Experiment Tank Publication nö. 106,1971.

Huse,' E., "Trykkimu1ser fra kaviterende propell", Paper presented at "Nordisk skipsteknisk møtç", Abo, Finland, 1971.

[12] CRC Standard Mathematical Tables. 1960

APPENDIX

Effect of tunnel wails upon measured pressur fluctuations.

The pressure signal recorded by the. pressure transducers

may 'e p1it into two parts,

1) the '!direct pressùre. wave" induced by propeller and

cavIty, propagating directly to the field point (pressure transducer), and

the refleòted pressure wave inuced by propeller and

cavity, propagating to the fïeld point by one or more rèflections at the tunnel walls.

Fig. 16 shows the

test section arrangément used for t-he

measurements described in Section 3 of this rèport. The width of

the test section is 0.70 in and its height to the horizontal p1te at WL 24 is 0.61 in.

A complete theòretiçal calculation o- the wail effect is

in. the present case impossible, mainly becäuse f insüficie'nt

knowledge of the mechanical impedance of the tunnel walls. However,

a theoretical discussion including some numeriCa]. reu1ts may still

by useful in providing a better understanding f the physical

phenomená involved.

Let us first assutne the'tun,nel walls to be infinitely stiff. In this case, by assuming thelengtli of the tunnel test section to be infinite in axial direction, the hydrodynamiC problem