Pressure fluctuations around a marine propeller results of calculations and comparison with experiment

FRtN

STATE NS SKEPPSPROVNINGSAN STALT

(PUBLICATIONS OF THE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)

Nr69 GÖÏEBORG 1971

PRESSURE FLUCTUATIONS AROUND A

MARINE PROPELLER

RESULTS OF CALCULATIONS AND

COMPARISON WITH EXPERIMENT

BY

STOCKHOLM.

PRINTED IN SWEDEN BY

In the present report some methods for the calculation of the

pressure field around a propeller in the free stream case are briefly

outlined and applied to some cases for which experimental iriformat!on

is available Further the extension to the case of a propeller in a wake is made and results of comparisons between calculations of this kind and experimental results obtained 'with propellers in behind conditiön are presented and discussed.

Finally the influence of some parameters on the pressure fluctua-tions in different points around a propeller is calculated for propellers in open water condition and in different wake distributions.

Some conclusions drawn from the results are summarized in Sec-tion 9.

1. Introduction

As a first step on the way to estimate by calculation the vibratory

forces on the hull, induced by the propeller, methods have been

developed for calculating the pressure field aouñd a propeller in a

free stream. Important contributions to this subjéct have been made

by BRESLÏN et al. [1-s], Poni. [6-7] and KERWIN [8].

Regarding further steps BRESLIN [9] has shown that the amount of

computer work necessary fOr including a realistic source and sink

representation of the hull for calòulation of the interference is enormous.

To take care of the influence of the wake distribution on the pressure

field is less complicated, as shown by TSAXONAS et. al. [10] and BAVIN

et al. [11]. Further the integrated forces on simple bodies have been

calculated by BRESLIN [3] and Po:aL [6].

In the present report methods for the calculation of the pressure field in the free stream case are applied to some cases for which experimental mformation is available Further the extension to the case Of a propeller in a wake is made and comparisons made with

experimental results obtained with propellers behind a hull.:

Finally the influence of some parameters on the pressure fluctuations

AD/AO = developed blade area ratio

2R2C

= T

Z _{non-dimensional coefficient for blade frequency pressure arn}

plitude

= cosine coefficient of the blade frequency harmonic of pressure fluctuation B = sine coefficient of the blade frequene harmomc of pressure fluctuation

dD

CD - drag coefficient (p/2)ldbV2

CDO minimum drag coefficient

= amplitude of blade frequency pressure fluctuation

D propeller diameter. dD = drag of.profile

r'

O = = non-dimensional circulation coefficient

irD V4 V

J

= = advance ratio Of propeller nD

= magnification factOr fOr influence of wake on pressure fluctuation. See

Eq. (14).

LUZ

k = - =redúced frequency for profile

= torque coefficient pD5n2 T K = . thrust coefficient pD4n2 C

= __4- - non-dimensional coefficient for blade frequency pressure amplitude

pD2n2

= length of profile 6r Îlade sèétioñ

n = number of revolutions

p - static préssure

= static pressure in undisturbed flow

P = propeller pitch

P = propellérpitch in ideal flow = j.V42 = dynamic pressure

Q - =torque

r = radius of field point

D

R

2

= maximum thickness of blade section at 0.6 R

T =thrust

U4 = induCed axial velocity from propeller in a field point induced radial velocity from propeller in a field point

U. =. induced tangential veloôity from propeller hi a field point

V = inflow velocity for profile or blade section q4

Fig. I. Notations.

VA advance velocity of propeUer w = wake

x =r/R

z, y, z = coordinate system for propeller and field point.. See Fig. 1.

z number of blades

angle of attack for profile or blade section

y = position angle for field point

f

= circulatiòn

4Q = torque variatión, single amplitude

¿IT = thrust variation, single amplitude = propeller efficiency

O = blade position angle p density of water q, = phase angle = angular velocity Subscripts = loading contribution t = thickness contribution z = blade frequency

3. Free Stream Case. Methods for Çalculation

The contribution from the propeller to the pressure field is twofold.

First, the vortex systems emanating from the blades due to the

loading induce velocities which contribute to this pressure field.

Secondly the thickness of the blade sections, which m this connection can be represented by a source-sink ditributiori, induces velOcities.

Breslin's Method

The most simple method for calculating the loading and thickness contributions to the pressure field is that of BRESLIN, as it is outlined in [3-5]. For the contribution of the loading BRESLIN starts froma

velocity potential and in a purely analytical manner arrives at the

blade frequency content of the pressure field.

It is assumed that the propeller is lightly loaded and a radial

distribution of the bound circulation

r(x)=x(ix)

is used. Further the concept of "equivalent" or "effective" radius Xe

is introduced and the assumption x,= 0.6

is made.

When determining the ontribution from the thickness distribution a blade form i assumed which corresponds to a sector in the projection

on the ,-z-plane, the angle of the leading edge being + O and that of the

trailing edge O. The blade sections are assumed to be biconvex, giving finite slope at the leading and trailing edges and a constant radius of curvature. Further the sections are assumed to be thin and the concept of effective radius is again used.

The essential merits of BRESLIN'S procedure are that, because of

the rather drastic simplifications introduced in the final stages of

his deductions, the resulting equations for calculating the amplitude

and phase angle of the blade frequency pressures are rather simple.

As could be expected, the impression is given in a recent comparison

with experiment, carried out by Dxy [12], that the more rigorous

approach followed by KERWIN [8] gives more reliable results.

Kerwin's Method

The method worked out by KERWIN is an extension of his lifting

surface method for the design of propellers. When designing

con-ventional propellers according to lifting surface theory, the customary

procedure is to calculate the induced velocities in different points

over the blades and to determine the camber of the blade sections by

integration of the velocity fiêld. BOth the contribution from the

vortex system, corresponding to the loading of the propeller, and that from the source and sink representation of the thickness distribution

of the blades, have to be included. Regarding the contribution from

the loading, the numerical calculations can be simplified considerably if. an approach introduced by PiEN [13] is folIowed By doing so the

integration of the contribution from the free vortices of the propeller to infinity downstream can be handled by using lifting line methods.

When calculating the velodities in different points around the

pro-peller PIEIc's approach is no longer useful, as the lifting line procedures

available at present only apply to the case of calculating the induced velocities along a line through the starting points of the free vortices, i.e. at the lifting line itself.

A detailed description of KERWIN'S procedure in this application is

still lacking. Of particular interest is the way in which the influence 'of the trailing vortices is handled. Regarding the contributions of the bound vortices and the thickness, they can be calculated in a similar

way to that described in [8] and [14] for the propeller design

appli-cation.

The SSPA Procedüre for Calculation of Effect of Loading

In one of BRESLIN'S papers [3] it is shown that, for a lightly loaded

propeller, the contribution of the free vortices in the wake to the

pressure fluctuations is zero.

in Fig. 8 of DNNYs paper [12] the contribution from thebound

vortices and the lifting line trailers, as calculated by KERWIN'S method,

are shown separately. From this figure it is èvidéit that, if the

in-fluence of the bound vOrtices and the part of the free vortices in the blade is coisidered, almost the whole loading contribution is obtained.

Upstream the propeller also the phase angle is obtained with reasonable

accuracy in this way. Having thesê possibilities of simplification in

mind, PLEN's approach could again be considered and a lifting surface

program for design purposes, available at SSPA, wasextended for this

application. This program [15] is based on the PIEN approach and is

rather similar to that described in [16].

The esseñtiál work to be done for covering this application is to

optimise the numerical procedure to handle the case of a point outside

the blade. When the indúced velocities have been determined, the

pressures can be calculated by the use of the BERNOTTLLI equation.

If we use a coordinate system moving with the velocity VA, cor,

this runs as follows:

Po ±- [VA2 +(wr)21=P + - [(cor ±U)2 +( VA +A)2 +UR2] (1)

where

VA=advance velocity of propeller w = rotational speed of propeller

jA, Uf,., UR=axlal, tangential and radial components of the mduced

velocity in the point P

p = static pressure in the point P Thus we get f r the pressu±e difference

4P

PoP

A2 U2 UR2 2UA 1Tx U

T72+172+T72 +

T7

+

(2)

qA qA A A A A A

In the scheme followed, this pressure difference is calculated for 25 equally spaced positions of the blade of a one-bladed propeller relative

to the point P. The harmonics of the pressure field are obtained by

carrying out a Fourier analysis of the result and that harmonic picked out whiòh corresponds to the desired number of blades.

When this program is used for off design cases it is necessary to have a program available for the solution of the "inverse problem" i.e. the problem of finding the angle of attack of the djfferent blade sections, the hydrodynamic pitch angle at different radii and the

radial circulation distribution.. Such a procedure has also to be used when calculations are to be made for a propeller for which the design

principles are not known. In the present report a relatively simple method has been used for this purpose, which is based on a lifting

line approach using GoIDSTEIx's x-method. The method is described in [18]. If the propeller to be analysedhas not been designed according

to the vortex theory, it is necessary, when usmg this procedure, to

have results from open water tests available. In such a case the

calculations are made in two steps. First, starting from the pitch distribution, the Kr-values are calculated for four different advance

ratios, assuming shock free entrance for the blade sections, and com-pared with the open water curve. When this procedure ha converged,

the angle of attack for the different blade sections is calculated for

different J-values exactly ii accordance with the procedure described in paragraph four of [18]. This give the radial distribution of hydro-dynamic pitch and circulation, necessary for usmg the lifting surface prógram described above.

As a check of the reliability of, the results when using this procedure,

the values of 'Kr, KQ .and ìo for different J-values can be calculated.

Results of such akulations for some of the propellers used in the present investigation are compared with open water results in Figs

O O O O 8 O Q Q o o

Open .01er (est, Inverse CeIcyIot,On.

O Innen. cole. sflrcftfreeentr ETA.

Oenny, pop'lF8

'b

0.O08

I

o000

I

Pohl prop prO

/

/

.0008-0.0I2 c,.00I ¡0Ko (TAO Tcchmindj prom2'Jh

H

2 0 06 08 J 06 0.8 ¡.0 IO J 03 O 07 ¡J

Fig. 2. Comparison between results of open water tests and "inverse" calculations.

Experimental values by DENNY, TACRMINDJI and Po.

2 and 3-. For the drag coefficient, which is Of importance when cal-culating KQ and 1o, the value

CD=CDQ±L7cx2 (3)

has been used, independently of the thickíiess ratio of the blade

sections. For the minimum drag coefficient CDO, different values have

been úsed as is indicated in Figs. 2 and 3.

As can be seen in Figs. 2 and 3, good. agreement between calculation

2 .1

Q

J SlIp 6.5 b1Od

CO0o08U

100KO10K7 . 5619 8._6 bIod.,.

I

2'

/

A

.I_

i_' ,"

I

i i

Invent 001K S9006free enO,

, OS QL 00! SlOp C. -00012 008

u

50,90.' 210dm.

C8u

O. 05 04 07 0.8 0.9 J 06 0.7 08 09 10 -III

Fig. 3. Comparison between results of open water tests and "inverse' ca1cu1ations Experimental valües from SSPA.

4. Results of Calculations, General

in the notation of Fig

1, the pressuré j,

in a point P, due to

loading, can be exFessed

p1(x, r, y, t)= Am1(X, r) cos m(Oy) +Bmi(X, r) sin m(Oy) (4)

Ït1is.well. known that, if z-1 equally spaced blades. are added in,

the corresponding. expression of the pressure will be

p,(x, r, y,. t) [zA kzl(X, r) cos kz( Oy) +zB1(x., r) sin kz( Oy)] (4a) AkZI and BkZ bemg the harmonic coefficients of the pressure signal from .a single' blade.

IOKT

100

IOKT lOO

In. the present report higher harmonics will not be considered. Thus only those terms which correspond to blade frequency, i e number of revs, times number of bladés, have to be considered, which leaves

p21(x, r, y, t)=zA21(x, r) cOs z(O-=y)±zB1(x, r) sin z(9y) (4 b)

For practical purposes one is interested in the amplitude C, and phase angle of the maximum value relative to the position of the

propeller blade. This gives the expression

p(x, r, y, t)=C2, cos (z(Oy)--.j)

(5) (6) where B1

ç'= arctan -,---

(7) z :LZ1

Analogous expressions óan be deduced for the contribution of the

blade thickness, gwmg the coefficients P21, A21, B21, C, and q

Finally the total pressure signal is given as

p2(x, r, y, t)=Ccos (z(Oy)p2)

(5 a) CV(Azi+A::)2+(Bz,+Bzi)2 (6 a) where B21 +B21

- arctan

A21 +A1

5. Free Stream Case. Comparison Between

Calculations and Experiment

Measurements of the pressure field around a propeller in a free

stream have been carried out by, among others, TACHMLNDJI and

-DIcKERsoN [17] POHL [7], BRESLIN and KowAisKI [5] and DENNY[12]

In the present report comparisons will be made between results of calculations and the experiments made by DENNY, Poiii and TAd

MINDJI and DICKERSON..

Eaperiments by Tachmindji and Dickerson Ii7

These experiments were carried out with four different propellers

m a cavitation tunnel having a test section diameter of 12 inches

The pressure gauge, having a diameter of 0 012 meters, was located m

the free stream, external to the propeller s1ipstieam. In the present report the results with. propeller No 2714 have been used for

com-parison. This propeller has the following data

D=0.2075 meters

zz3

P/D= 1.059 (constant)

AD/Ao=0.718

The open water test results for this popeller, taken from [17], are shown m Fig 2 together with some spots obtained by mverse

cal-culations.

In [17] the measured amplitudes of the pressure oscillations at

blade frequency were given as

z

(8 " _pD¼2

where C=pressure oscillation (single amplitude) at blade frequency. The same representation has been used for presentmg these results in the present report.

The results of a comparison bet%een calculations and experiment

are shown m Fig 4, giving amplitudes and phase angles for three

different values of the advance ratiO.

For the contribution of the loading to the amplitude both BRESLLN'S

original rtethod and t1e etiod developed at SSPA have been used.

Ecxrpeñrnents by Pohl

Porn.? s experiments were conducted in a circulating water channel with free surface, 2 meters of breadth and 0.6 meters of depth. In the

bottom of the chañnel three pressure gauges were placed, having a

diameter of 0.02 meters; Five different propellers were tested of which propeller No 2 was used for the comparison made here Data for this propeller are

D=0.2 metérs

z==4

P/D=0.775

AD/AQ=0.524

In Fig 2 the results of the open water tests with this propeller are shown together with the results of the inverse calculations. In his

prçsentation of the results of the measurements of the blade frequency aniplitude Porn uses the definition

A

2R2C

T (9)

which is used in Fig 5 of the present report, where the results of Poni's measurements with this propeller are compared with results

1i

'k

FMI

iiI

OJO 00/ Upstream

I3

007 0.06 0.0 002 00! -0.5 -Q25 Upstream 025 05 Downstream 0 025 05 Downstream XIR Upstream JO.6 JO.8 Downstream X/R X/R 0 -05 -025 Ups ream 0 025 05 Downstream XIR 0 025 05 Downstream XIR 05 0.25 0 025 05 Upstream Downstream XIR

Fig. 4. Pressure fluctuations around a propeller. Free stream case. Comparison between calculations and experiments. Experiments by TACHMINDJI and DICKERSON

(Prop. No. 2714) r/R=1.196.

Not mic/az Loadinç

- Eperim Espenm

Breslin Breslin - BFesliff SSPA

.1.03

-IO -05 0 05 - 10

Upfr.A, O,.M-.,,, XIR ApR ,-.N,,, 8r.Ni,, ß,eN1,, 53RA APA .1.075 IO 0 05 1.0 -I.0O.5 O 0.5 I.e NIA

Fig. 5. Pressure fluctúations around a propeller. Free stream case. Comparison between

calculations and experiments. Experiments by Pdar (Prop. No. 2) z/R=1.202.

Experiments by Denny t121

DENNY's experiments were run in a 27-inch-diämeter closed jet test

section of a cavitation tunnel The majority of the testswere carried out with two propellers, 4118 and 4119.

The oiiiy difference between these propellers was that propeller

4119 had twice the thickness of propeller 4118 and a slightly different

pitch due to the larger thickness effect. Thereby the contributions from the loading and thickness to the pressure fluctuation could be separated. Data of the propellers are

D=0.3148 meters

z= 3

P/D= 1.08 (at 0.7 B) A0/A0=0.60

The open water test results with propeller 4118. are shown in Fig. 2

together with results of the inverse calculation. The design calculations

for the propellers were carried out using the mduction factor method

according to LERBS and lifting surface correctiOns of KERWIN,

including-thickness effect.

The pressure fluctuations were measured by seven transducers Dynisco PT25-10 placed in a plate near the propeller, having the

AXIAL DIRECTION. Y/A .0 HORISONTAL DIR3CTION. XIA .0 ApR

dimensions 0.483 x 0.457 meters.

The results obtained are compared with calculations in Figs. 6-9. In this case the representation 2 K,,3 according to Eq. (8) has been

used, as in DENNY'S report.

In addition to the results obtained by using BREsIn's method and

the SSPA scheme for the loading contribution, also results obtained

with KERWIN's method are included, the latter being reproduced from

DENNY'S report.

Discussion of the Results

A general tendency of the results is that the amplitudes are

over-estimated by the calculations at high values of J. From the results of

DENNY'S experiments it is clear that this is mainly due to

over-estimation of the loading contribution. This also applies to KERWIN'S

method which is the most rigorous one.

At lower values of J there is a tendency that the amplitudes are underestimated when BREsu's method is used. This may be due to the fact that, when the loading contribution is calculated by this method, the propeller is assumed to be lightly loaded which should

mèan that the induced velocities are underestimated.

When the loading contribution is calculated according to the method

developed at SSPA the agreement with experiment is improved con-siderably at low J-values compared to BRESLIN'S original method. When compared to PENNY'S experiments the SSPA method shows

however bad agreement in the cases where r/R= 1.3.

Both KERWIN'S and BRESLIN'S methods for calculating the thick

ness contribution compare well with DENNY'S experiments.

In the

case of BRESLIN'S method this could be regarded as remarkable, as

this procedure is very simple compared to that of KERWIN.

What regards the phase angles, all three methods agree well

up-stream the propeller. Downup-stream the propeller only KERWLN'S method

gives values which compare well with experiment. The main reason

for this seems to be that, only when the loading contribution is

cal-culated according to KERWrN'S method, is the influence of the trailing

vortices taken into account. This is illustrated by Fig. 8 in [12] from

which it is evident that, downstream the propeller, the contribution

of the trailing vortices to the phase angle is predominant in contrast

to what is the case üpstream the propeller, where the finalvalue of the phase angle is almost identical with that obtained when only the influence of the bóund vortices is considered.

O. 2Kg, 3 004 O.Oj 0.02 001 008 007 0.06 002 001 0.08 0.07 006 / I p08 2r _3/i 004 003 :,$ -W-05 O 05t015

Upstreath X/R Downs treom

15-1.O-Q.5 0 05 1 1.5

Ups (ream XIÑps (ream

001

o

-1.5 -10 -O5O 05 1.0 1.5

Upstream XIR Downstream

J1O.833 =lO TO TA L, THICKNESS LOADING 40 G20 lead0 60 -21 -4 05 ,Qf5 X_R. /5 -t0--05QQ5 1.0 1.5 X -. 'R

Fig. 6. Pressure fluctuations around a propeller. Free stream case. Comparison between

calculations and experiments Experiments by Dx1cNY (Props Nos 4118 and 4119) J=O.833. r/R=,1.1O.

Notation Loadir,g Th,ckness Ecperimen( EioeHment

- Kerwin Kerwin

Breslin Breslin

-2

-15 -lO-OS 0 05 10 15

Upstream X/R Downs fr earn.

004 0.03 2(K) a02 40-05 0. UpstreornXlR Dotrèâm 0 aot -9/ o o o. o

îsw as o

051015

Ups freamX/R Downs Iream TOTAL THICKNESS LOA O/NG lead lag 60 40 20

Ld o

_tag O -20 JDO.5 O OSLO £5 XI,?

Fig. 7. Pressure fluctuations around a propeller. Freestream ease. Cothparisonbetwèen alu1ations and.eiperirnents. Experiments by DENlv (Props. Nos. 4118 and' 4119)

J=b.833.'r/R=1.3O

Nootigp 4ojng

Ikns

Ex,dt Exp,ê.l

Kerw,n Kerwin Breslin Breslin o SSPA Breslin'

¿k 2(KP 3-2 ( 016 014 01 0.10 0.08 _{TOTAL 20} lead5 0.05 0 0.04 1 -20 008 007 006 o.ôs 0.04 0.03 002 002 _-40 o

---

_-60 454.0-0.5 0O5 10 15 -15 UpstreamX/ Oownstreath 0f 0.14 0i2 008 0.ò 0.0, S40-05 0 0.5 10 1.5

Ups ti-earn IpDownstream

J Z=o:6ò :?' THICKNESS 60 60 40 20 LOADING tog 004 -20 002 ' ,40 0---

--- -

-60 510O50''05 10 15 -1.5-ZOO.5 0 05

Upstream X/R Downstream . XIR

Fig. 8. Pressure fluctùations. around apropeller. tree stream-case. Comp i On bt*een calculations and, experiments. Experiments by Dx (Props. NOs., 4118 and :4119)

J=O.Ò, r/R=1.1Ô.

1.0 1.5

Notúl ion Loading Thickness

Esperiment Experiment --- Kerwin Kerwin Breslin &eslh-, O SSPA Bresliñ 05 10 1.5 -IO .05 O X fi

21K 008 007 006 005 004 0O3-002 001 o -15 -10 -05 0 0.5 ¡.0 15

UpstreamX/ Downs tream

004 003 002 o -1.5 -(.0 -05 0 0.5 ¡0 ¡.5 Up stream X/R Downstream o o

00

001 o -15 10 -05 0 05 lO 15 L/pstreamXlR Downstream 0/ 'e j.I.O.60 qs1.3 lead TOTAL THICKNESS LOA O/NG 6 40 20 -AO 4 2 -15'10-0S' o' 05'iD 15 X/R 60 -60 1.5 10 0.5 0-0.5 X/R iO 15 -20 -4 -15 -10-05....051.0 1.5 X/Ñ

Fig 9 'Pressure fluctuations around a propeller. Freè stream case. Comparisôñ betweeñ

calculations and experiments Experiments by DEI.iirx (Props Nos 4118 and 4119) J=O.6. r/R=1.3O

Notation Loading Thickness E.cpdriménf Ex4'rimeñl Kerwin korwin Breslin ' Breslin - 'O ' SSPA -0 007 0.06 005 4

6. Propeller In A Wake. Methods of Calculation

Bavin's Method

When a propeller is working in a wake, the blade circulation at a certain radius is no longer independent of the position of the blade. The cyclic variation of the strength of the bound circulation in the

blades and accordingly of the strength of the free vortices behind the

blades, can be expected to affect the pressure field around the

pro-peller considerably.

This problem has been treated in an analytical manner by BAvIN

et al. [11]. Starting from Bnxsin's expressions for the contribution of the loading in the open water case, they arrive at an equation for the pressure signal p at blade frequency which is built up by a set of

terms made up of all possible combinatiöns of loading and spatially function harmonics which sum to the blade frequency. In other words the expression is of the type

Piz

EI=O=O

[Ampi sin [(m+n)0my]+Bmn cos [(m+n)0my]

±mn sin [(mn) 0my] +Dmn cos {(mn)0my]]

(10)

where m and n have to satisfy the following relations

m+n=±kz

mn=±kz

m being related to the Fourier coefficients for the pressure signal in

the open water case, see Eq. (4) and n being related to the Fourier

coefficiénts of the peripherical variatiön of the circulation

or

r(x, 9)=

[A(x) cos i0+B(x) sin nO] (11)

F(x, 0)= L' C(x) cos (nO ç(x)) (12)

From Eqs. (10) and (11) it appears that all the harmonics of the

wake lower and equal to z influence on the magthtride of the pressure field of a z-bladed propeller. This is in contrast to what is the case for the thrust and torque variations, to which only the harmànics n=1cz contribute.

-The success of calculations of this kind depends strongly on how

accurately the amplitudes C(x) and phase angles (x) of the unsteady

circulation can be determined. Especially the determintion of the phase angle is difficult and requires a nonstatiounry lifting surface

theory of the kind developed by TSAXONAS -et al. [19]. Very few

laboratories are equipped with such a sophisticated tool and the

others have to rely on quasisteady methods with phase angles based on 2-dimensional or 3-dimensional profile theory.

What has been said abOve applies only to the loading contribution.

The thickness contribution is the same in a wake as in the open

water case.

BAvIN, by using a simple non-stationary theory for the

deter-mination of the amplitudes and phase angles of Eq. (12), arrives from his calculations at the result that, for a 5-bladed propeller in the wake of a single-screw ship, the total blade rate pressure, amplitudes are,

in that particular casé, 40 to 60 per cent larger in the nonuniform case than when the propeller operates in a circumferentially uni-form flow.

The SSPA Method

As was shown earlier, better results were obtained at low values of the advance ratio in the uniform flow case, if the SSPA method was

used for the, calculation of the loading contribution than if BRESLIN'S

method was used. Thus, when extending the calculations to the case

of non-uniform flow, it was decided to use this method as a basis,

instead of that of BRESLIN.

As no nonstationary lifting surface method as available, the

in-fluence of the non-uliiformity of the flow was calculated by the use

of a quasi steady procedure, based on the same equations as the

"inverse" procedure, described in paragraph 5. This quasi steady

procedure has been outlined in [18].

In the input into the program for calculating the pressure

fluctua-tions the influence of the wake variafluctua-tions can be considered by the use of a magnification factor k defined by

(13)

where

p(0y)=local value of pressure in wake p3(9y)=local value of pressure in free stream

The value 1c( 9, y) should give the integrated influence of the wake on the circulation of the propeller. This could be done by for instance

using a value f(Xe, O) of a representative blade section. In the cal-culations of the present report, however, the thrust value has been

used, as this was considered to give a better measure of the integrated

influence of the wake: Accordingly, by use of the "inverse" procedure,

the quasi-steady values of the thrist T( O) at 25 different angular

T(0)

kwi(OY)=p (14)

mean

determined

The program for calculating the pressure fluctuations was then

used in the usUal manner. Thus, as input values for the radial

circula-tion distribucircula-tion the values 1(x) corresponding to the mean wake

were used. Before making the final FOURIER analysis the calculated

pressures were multip1ie4 by magïiification factors in accordance with Eq. (14).

It is the author's experience that the amplitude of the non-stationary

thrust is obtained with reasonable accuracy from quasi-steady

cal-cuJations, provided the blade area ratio is not too large. One evidence to support this is given rn Fig 10, which shows the results of a corn

parison between thrust and torque variations obtamed by

expen-ments, unsteady lifting surface theory and quasi-steady calculations. The diagram in Fig. 10 was essentially reproduced from [19], the

results of the quasi-steady calôulations, carried out at SSPA, being

added afterwards.

From Fig. 10 it is evident that up to a blade area ratio of about

0.6 the amplitu4es obtained by using the quasi-steady method agree

well with the experimental results in this case. It is anticipated that

the same holds for the non-stationary circulation, which is of interest in this context.

However, the phase angle between the position of maximum

cir-culation and the blade position can nOt be obtained by quasi-steady

cálculations.

In order to mvestigate the mfluence of the phase angles these were calculatòd according to the following fórmula, taken from [20], giving the complex circulation of a flat plate two-dimensional airfoil, subject to a sinusoidal gust G

- i[J0(k*) - iJ(/c*)]

41C -1/ 2

= /*

[K0(ik) +Ki(ik*)] (15)

2iryA +x2

where wi k=reduced frequency= 2V

- Unsteady lift. surface thedry (8 strips)

(/6 stripS)

NSRDC exn data D Quasi-steady cala. SSPA

T- j4

0.08

0.06

0.04

0.02 _{Expanded area ratio}

Reduced frequénc k at 0.75R

oided, tab.

o

Expanded area ratio

03 0.6 0.9 /2

0.9 1.8 3.6

Reduced frequency k at OJ5R

Fig. lO. Thrust and torque variations of a propeller in a wake. Comparison between experiments non stationary lifting surface calculations and

- quasisteady calculations.

V=inflöw vélocity

J, J1-= BESSEL functions of the first kind

K0, K1=modified BESSEL functions of the seconc kind

A=J/7r

-The procedure followed at these calculations was the following

A Fouiui analysis was made of the values k( O, y) and the phase

angles of the different harmomcs calculated New values k2( O, y) were

then calculated using an equation similar to Eq. (1l)

these new values fed - into the computer and the same type of calcülations

carried, out as when sing k1. . -

-03 -J --- - 0.6---.- - - .- 0.9-Ì 1.2 09 - /8 27 3.6 0.012 0.010 0.008 0.006 0.004 0.00 o

Calculations and Experinients

Work done by BRESLIN on calculations of propeller induced

vibra-tory forces on bodies [9] shows that the amount of computer work necessary for including a realistic source and sink representation of the body in the calculation scheme is enormous. One is therefore

tempted to try to extend methods of the kind discussed in Paragraph

6 for calculating the pressure field around a propeller in a wake to

cover the case of a propeller behind a ship. Accordingly, the method

described in the preceding paragraph was used for calculating the

vibratory pressures at a few points of the hull of some merchant ship models of different types The computed values were then compared

with experimental results published by HUSE in [1] and results of

measurements carried out in the towing tank at SSPA.

Huse's Experiments

RUSE'S experiments were carried out in a towing tank witha tanker ship model having a four-bladed propeller. Pressure gauges were

placed in 34 points distributed over the afterbody, in the centerline and on both sides. The longitudinal position of the propeller was

varied during the tests. For a few points calculations have been made

using the method described in the preceding section. The results of

these calculation are compared with experiment in Fig. 11. In the diagrams in Fig. 11 the following results are included:

Experimental values.

Calculated values for a popeller in uniform flow. BRESLIN'S method

for loading and thickness contributions.

Calculated values for uniform flow. SSPA method for loading

con-tribution, BRESLIN'S method for thickness contribution.

Calculated values for a propeller in a wake, SSPA method, mag-nification factors k2( O, y). (Phase angle from 2-dimensional profile

in a gust.) BRESLIN'S method for thickness contribution.

Calculated values for a propeller in a wake, HUsE's method. Results

taken from [2Ï].

From the diagram in Fig. 1,1 it is evident that the methods used in

the -present report, as 'wèll as HuSE's own method, in this _{case give}

amplitudes which are in good agreement with those obtained at the

0o2 00l o -'o -30 -'o 70 08 -06 04 -02 0 - ₁₀

\

o o 56

N

-06 04 -02 0 02 04 06 -IO 8

-Fig. 11.Blade frequency pressure fluctuations induced by a propeller on a ship modeL Experimental results from HUSE s investigation [21] 4 bladed propeller Pomts C1 C2

- P3, and S8.

-No1o1io,r ThiCk000 LoOdiOq lIft. 0,0110

- 0l0,001t EOpeìelY00

8.0,011,, 8,0,0110

-,--- 800lln SSPA - No 81,001/s -- SSPA Ye.

1:OT. HusC - HUSe. - - Yes

ssj -3 0P3 -áO 20 o 30 l(p4 K04

by the calculations. The influence of the peripherical wake variations, as it is borne out by the calculations using the SSPA method, is. very small in this particular case.

Expèriments Carried Out at SSPÄ

The measurements at SSPA were carried out in conneetiòn with

routine tests in the towing tank, the pressure gauges being placed in standard positions which, for single screw ships, were in the centerline of the ship. See Fig.. 12. Phase angles were not measured.

The results of the comparison between calculations and experiment for the different models are shown in Figs. 15-16. In Table 1 data for

the ships are given, in Fig. 13 the wake distributions at 0.7 Rare shown.

The harmonics corresponding to the wake distributions in Fig. 13 are shown in Fig. 14 and the positions of the pressure gauges are given in Table 2 in the notation of Fig.. 12.

It should be mentioned that in the case of the ships C1, C2 and D no wake measurements were carried out. The wake distributions for these ships, shown in Fig.. 13, have been estimated from similar ships.

In the case of ship C the wake shown corresponds to the case At the corresponding calculations for Ship C2 the wake variations

have been assumed to be the same as for Ship C1.

The calculated values in Figs. 15 and 16 also include results of cal-culations for a propeller in a wake according to the SSPA method for the loading contribution, using magnification factors k1( O, y), i.e. no phase angle is assumed for the influence of the wake.

From the diagrams in Figs 15 and 16 the following conclusions can be drawn:

Calculations assuming undisturbed flow give lower values for the pressure variations than those obtained from experiment.

Calculations considering the wake variation give higher values than experiments in most cases, if the phaseangle between bladeposition and maximum amplitude of unsteady circulation is not considered.

.e) When the phase angle between blade position and circulation is

considered by using values applying to 2-dimensional profiles in a gust, the calculated and measùred values of the pressure

fluctua-tions are of about the same magnitude, except in the case of the

twin-screw ship D, where the calculated values are lower than the experimental ones in all points except one.

Regarding the discrepancies between calculated and experimental

SHPA.8,,dC1

SHIP C2

SHIP O.

SHIP O

Fig. 12. Measurements of pressure fluctuations induced by a propeller on a ship model.

1)

At 0.6

R.

TABLE 2.

TABLE 1.

Note. All distances are made dimensionless on

the

basis of the propellèr radius R. Foi definitions, see Fig.

12. Ship Type Approx. Length LVL in meters Block- coeff. Speed knots Number of screws rudders blades Diam D metres

No. of revs, rim

Pitch _ratio1) P/D Blade area -ratio L Mean A Cargo 150 0.65 18 1 1 4 5.70 F10 0.94 0.65 18 1 1 5 5.66 107 0.97 0.6Ó 18 1 1 6 560 108 0.95 0.73 B Tanker 300 0.85 15 1 1 5 8.90 82 0.69 0.53 IS 1 1 6 8.60 82 0.74 0.59 C i Cargo 160 0.55 22 1 1 4 6.00 134 0.98 0.70 02 Cargo 160 0.55 22 1 2 4 6.00 142 0.98 0.70 D Cargo 160 0.55 22 2 1' 4 5.00 1:32 1.21 0.52 ship Number of screws blades Gauge A Gauge B1 XA RA -B1 RBI Gauge B2 X82 R88 Gauges G z1 z3 z4 z5 Gauge D _Xp A 1 4 0.754 0800 Ó.572 1.345 0.592 1 5 0.760 0.806 0.580 1.353 0603 i 8 0.768 0.814 586 1.367 0.620 B i 5 0.764 0788 0.640 1.160 0.604 -0468 1 6 0.791 0.815 0.663 1.201 : 0.660 : -0.484 1 4 0.700 0800 0.610 1.052 0352 05 1' 4 1.883 0.800 L183 1.358 0.400 L542 0640 D 2 4 0.840 0.8Ó0 0.832 L070 1.21ff

0.3 29 02 0 90 90 90 0 0. 11 Harrnon,c3

Fig. 14.. Ship models of the SSPA investigation. Wake harmonics corresponding to

waké distributiäns of Fig. 13:

k

w_

-

..

5 30 '5 60 90 .105 /20 '35 F50 /95 18.0_e

Fig. 13. Ship mädels of the SSPA investigatión. Wake distributions at the propeller

blade section 0.7 R. .2

0.04 0.03 002 0 ao OOY 0.02 9i 0. ci z-4 ShiPA cargo ship 2, A C,: Ship B tankèr Z5 \ \ \ N

'

* \\ .: \ A A(/o LOdig fr,fl.,oke

= SSPA &ii,,

__p

o C, A O C, B, A

Fig. 15. Binde frequency pressure fluctuations induced by a propeller on a ship model. SSPA investigation. Comparison between calculations and experiment. Ships A and B.

Data iñ Tables i and 2. 0.0/

4'

.---

₉

i

COST

Kpz

Gonventiòñòl single screw Ship C

fast cargo ship

C,

Sho D

fast cargo twin screw

C3

Single screw witt, 6rgë clearancès

Fig. 16. BÏade réquêncy pressure fluctuations induced by a propeller on a ship model. SSPA investigation Comparison between calculations and experiment Ships C1

C2 and D.. Data in Tables 1 and 2.

dláti LôOding kTi mf!. wakê.

fxpr,m fperim Yes s D-- Breslin Breslin No SSPA Breslin No

k SSPA

ßreslin QÜasieacy Thrust COIC. Nophaseaflg(e 4 SSPA Breslin Sà4.bùt Phase angle 2 dim pro f. theory do,. O

tion carried out by BAvIN :et al. [11].

In BAVIN's case a propeller in a wake behind a screen was

investi-gated. In the position of maximum wake the ratio Kp9/Kp

varied between about 0.5 to about 1.2 within the range 0.6<x/

.R<0.4. In the position of minimum wake KpmeaglKpcaic ranged from

about about 0.4 to 0.8.

RUSE reported ratios Kp/Kpa ranging between 0.5 and 1.0

with extreme values of 0.35 and 2.1. Part of the dispersion in this ratio could, according to RUSE, be attributed to the fact that the

image effect varied for different transducers, dependent on the

curva-ture of the surrounding surface. The value Kp,8/Kp=1 .0 cor

responds to a plane surface of infinite extent and it is reasonable that

this assumption does not hold for all transducer positions.

Having these facts in mind more definite cônclusions could be

drawn from the comparison reported in Figs. 15 and 16 by presenting

meas.

Fig. 17. SSPA investigation. Ratio of measured and calculated blade freqiency pressure fluctuationifor.ingIescrew ships. Mean values for different ship-propeller combinations

the results in the form of mean values for each ship-propeller

con-figuration in the way shown in Fig. 17.

From the diagram in Fig. 17 the following conclusion can be drawn:

When applying calculation methods, deduced for the case of

uni-form flow, to the case of a propeller behind a ship, there is a risk that the beneficial influence of increasing the number of blades is over-estimated (Ships A and B). The same applies when the influence of

the clearance is calculated (Ships C1 and. C2).

8. Calculation of the Influence of Some Parameters on the

Pressure Fluctuations Around a Propeller

In this paragraph the influence of the following parameters will be

considered:

distance from propeller radial circulation distribution peripherical wake distribution advance ratio (pitch ratio) number of blades

blade area ratio

thickness of blade sections thrust coefficient K

As reference for the variations of the parameters mentioned above the pressure fluctuations calculated for a propeller with the following data were used

Number of blades z=4 Advance ratio J=0.5

Thrust coefficient KT=O.l9

Circulation distribution=BETz optimum Pitch = constant

Blade area ratio

_4D/4o=°

Blade form, see Fig. 18

Thickness of representative radius S0 6/D=0.019 (0.6 R).

The pressure fluctuations were calculated m points A and C, of

Fig. 12 with

XA=l.l, 1.3, 1.5

Z1=0.5, 0.7, 09

The amplitudes given in the diágrath in Fig. 21 and Figs. 24-30 are single amplitudes for a propeller in an unbo ìded fluid.

A/A/oO

Fig. 18. 1ade form and general appearance of base propellers for calci:ilations of the

influence of different parameters on the pressure fluctuations around a propeller

Ìnfluence o Circulation Disti-i bution

FOr the case J=O.5, K.=O.19 the pressure fluctuatioñs in the

ui-form flow case have been calcUlated for the th±ee different circúlation

distributions illustrated in Fig 19, representing different degrees of

unloading of the blade root and tip relative toBETZ'optimum distribu

tion The correspondmg pitch distributions are shown in Fig O Tri the optimum case, GOLDSTEIN'S x method has been used for the calculations, in the two other cases the hydrodynamic pitch has been calculated by the use of an induction factor method, described in [18]

From the diagram m Fig 21 it is evident that the magmtude of

the pressure fluctuations is only slightly affected by the circulation

distribution and pitch distribution. Influence of Wake Distribution

The results reported in Paragraph 6 show the importance of the

influence of the wake on the amplitude of the pressure fluctuations. As this influence might affect the tendency of the miluence of other parameters such as the number of blades, blade area ratió and so on,

it seems motivated that the calculations of the influence of these parameters are carried out for the uniform flow case as well as for

the case of a propeller in a wake. Before starting such calculations, a comparison of the influence of different wakes should be carried out to make sure that a representative wake distribution be selected fOr

this purpose.

Accordingly, for two different advance ratios, J= 0.35 and J = 1.0, the amplitudes and phase angles were calculated for unifOrm flow and four wake distributions at 0.7 R, shown in Fig. 22.

The magnitude of the harmonics of these wakes is shown in Fig. 23. The amplitudes and phase angles of the pressure fluctuations, ob

tamed from these calculations, are shown in Fig. 24 and 25. From

these figures it is evident that the influence of the wake is more marked

at J= 1.0 than J=0.35. Further the influence of the wake is more

pronounced for the lower value, r/R=0.8 than for points above the propeller (rIB >1). A comparison with Fig. 23 shows that the 4th harmonic is of about the same magnitude for alJ flip wa]es. The differences between the amplitudes in the différent wake fields can

thus be attributed to lower wake harmonics. This fact has been

pointed out by other investigators, see [1 1].

As has been pointed out earlier in the present report the' calculated amplitudes in the case of a wake have a lower degree of accuracy than those obtained for the free stream case.

Influence of Advance Ratio

The influence of the advance ratio is shown in Fig. 26 for the open

water case' and for Wake IV. This influence is most marked on the

phase angle,: the main reason being the change of reduced frequency with changing J.

Influence of the Number of Bladés

In Fig. 27 the préssure fluctuatiOns are shown for propellers having

e 50 u 08 06 &ade sect,on A nR

Fig. 19. Circulation ditributions for calculating the influence of circulation ditribution

on pressure fluctuations in uniform flow.

0,tn 2

0.5 0.6 0.7 08 0.9

ßd bo,,nIR

Fig. 20. Pitch distributions corresponding to the circulation distributions in Fig. 19 J=O.5, KT=O.l9.

I,

7

I

0fr2.

IFA

:1V!

r

. -0) 0.0 03 a,. I.

4Ø degr 35 30 25 40 degc -25 -20 -15 K4 0.03 002 00! o Kp4 0.03 a02 0.01 r/R0.8 1.! 1.2 /3 _{/4 /5}

Vertical position nR of point P relative to shaft centerline

0/str 2 0/str 2

0/str! Opt.disfr D/str...

Disfr3 0/str I

05 0.6 0.7 0.8

Positian of point P ahead of propeller 0.9

Fig. 21. Amplitudes and phase angles for pressure fluctuations around a propeller in the uniform flow case, as calculated for different circulation distributions according to Fig. 19 and corresponding pitch distributions according to Fig. 20. J=0.5, Kr=O.l9.

b

Slug B of f;glO SIt/pC of f,619

s

Fig. 23. Magnitude of wake harmonics, corresponding to wake distributions in Fig. 22.

Notot,oa p. of 0,0k.- ---Noie I

I

N

arge anker 14110e-e.

Lar e tanker ballast can. Past cargo sap Post carga Si,,p

,P of f9l Sh,pC aff.gJ ¡.0 1g - -

-k'

N - -- II-O 5 30 45 60 75 90 #05 ItO - #35 #50 165 tsp Blade pos,t,on Q

Fig. 22. Wake distributions at 0.7 R for calcWating the influence of wake distribution on the pressure fluctuatiòns.

NOlaton Type of wOke Nate

z

Large tanker, fully loaded Lar etanke,- ballast Cond Font sorgo

--40 degr Kp4 - 0 -0.06 -50- 005 -40-0.04 -30- 0.03 -20-0.02 -io- aol 40 deq /( 4 0.06 0.02 - 0.01

- Con sto-nt flow

Constant f10

Woke if

Wake 27E Wake

X0

Amplitude Phase onçle

Wake 7E Woke Wake

r/RO.8

Constant flaw Wakel

Wake]7E Wokel

- ---

Constant flaw

__

Wake J=035 WakeifI 0.05

-j

-0.04-

-- .

-

W k e

-0.03- _{_..} Wake

Position .' of point P ahead of propeller

Fig. 24. Amplitudes and phase angles of pressure fluctuations in uniform flow and four different wakes. J=O.35 Unbounded flow.

Wake.I Wake.7E o

1.1 /2 - /3 1.4 1.5

Vertical position nR of point P relative ta shdf t cnterline

0.5V 0.6 0.7 0.8 -60 -50 -40 -30 -20 -, Q o

40 degr Kp4 -70-0.0 7 -50 006 -50- 0.05 -40- 004 -30-00.3 -20-1102 -10- 0.01 Wake

j-Constant flaw Wokel Wake H Wake Wak 0- 0 -1.1 1.2 1.3 1.4 1.5

Vertical position nR of po/Al P relativé Ea shdft centerline 40 dègr Kp4 -70- 007- -6Ò-006--50- 00540004 30003 20002 --Id o - 001 Woke E

--- I.

Wake lT ConsEant flow Wake I 0 Wake E Wake .77 05 06 07 - Amplitude ---Phase anglè' Wake J =1.0

==-WZ

Constant flaw Woke.

J-

n/R-08 J1.0

Woke 277 Constöht flow

-Wake I

Position x of point P ahead 06of propeller 09

Fig. 25. AÎplitúdés ad phase angles of pressue fluctuations in uiform flow and four different wakes. J= l.O Unbounded flow.

J-10

07

x.o

11 /2 /3 14 1.5

Vertical position rIP of point P relative to shaft centerline

rIP.. 0.8

05 06 0.7 0.8

Position x of point P ahead of prcpèiler

Fig 26 Calculated amplitudes and phase angles of pressure fluctuations in uniform flow and Wake IV. Influence of advance ratio J. Unbounded flow.

Notation

--

Flow

C&stont Amplitude

- Wake/V Amplitude

----Constant Phase ongle

Z Kpz -60 -50 -40 -30 -20 -10 ZØ /(pz x-0 o 1/ 1.2 13 14 15

Vertical p0s1lio rIP of po/nt P relà live lo shaft centerline

0.5 0.6 0.7 0.8 0.9

Position .x of point P ahead of propeller

Fig. 27. Calculated amplitudes and phase angles of pressure fluctuations in uniform flow and wake IV Influence of number of blades Unbounded flow

Not al/on Flow .. -

-Constant AÑz(itud

- -, Wake/V Athplitude.

- -

Contanl Phasè anule Woke/V Pnase angle

-60 -50

-0

-20 -10 4Ø Kp4 002 001 x=o A9/A0 1.00 1.1 /2 /3 /4 /5

Vere/col position nR of point P relotive to shaft centerline

/A0=iO0

A/A-0.75

0

0.5 0.6 0.7 0.8

Position x of point P oheod of propeller 0.9

Fig. 28. Calcúlated amplitudes and phase angles of pressure fluctuations in uniform flow and Wake IV. Influence of blade area ratio. Unbounded flow.

Notation

-f.low

Con.slont Amplitude Woke/V Amplitude

Constant Phase angle Wake/V Phaseorqte

dR0.8

0.05 0.04 ao-3 0.02 0.0/ o J1.0 J07 JO35 J=07

-506/O qo38

-J,.0

----J---

-.-:-J 07

Vertical posieion nR of point P relative to shaft centerline

Fig. 29. Calculated amplitudes and phase angles of pressure fluctúations iñ uniform

flow. Influence of thickness of blade sections. Unbounded flòw.

0.5 06 0.7 0.8 o.c

Position s of point P ahead of propeller

---Phase ongle 0.06 1.1 12 1.3. 1.4 15 -40 -30 -20

40 ¿(p4 003--40 0.02 30- -20-001 10- -50-003

40

.30 20 10 0,02 0.01 Kr Q19 K0JO Amplitude Phase ai',gle X=0 KT019 K7-0/0 ii 12 /3 14 - 1.5

Vertical position rIP of point P relative to shaft centerline

Position s of point P ahead of propeller

Fig. 30. Calculated amplitudes and phase angles of pressure fluctuations in uniform flow. Influence òf propeller th±ust coefficient K. Unbounded flow.

Advance ratio J=0.5.

0.9

z=4,5 and 6, as calculated for the open water case and for Wake IV. For points ahead of the propeller the beneficial influence of increasing the number of blades is smaller in the case of the wake field than in the open water case, as was stated earlier in this report.

influence ofBlade Area Ratio

In Fig. 28 the pressure fluctùations are shown for four-bladed

pro-pellers with different blade area ratios, as calculated for the open

water case and for Wake IV. The diagrams show that the amplitudes

are about the same for AD/AQ=0.5 and 0.75. If the blade area ratio is increased to 1.0, a decrease in the amplitudes is obtained for the

points ahead of the propeller.

influence of Profile Thickness and Thrust Coefficients

An impression of the magnitude of the influence of profIlé thickness

and trust coefficient KT for four-bladed propellers is obtained from Figs. 29 and 30 respectively.

9. Conclusions

From the results of the calculations and experiments presented in

the present report the following conclusions may be drawn:

The calculated amplitudes of the pressure fluctuations at blade frequency are of the same magnitude as those obtained from ex-periment in the open water case as well as for a propeller in the

wake field behind a single screw ship.

The corresponding phase angles are predicted with reasonable

ac-curacy by calculations in the open water case. In the case of a

propeller in behind condition the material of comparison is too smali.

For the only twin screw case investigated bad agreement was

ob-tained between calcúlated and measured amplitudes of the blade

frequency pressure fluctuations.

4 If the influence of the wake field is not considered at the calcula-tions of the blade frequency pressure fluctuacalcula-tions, there is a risk that .the beneficial influence of the increase of factors like

clear-ances and number of blades is overestimated.

5 The influence of the wake field on the results of the calculations

increases as the advance ratio increases.

Finally it should be emphasized that all the results of the present

report apply to noncavitating conditions only. Recent experiences in full scale and model testing indicate that the conditions are changed

io. Acknowledgement

The author wishes to express his gratitude to the Hugo H a

m-mar Foundation for International Maritime Research

and the Martina Lundgren Foundation for Maritime

R es e arc h. for financing the present investigation.

The authör will recognize his thanks to Dr HANS EDSTRAND, Director

General of the Swedish State Shipbuilding Experimental

T a n k, for having stimulated and granted the work with this study.

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Research Vol. 1 No. 4, March 1958.

BRESLIN, J. P., TSAXONAS, S.: "Marine Propeller Pressure Field Due to Loading

and Thickness Effect." Tran8. SNAME, Vol. 67, 1959.

BREsLIN, T. P: "Review and Extensiòn of Theory for Near-Fe1d Propeller-Induced Vibratory Effects." Fourth Symposium on Ñaval Hydrodynamics,

Washington D.C. 1962, ONR Pubi. No. ACR-92.

BRESLIN, J. P.: "Review of Theoretical Predictions of Vibratory Pressures and Forces Generated by Ship Propellers." The Second Interñational Ship

Strut-tures Congress, DeIft, The Netherlands, July 1964.

[ui] BRESLIN J P KOWALSKI T Experimental Study of Propeller Induced Vibra

tory Pressures on Simple Surfaces and Correlatiòn With Theoretical

Predic-tions." Journal of Ship Research Vol. 8, Dec. 1964.

Poni K H Das instationare Druckfeld in der Umgebung emes Schiffspropellers und die von ihm auf benachbarten Platten erzeugten periodischen Kräfte."

Schi!fstechnik Bd6, 1959, Heft 32.

Poni, K. H.: "Die durch eine Schiffsschraube auf benachbarten Platten erzeugten

periodischen hydrodynamischen Drücke." Schiffstechnik Bd7, 1960, Heft 35. KERWIN, J. E., LEOPOLD, R.: 'A Design Theory for Subcavitating Propellers."

Trans. SNAME, VÓZ. 72, 1964.

BRESLIN, J. P., Eivo, K. S.: "A Method for Computing Propeller-Induced Vibra. tory Forces on Ships." Proc..Fir8t Conference on Ship Vibration, Stevens Inst.

of Technology 25-26 Jan. 1965. David Taylor Model Basin Repp?t 2002,

Aug. 1965.

TSAONAS, S., BRESLIN, J.P., JEN, N.: "Pressure Field Around a Marine Pro-peller Operating in a Wake." Journal of Ship Research Vol. 6, No. 4,

April 1963.

BAVIN, V. F., VASEREVICÌ, M. A., MINIovIcH, I. Y: "Pressure Field Around a

Propeller Operating Itt a Spatially Non-Uniform Flow.' 7th Symposium on

Naval Hydrodynamics, Rom Aug. 1968.

DEiY, S. B.: "Comparisons of Experimentally Determined and Theoretically Predicted Pressures in the Vicinity of a Marine Propeller." Nával Ship

Research and Development Center Report 2349, May 1967.

Pmne, P. C.: "The Calculation of Marine Propellers Based on Lifting Surface Theory." Journal of Ship Research, Vol. 5, 1961.

[4] KEEWIN, J. E., LEOPOLD,. R.: "Propeller Incidence Correction Due to Blade

Thickness." Jour-nat of Ship Research, Vol. 7, 1963.

[Ï5] JoENssoN, C.-A.: "Tilampning och experimentella verifikationer av en teoretisk propellerberäkningsmetod." SSPA General Report No. 11, Göteborg 1965

(in Swedish).

CNG, H. M.: "Hydrodynamic Aspect of Propeller Design Based on Lifting Sur-face Theory Part i Uniform Chordwise Load Distribution David Taylor Model Ba8in Report 1802, Sept. 1964.

TACHMINDJI A J DICKERSON M C The Measurement of Oscillatmg Pres süres in the Viciinity of Propellers." David Taylor Model Basin Report

No. 1130, April 1957.

JOENSSON, C.-A. "On Theoretical Predictions of Characteristics and Cavitation Properties of Propellers." SSPA Pubi. Nó. 64, Goteborg 1988.

TSAKONÄS, S., BRESLIN, J P., MILLER, M.: "Correlation and Application of an

Unsteady Flow Theory for Propeller Forces." Tran8. SNAME, Vol. 75,

1967.

BROWN, N. A: "Periodic Propeller Forces in Ñon.Uriiform Flow." MIT Dept. Nay. Arch. Mar. Eng. Report No. 64-7, June 1964.

HusE, E.: "The Magnitude and Distribution of PropellerInduced Surface Forces on a Singlé-Screw Ship Model." Norwegian Ship Model Exp. Tank Pubi.

No. 100, Dec. 1968.

Contents

Synopsis.

.. ...

3

1. Introdíiction . . . . r 3

2. List of Symbols .. . 4

3. Free Stream Case. Methods for Calculation 5

4. Results of Calculations. General . - . 10

5. Free Stream Case. Comparison Between Calculations and Ex

periment

...- -

11

6. Propeller in a Wake. Methods of CalculatIon 20

7 Propeller m Behmd Condition Comparison Between Calcula

tions and Experiment -. 24

8. Calculation of the Influence of Some Parameters on the

Pres-sure Fluctuations Around a Propeller 33

9. Conclusions 46

10. Acknowledgement .47