The magnitude and distribution of propeller induced surface forces on a single screw ship model

THE M AGN lIUDE AND DISTRIBUTION

OF PROPELLER-INDUCED SURFACE

FORCES ON A SINGLE-SCREW SHIP

MODEL

by

E. HUSE

NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION NO.100

THE MAGNITUDE AND DISTRIBUTION

OF PROPELLER-INDUCED SURFACE

FORCES ON A SINGLE-SCREW SHIP

MODE L by E. Huse Contents page Abstract, _i List of symbols. i i. Introduction. 5

Theoretical calculation of free- 6

space pressure field.

Description of experiments. 13

Comparison between theory and 18

expêriments, introduction of solid boundary factors.

Main conclusions. 21

References. 22

Appendices. 24

Tables. ₃₆

ABSTRACT.

This report presents' the results öf experimental investi-gations of propeller-induced pressure fluctuations on the. after-body of a tanker model. The corresponding free-space pressure field due -to blade thickness, stationary and dynamic lift on the. blades is calculated numerically. The éffect of the solid bound-ary. of the àfterbody 'is determined, by comparison between experi-mental results and theoretical free-space pressures. Empirical

"solid boundary factors" are. introduced in order to make free-space pressure calculations- a more effective, tool for practical prediction of propeller-induced pressure fluctuations on the

afterbody. .. .

-LIST OF SYMBOLS.

B breadth, -.

C(..t) generally a function of time,

D . propeller diameter,

Dsf distance between source point and field point,

Fvb vertical bearing force,

F vertical surface force,

-vs

F(e) lift distribution over chord,

I-I 'function describing helicoidal lifting surface,

K

,K

pt PS K

,K

pd ptot

non-dimensional coefficients of instantaneous theoretical free-spaòe pressure due to blade thickness, sta.tionary lift, dynamic lift, and their sum respectively,

La lift pr. unit area of blade,

length-between perpendiculars,

P -. pitch, . , .

2-S strength of pressure pole,

Sd strength of pressure dipole,

T mean propeller thrust,

T draught (Fig. 3.1.),

U velocity of undisturbed flow at field point,

chordwise velocity at propeller blade,

UT total velocity at field point,

Z number of blades,

b propeller-skeg clearance (see Fig. 3.6.),

c chord,

c tip clearance (see Fig. 3.6.),

cpmt single amplitude of m'th order component of

T? cpms V? cpmd 't cpmtot

Kt0t

cv5 non-dimensional value of Fvs cvb 'T

Fb,

c experimental blade-frequency pressure single amplitude of -bladed propeller,

x, r', e cylindrical coordinates of source point,

x, y, z rectangular cartesian coordinates of source point,

n propeller shaft revolutions pr. second,

unit normal of helicoidal lifting surface, +

pp r rh rt s s-t t

u, u, u

vn n

_X'

n n

r' e components of in -, r-, and e-direction respectively,

free-space pressure signal,

pressure field of pressure pole,

pressure field of pressure dipole,

radius,

hub radius,

tip radius,

chordwise coordinate,

chordwise coordinate of leading edge,

chordwise coordinate of trailing edge,

time,

instantaneous value of dynamic component of

thrust on one blade,

field point perturbation velocities in -, p-, and C-direction respectively,

induced velocity normal to blade section,

angul4r propeller position, positive in the

direction of rotation (=-y,se.e Fig. 2.l.a),

pitch angle,

angular proeller position (see Fig. 2.l.a),

angular position of blade number M,

cylindrical coordinates of field point,

rectangular cartesian coordinates of field point,

chordwise coordinate (see Fig. 2.1.a.),

B

_-L_

- value of O at leVading andtrailing edge respec-tively,

0003

00.1, value of O at 3, lO, 25., and 80 percent of chord,

0O25 008

measured from leading edge,

K rake angle,

density of water,

source strength pr. unit area,

T(S)

blade profile thickness,

T01

profile thickness at 10 percent chord,

n' ''

phase angles of 9pmt' .cpms pmd and Cpmtot

-ind' mtot \ respectively,

phase angle of

dIfference between experimental phase angle and

diff the corresponding phase angle obtained from theoretical free-space pressure field calcula-tion,

velocity poténtial due to thickness. of one propeller blade, .

vélocity potential due to thickness of Z pro-peller blades.

induced by the propeller.

First we have thé "bearing forces".

They originate from the inhomogeneous wake field and are

tranS-mitted to the hull through the propeller shaft bearings.

Secondly

we have what is called "surface forces", which are caused by the

pressure field surrounding each propeller blade.

When the

pro-peller rotates in the vicinity of the hull, these pressure fields

indüce vibratory forces directly on the surface of hull and rudder.

Propeller-induced forces have been the source of

trouble-some vibrations on board many ships.

Therefore they have also

been ¡nade the subjéct of extensive investigations by many authors.

An exhaistive list of publications in this field up to 1966 is

given in (I).

(References at the end of Section 5).

The present report deals exclusively with surface forces.

Measurements by Pohl,

(2), in a free-surface water channel, by

Tachrnindji and Dickerson,. (3), and Denny,

(1.f), in water tunnels

have provided a considerable amount of experimental information

on the free-space pressure field around a propeller working in

uniform flow.

Present-day theories are able to predict this

free-space pressure field in uniform flow,

_('fl.

However, they do not

predict the pressure fluctuations n a ship hull satisfactorily.

The two main reasons-for this are, first, the solid boundary of

the hull makes the assumption of free space invalid.

Secondly,

the t.ake field is not unifórni.

As

rtiatterof fact, it is not

even óonstant in time, but fluctuating due to turbulence.

Experi-mentally measured pressure fluctuations on a hull have been

re-ported to differ by a factor Of 5 to 8 from theoretical

calcula-tions based on unifOrm flow, (5).

Fortunately,.a relatively good agreement between model and

full écalé measuremnts, (5), seems to indicate thatvaluable

information about surface forces can be obtained from model scale

measurements.

This report presents the results Of some preliminary

in-vestigations into the magnitude and distribution of surface forces

6

the correlation between experimentally measured pressure fluctu-ations and the theoretically calculated free-space pressure. field..

The theoretical caici4ltiOns presented here also include the effect of dynamic lift upon thé pressure field. The resulting

discrepancy bétween theöry and experiment is expected to be Ïnainly due to the effect of the solid boundaries of hull añd rudder. We

shall therefore use the notation "solid boundary factor" S for the ratio beÑeen experithental and theoretical free-space pressure

amplitudes.

Theoretical òalculation of the image effect of the solid

boundaries is possible fOr bòdies of simle g.éothetry, é.g. n in-finite plane surface or a cylinder, (io). For a more complicated geometry, like the afterbody Of a ship, a numerical solution sed

on potential theory is still possible in principle, (ii), but

pro-bably too comlicated îd time-consuming to be of prcticl

in-i-terest.

In the aúthor's opinion a practical method to preiçt surface forces in. th.future will be to apply free-space pressure field theory, taking image effects into account by introducing

empirical "solid boundary factòrs". One of the òbjecs Of the

present report is to provide sorné empirióal data for sûch use.

2.. THEORETICAL CALCLtLATION O FREE-SPÄÒE PPESSÜRE IELD.

The free-space pressure field around a prope1i.r, i.e.

t1e pressuré. field that would be preent fthee were no solId

boundaries in the. vicinity of the propeiler, may be .ssumed to originate frôm the following three effects:

blade thickness,

average or steady lift on blade sction,

fluctuating lift or blade sèctiondue o non-nifor.rn wak field.

Assuming lInearity the contributions from these three effects nay

be superposed to give' the total free-space pressure field.

The linearized tréatmert of the pressuré field is. wel known, (6)and (7). fi Appendix i of this report the following

formula is derived,: giving the pressure fielddue to blade

thick-nes,

r _er(r) K Pt i 4 ir'n -3/2 z

_f

2. 2 2 3T (P +L r )

-3s K PS T

zf

j 0.9 Q5 _{KTr pll}

''t

'h

Ml rh

_e1(r) 3

_22,,_ 2

2 (1 -2r_rh_r _{)2) F(e)} r._rh

(ee1

-

Kr)2+p2+r2_2rp.cos(e_O_y_(M_1))}3/2.

Ml

rh

601(r)

-Kr)2 + + _{- 2pr.cos(c-e-y-(M-l)--))3"2} {rrnrpsin(e-e-y-(M-1)-) +

. (_P-- - Kr)}dedr

JP2+22

t4ir nD M=1 rh

((tP

_;°- Kr)2 + r2 + p2 .- 2.rpcos(c-0003-y-e r)}.dr (2.1)

(2.1) has been derived by representing the blade thickness

by .a continuous sourcesink distribution along the mean line of

the profile, .ihich is assumed to be symmetrical.

The non-dimensional coefficient K5 of the instantaneous pressure due to stationary loading is derivedin Appendix 2,

where 8 (2rrr(-P f--Kr) + 2TrrK(r_p.cos(s_e_y_(M_141L)) - Pp sin(c_0_y_(M_1).!L)).d0.dr. z for e1 < e

e08,

F(e) :_0)/(0t_00.8) for 00.8 < e et' (2.2)

(2.2) has been derived by representing blade lift by a

continuous layer of pressure dipoles along the mean line of the

profile. Circulation distribution over the radius is assumed to

be elliptic. The chordwise d:istribution of lift is assumed tobe

that of a NACA a 0.8 mean line, at ideal angle of attack.

A propeller working in a non-homogeneous wake field will be subjected to fluctuating hydrodynamic forces in addition to its

stationary or meañ thrust and torque. This dynamic component of

lift on the blades also contributes to the total pressure fie1d of

the propeller. An exact mathematical calculation of this

contri-bution is very complicated, the main problem being the determina-tion of-the instantaneous lift distribudetermina-tion over thé blade (see

for instance (8) and (12) ). We shall here approximate the

chord-wise load distribution by a line distribution of pressure dipoles

at the, quarter_chord:. By assuming elliptic circulation distribu-tion over the radius, the non-dimensional coefficient Kpd of pres-.. sure due to dynamic lift becomes (Appendix 3),

r r 1<

22

pn

D (r _rh )cos K M=l 2r-rh-r 2 (i- t) ₎ r t-rh e -3/2

{(-p

Kr)2+p2+r2_2Prc05(C_0025_IM)}

.0 (2irr( p_0.25

_{_Kr)+21TrK(r_p.cosE_eO25-y.»}

- Pp.sin(e_e025...IM)).dr. (2.3)

td(YM) is the instantaneous value of the dynamic component of

thrust on one blade at àngular position

1M' This value may be

obtaifled tieoretically by application.of one of the usual methods for calculating thrust fluctuations from wake field data. It may also be measured directly ön a cale model with a suitable dyna-mometer (9).

At The Nrwegian Ship Model Experiment Tank the numerical

calculation óf K, K5 and

_{Kd from (2.1), (2.2), and (2.3) has}

been pi'grarnmed in FORTRAN IV for a UNIVAC 1107 computer.

Integration of (2.l) over the chord is carried out by 10 intervals Simpson integration. _{The slope 3t/s in (2.1) is} ob-tained by reading profile thickness at various chordwise stations as input data for the program. _{Then the slope is cálculated by a}

stepwise approximation of

T(s)

by a second-order polynomiaL The

required computer time on UNIVAC 1107 for calculation of

_{Kt is}

00 milliseconds for each field point and each propeller position (for four-bladed propeller).

- The numerical evaluation of (2.2) is carried out by a 10

intervals Simpson integration over the chord followed by an 8

intervals Simpson integratiOn over the radius. Computer execution time, is' 320 milliseconds for each field point and propeller posi-tion.

(2.3) is evaluated numerically by _{an 8 intervals} integra-tion over the radius. _{Computer execution time is 70 milliseconds.}

The computer program reads field point coordinates from

data cards. For each field point the pressure coefficients K , K and their sum K is calculated for twelve different

ps pd ptot

angular positions of the propeller. In the case of a four-bladed propeller this means that the coefficients are calculated at y intervals of 7.5 degrees. _{Harmonic analysis is performed on the} resulting coefficients to obtain amplitude and phase angle of the

blade-frequency and twice-the-blade-frequency _{content of both the}

total pressure signal and the thickness, stationary lift, and

dynamic liftcontributions. _{Complete calculation and analysis pf}

the pressure in this way requires a computer execution time of 10 seconds for each field point.

10

-The results of some calculations with this-program are given in Fig.s. 2.6 through 2.14. The calculations havé been carried out for the propeller shown in Figs. 2.4 and 2.5. The results presented in Figs. 2.6 through 2.14 have been. obtained for thrust, shaft c.p.s., and speed cörresponding- to service conditions for the tanker model described in Section 3 of this report. For this tanker model and propeller the thrust fluctuations on one blade have been measured, (9), the result shown in Fig. 2.3. The

thrust fluctuations required to calculate Kpd from (2.3) have been taken from Fig. 2.3.

The choice of field point flow velocity U in (2.1) repre-sents a pecial problem when the pressure field of a

wake-opera-ting propeller is to be evaluatéd from free-space pressure caicu-lations. The field point is usually a point. on the surface of the hull whére the. actuál flow velocity is zero. Moving outwards from

the hull the flow velocity increases to a value eqúal to ship

speed outside the boundary layer of the hull. or U equal to zero the linearisation applied in deriving (2.1) becomes iñvalid. As

a matter of fact, the validity of theoretical formu1e based on

potential theory is always questionable when they are applied to

describe the flow field around the .afterbody of a ship. Their accuracy or usefulness can only be proved by comparison with ex-periments. Returning to (2.1) we find that fortunately the value of U is not so important after all,

because-for small values of the terms.containing U in (.2.1) are relatively small compared to the terms to which they are added in (2.1),

for higher values of (k/DI>0.5) the pressure due to blade thickness becomes small compared to the pressure due to dynamic lift, the latter being independent of U.

Calculations have been carried out for the 4-biaded pro-peller and ship modél mentioned above with U pu equal tb zero and equal to inódel speed. For most field points the discrepancy bet-wêeîi total pressure amplitudes in the two cases is less thán 3

percent. The maximum discrepáncy for any of the field points tested is 18 percent, obtained for /D 0.33 and pID 0.86.

of Kt() K(a), KPd(a)

and their sum K.0(a)

phase angles being defifled by the following expressions,

:m;0 K

()

c sin (ma. + ) PS pms

mO

K (a) c pd pmd

m0

sin (ma + pint mt sin (ma +

K(cL)

= _Cpmtot sin (ma

+ mtot

m0

where a. is angular position of blade number one, positive in the direction of rotation (a = -y).

In the above equations the amplitudes will, for a four-bladed propeller, be zero for all values of in except m 0, 4, 8,

12, 16 V-- etc.. _{In Figs. 2.6 through 2.14 only the blade}

fre-quency content (m = 4) is shown. _{The théoretical free-space}

pres-sure fluctuations shdwn in these diagrams have been calculated

foi thrust, speed, and shaft ç.p.s. corresponding to service

con-ditiöns for the tanker described in Section 3.

Fig. 2.6 shows thé effect òf axial clearance upon the pressure fluctuations, the radius kept çonstant. _{The total} pres-sure amplitude c has its maximum at an axial distance of

'p4'tot

-about 15 percent of diameter upstream of propeller. For /D

values between +0.4 and -0.4 blade thickness represents the, main contribution to the total pressure amplitude. V At larger distances

up- and downstream the contribution from dynamic lift becomes the most important.

12

-Fig. 2.7 shows the effect of tip: clearance upon the total pressure fluctuation for three values of /D. In Fig. 2.8 the contributions from blade, thickness, stationary. lift, and dynamic lift are shown separately.

Figs. 2.9 and 2.10 show' the influence of axial clearance upon the pressure fluctuations at field points upstream of pro-peller. including rad:ii less than propeller radius. Figs. 2.11 and 2.12 show for corresponding field points the pressure

fluctu-ations versus radius,.

Fig. 2.13 shows how the pressure fluctuations due to

dyna-mic lift dèpend upon the angular position s of the field point, radius p and axial position kept constant. The pressure ampli-tudes due to blade thickness and stationary lift are, of course, independent f s.

Teir valueè at

/D =0.39 and p/D = Ö.86 are

c = 7.6 lÖ

p'4t

pLs

Fig. 2.l shows how 'the total pressure fluctuations depend on s.

Calculations have also been caried.out f o' combinations of thrust, shaft c.p.s. and flow velocities corresponding to

Variçus speeds of the model described in Section 3. The resulting'

pressure coefficients turn ot

tobe practically indepèdènt 6f

model. speed.

The computer program also yields the second order component of the pressure fiuct'uatiönè Cm 8, equations (,2.'4)).,. This

coth-ponent turns out to be considerab].y lowèr than the bl&de frequency component. The ratio is lowest for field points

near thè circurnfe±ence of the propeller disc. Èor Û the ratio increases 'from about 3.5 for p/D 0.6 to about 100 for pID = 1.3. At a constant radius p/ = 0.86 it increaès from about 15 for

3.

DESCRIPTION OF EXPERINTS.

Description ofrnodel and testing technique.

The magnitude and distribution of pressure f1uctuatons

have bèen mea.sured on the aftérbody of a l:40 scale inödel o

a

tanker of the following main particulars,

lengthbett.een perpendiculars

250.00 in

breadth

1.12.00 m

draught

15.214 m

blòckcòefficient

0.80

displacement

128552 in3

shaft horepower

30 000 hp

shaft revolutions

110 r.p.m.

ship speed

17 knots

propeller diameter

8.18 in

The form ofthe afterbody is shown in Fig. 3.1.

All measurements

were carried out at DWL, with the mödel self-propelled in the

towing tank.

The rudder was fitted as shown in Fig. 3.5.

The

propeller isshown in Fig. 2»t.

The pressure fluctuations were measured with pressure

trans-ducers mounted flush With -the surface of the hull.

The transducer

is shown in Fig. 3.2.

The relative displacement between the

fer-rite core, fitted to the membrane, and, the coils, fitted in the

transducer housing, is measured by the indicl.ve half-bridge

prin.-ciple.

In connectioi-i with a Hottin.ger KWS/6T-5 carrier frequency

amplifier and a CEC, Type 5_l2L1 galvanomenter recorder the lowest

pressure amplitudes that can be record?d are a few tenths of a

millimeter water column.

The lowest resonant. frequency of the

transduòer membrane sbmerged in water is 260 cps.

The relatively

small thicknessof only 15 mm makes the transdtcers well suited

for mounting in the rudder or in the skeg of -the model.

Fig.

3.'l

shows a picture of the transducer and Fig. 3.5 some of therti fjtted

The relatively large transducer diameter of 22 mm makes it

impossible to record the pressure at a definite point on the hull.

Instead we get a weighted mean pressure over a finite area.

How-ever, since the deflection of the ferrite core is most

sensitive

to pressures near tie centre of the membrane, and not so.much.to

pressures near. the circumference where the membrane is soldered

to the housing, we may expect the dirneter of the "effective"

pressure-sensitive area to

of the order of 10 mm.

Fig. 3.3 shows the instrumentation setup.

The propeller

angular position indicator consists of a segmented ring in. a slip

ring assembly fitted to the propeller shaft.

The oscillograms have been analysed by transferring them

to punched tape by a curve follower for subsequent harmonic

ana-lysis on a digital computer.

The main disadvantage of this

test-ing technique is that it is very time-consumtest-ing.

Due to

turbu-lence and instabilities in the wake the pressure signal changes

from one revolution of the propeller to the next.

In order to

obtain an accurate mean value for the pressure fluctuations the

records from several propeller revolutions have to be analysed.

An improved testing technique based on measuring the

c'rosscorre-lation between the pressure signal, and a known reference signal

is presently being developed.

This teòhnique,, which filters out

the effect of turbulence, will be used in future investigations

of pressure fluctuations at The Norwegian Ship Mode.l Experiment

Tank.

Presentation of experimental results.

The results presented in Figs. 3.7 through 3.18 have been

obtained by harmonic analysis of' ò'scillograuis.

Eaòh point plotted

in the diagrams represents tiie mean value'of independent analysis

of four propeller revolutions, recordedat different times.

Short

vertical lines, one beside each point, indicate the standard

deviation calculated from the fÔür Ïneasurements.

DUe to turbu=

lence a certain level of pressure fluctuations will be' present

even, when towing the model with'the propeller removed,

Such dummy

runs have been carriéd out.

Analysis of the resulting oscillo-'

non-dimensional pressure amplitude c

of about 4lO.

turb

Fig. 3.6 shows the transducer positions _{on the skeg and} transom of the model. We shall reer to each transdúcer positIon

by its number, written in he circles in Fig.: 3.6.

The trnducers

are symmetrically positioned on the port and starboard sdeof

the model. _{Transducers oti the staboard side are denoted by an S} in front of the number,- those on the port side by a P.

The to

transducers directly above the shaft centerline are 'denoted by a C in front of the number.

Thè experimental results in Figs. 3.7 tirough 3.18 are presented as single-amplitude, non-dimensional _pressure coéffici-ents _{of the blade frequency content of the pressure igna1,} and the corresponding phase angles _{The phase angle is here} referre.d to .fieid point angu,ar position (and not to ertica1

blade as in Section 2) so that the blade frequency _{content is}

éx-pressed by

c- {i.(

+ E)

+ _}

pLI. _c

Thus, a _{value of 270 degrees means that the suction peak}

_occr

when the blade centerline points in the direction _{of the field}

point (when y ,E in Fig. 2.1).

For all the transducer positions shown in Fig. 3.6

measure-ments have been carriedout with the propeller in various _axial

positions. _{Figs. 3.7, 3.8, and3.9 show the resulting pressure.} fluctuations onthe transomn.vs. propeller, position, one curve for each transducer. _{The transducers Cl and C2 directly above the} propeller shaft give maximum amplitude when _{/D is about -0.12.} For transducer positions at larger distances from the propeller this maximum is not so pronounced _{There is another interesting} effect, however _{Comparison of-the starboard transducers 53,514,}

S5, S7, and SB with the corresponding transducers on the port side

shows that for propeller positionsgiving _{/D va1uesbelow -0.12}

the pressure amplitude is higher _{on the port than on-the starboard}

side'. For' _{ID values above '0.12 we have the cpposite 'tendency.}

This effect is even more äpparent in Fig. 3.10 which shows the

-pressure amplitude' .istributio over' one frame - for various propeller positions. _{The frame in question is the one with trarducers' 3,}

16

-7, and 8. The explanation.of this effect is that the tangential component of the wake field makes the thrust on one blade higher on the starboard side than on the port side (Fig. 2.3). Thus the

contribution from dynamic lift to the total pressure fluctuation will not be equal on the two sides. The change taking place at

a /D value of about -Ö.l2 is due to the fact that lift and blade thickness contributions are in phase with each other upstream of propel:ler but 180 degrees out of phase downstream. (see also Fig. 2.6).

Figs. 3.11 through 3.114 show the pressure fluctuations

measured on the skeg of the model. They are presented as functions

of the propeller-skegclaraflCe bID. The variation of b has been

obtained by changing propeller axial position. (For definition of b, see Fig. 3.6). For small values of bID the diagrams show a relatively rapid decrease of pressure amplitude with increasing b. For instance, when bID increase.s from 0.2 to 0.3 the amplitudes decrease by a factoi of about 2.5. Couiparisofl betwen Fig. 3.11 and Fig. 3.12 shows that on the skeg above the propeller shaft

the pressure amplitüdS on the starboard side are slightly higher

than those on the port side. Fröm Figs. 3.13 and 3.1'4 we see that below the shaft thé amplitudes are slightly higher on the po,rt

side.

Measurements have also been carried out for various model speeds. Pressure amplitudes vs. model speed are presented in Figs. 3.15 through 3.18. We see that the variation of pressure

coefficient with moe1 speed is very small. For män of the

transducer positions it is within the limits of the standard devia-tion of theméasurefrientS.

From thé viewpoint of practical ship design it is of im-portance to know not Only the, pressure fluctuations at various points on the afterbody. More important is the integrated effect, i.e. the total vibratory force on the afterbody. Thi',',n..be ob-tained by integrating the pressure signals over the afterbody, taking proper account of the phase angle at each point.

Thepres-sure fluctuations measured with the propeller i an axial position

corresponding to b/ 0.148 have béen' integrated numerically over the afterbody to obtain the resulting force fluctuation in vertical

direction. Expressed as a non-dimensional, cOefficient c defined by C = vs F vs Z- i

p nD

_w

where F5 is the ¿ing1e amplitude, bladefrequency vertical force,,

±he result is

C 2.0

vs

Due to the re1tive1y poor accuracy of the measured phase angles

(Figs. 3.7 through 3.1') the result of the integration is not very accurate. A rough eVauatiOn of the standard deviation to be ex-pected yields a factor of aböut 1.6, that is, c is bétween 1.3

l0 and 3.2

l0.

y

From measurements described in (9) t is possible to

esti-mate the blade-frequency vertical bearing fòròe FVbfor this model

and propeller. The corresponding non-dimensional coefficient Övb defined in the same wã.y as ç above, also becomes approximately

2.0

. l0.

_{However, F} and Fb are nearly 180 degrees out of

phase with each other so that their sum is nearly zero. Thus we may conclude t'hat for this model and propeiler the tip clearance c/D 0.27 is about the ideal one for reducing the total vertical excitation force on the propeller and, af±erody. BQth smalier and

larger valués of cID would probably result in higher ertjca1 excitation force. (Tip clearance c is defined in Fig. 3.6.,) It

should be noted that the total vertical excitation force Will show

a definite minimum for a certain vaìue of' c/D only on condition

that the surface force Fvs and the bearing force Fvb are out of

phase. From consideration of the wake fieI one may conclude that as a rule this will, be the case for ships propelled, by a single, 4-bladed screw.

The total transverse, horizontal uf ace force can also in principle be obtained by numerical integration of the pressure

over skeg and, rudder. In the present case, however., we have no

reliable méasuremnts of thé pressures on the rudder. Regarding

pressure fluctuations on the skeg, the amplitudes at corresponding

18

-angles have been measured with an accuracy which i.

hrd1î

suffi-cient to justify an integration over the sur±ace.

Until now, all the experimental results presented in this

section have been regarding the blade-frequency cbntent ofthe

pressure fluctuations.

The numerical harmonic analysi

of the

oscillograms also yields the sedond order component (at twice the

blade frequency).

Ït turns out to be of considerably sm11er

amplitude than the blade frequency content, in most cases by a

factor of 10 or more.

Due to turbulent noise the accuracy obtained

for

he scond order component is very poor.

'4. COMPARISOÑ BETWEEN THEORY AND ENPERIMENTSI, INTRODUCTION OF

SOLID BOUNDARY EACTORS.

As stated in the introduction one of the main intentions

with this report is to provide experimental införmation which, in

cdPrection With fred-space pressure field calculations should be

useful fór prac'tióa], prediction öf propeller-induced pressures on

the afterbody.

We shall here introduce the conept of "solid

boundary factór" S defitied by

.c

-c

S

turb

C

LI. tot

Where

is the nor-dimensiona1 pressure amplitude meured

ex-perimentally on the'

ode1 as described in Section 3..

_cturb

is a

correction for the turbulence contribution toc

'

as mentioned in

pLi.

Section 3,.

It has here been put equal to 0.11

10 _.

_cP'4O

is

the non-dimension1 'pressure amplitude calculated theoretically

as described in Section.2.. When S is not equal 1.0 the reasons

may be

the solid boundary of the afterbody makes the assumption

of free space invalid,

,

the assumpti:onof potential flow on which the theory is

based, is inconsistent with the viscous boundary layer

around thé 'afterbody,

pressure,

inaccuracy of experimental measurements.

Of these effects the first one is believed to be the most pre-dominant, thus justifying the name "solid boundary factor".

On the transom above the propeller the solid boundary

effect may be thought of as an "image effect". If the soli'd bouri-dary were aplane surface of infinite extént the method of images

yields S 2.0. Generally, for convex surfaces the theoretical effect of the solid boundary is to increase the free-space

pres-sure amplitude by a fäctor of between 1.0 and 2.0. An analytical calculation of the effect is ónly possible for very simple geome-tries, for instance a plane surface ór a cylinder (io).

Table i.l shows the solid boundary factor S for the various transducer positions. The S factors for the transducers on the skeg (transducers nurrther 10 to 17) refer to an axial prOpeller position bID 0.186. For the remaining transducers bID

0.8.

In order to indicate the accuracy of the tabulated S values the standard deviations are also included. They correspond to the standard deviations indicated in Figs. 3.7 through 3.18.

We see from Table t+.i that for transducers Ci and C2 S is slightly less than 2. That is, the experimental pressure ampli-tude is slightly less than twice its theoretical free-space value. This comparés very well with the value one might expect from con-sideration of the 'image effect". On the port side of the transom

(transducer positions Pl through P9) we also have S values of about

2.

It is interesting to notice that the values of S for the transducers on the starboard side f the transom are, within the limits of standard deviation, equal to those on the port side. The only exception is transducers number 7. Thus the free-space pressure calculation seems to explain the observed difference in pressure amplitude on port and starboard side as discussed in Section 3.

dis 20 dis

-cussed from the viewpoint of image effects. The effect of the

solid boundary can be treated theoretically only by numerical

solution of Laplace 5 equation with boundary conditions so corn-plicated as to make the numerical calculations very

time-consum-ing. However, since there is evidently a close relationship bet-. ween the free-space pressure field and the actual pressures on the

skeg, the introduction of the solid bOundary fadtor is justified

also here. As can be seen from Table '4.1 it is between 1.0 and 2.0' on the port side (transducers PlO through P17).

Regarding the pressures on the skeg Table '4.1 shows

con-siderable difference in S values when corresponding positions on

port and starboard sides are compared. This is explained by the

fact that the free-space pressure calculations do not account for

the lift on the skeg induQed by the propeller.

The last coÏumn.in Tablé 4.1 shows

diff which, is the

difference obtained when comparing the experimentally measured

phase angles with those calculated froth the theOretical free-space pressure signal,

diff c -' 4tot

+

Thus a negative

dif f means that the experimentally measured pres-sure signal is lagging behind the corresponding theoretical. free-space pressure signal at 'the same field point. We see that for

the transducers on the transom there is hardly any phase dif

fer-ence between theoretical and experimental pressure signal as far as,can be judged from Table 4.1. On the skeg, however, we see that below the propeller shaft the experimental signal is lagging

behind the theoreticalvalue on the port side and leading it on the starboard aidé. Above the shãf t we have the opposite effect. The reasòn for this hase shift is' probably the propeller-induced

lift on the skeg. ' .

'

Fig. 4.1 shows how the solid boundary factor for some of, the transducers on the skeg depends upon the pröpeller-skeg clear-ance bID. We see tht within the limits' of measurement accuracy

there is hardly any definite change in solid boundary factor with

5. MAIN CONCLUSIONS.

On th -transom of the model, non-dimensional pressure

ampli-tudesof up tO 0.016 have been measured.

On the skeg of fije mädel pressure anpl-itues up ±0 0.024 have been measured. They decrease rapidly with inóreasing pröpel.-1er clearance.

The non-dimensional pressure amplitudes are independent of

model speed.

To-tal surfacé force iti vertical direction obtained by inte-gration of the pressure over the afterbody is of the saine magnitude as the vertical bearing force.

Introducing the "solid boundary factor" S defined s the ratio between the experimental pressure amplitude, and ±he

theöreti-cal free-pace pressure amplitude, We generally obtain S

values from about 1.0 to about 3.0.

The value of S On the skeg does not depend appreciably on

pro-peller clearance.

The introduction of émpirical sólid boundary factors in connec-tiori with free-space pressure field calculations sems to be

an effective method fOr practical prediction of

propeller-induced surface forces.

At the Norwegian Ship Model Experiment Tan more extensive investigations with improved experiirìefltä.l tëchtiiqüe are being i'lan1ed. Thèse investigations are expected to provide more

syétematic empirical information on the o-1id boundy factors for various'tyes of propel]èr and afterbodies.

22

-References.

(i) Schwaneóke, H., "State of the Art on Propeller-Exited

Vibrations", 11th ITTC, Propeller Committee Report, (1966).

Pohl, K.H., 'Die durch eine Schiffsschraube auf

beachbar-ten Platbeachbar-ten erzeugbeachbar-ten periodischen hydrodynamischen Drucke",

chiffstechnk, Heft 35, Band (1960).

Tachmindji, and Dickerson, M.C., "The Measurement of

Oscillating ressures in the Viöinity of Proe11ers",

DTMB Report 1130 (1957).

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

Theoretical1 Predicted Pressuras in the Vicinity of a

Marine Prçpeller", Ñaval Ship Research and Develoment

Center, Report 2349 (1967).

Keil, H., "Méssung de Druòkschqankungen an der Aus nhaut über dein Propeller", Schiff und Haken, 17. Jahrgang,

Heft 12, (i965).

-Breslin, J.P. and Tsakonas, S., "Marine Propeller Pressure Field Due to Loadiig and Thickness Effects", Transactions SNAME, Vol. 67, (1959).

(7 Kerwin, J.E., " A Design Theory for Subcavitating Propellers", Transations SNAME, Vol. 72, (1964).

Tsakonas, S., Chen, C.Y.; and-Jacobs, W.R., "Exact

Treat-ment f the Helicoidal Wake in the Propeller

Lifting-Surface Theory", Journal of Ship Rèsearch, Vol. 1.1, No. 3, (1967).

Huse, E., "An Experimental Investigation of the Dynamic

Forces and Moments on one Blade of a Ship Propeller",

Ptoceedings, ymposium on Thsting Techniques in Ship Cavi-tation Research, Part II, Norwegian Ship Model Experiment Tank, pub1ication No. 99, (1967).

(io) Breslin, J.P., "Review and Extension of Th'eory for Near-Field Propeller-Induced Vibratory Effects", Fourth Sym-posium on Naval Hydrodynamics, (1962).

(il) Smith, A.M.O., "Recent Progress in the CalculatiOn of Potential Flows", Seventh Symposium on Naval Hydro-dynamics, (1968).

(12) _{Tsakonas, S., Jacobs, W.R. and Rank, P.H., Jr., "Unsteady}

Propeller Lifting-Surface Theory With Finite Number of

Chordwise Modes", Journal of Ship Research, Vol. 12, No. 1, (1968).

Free-space pressure field due to.hiade thickness.

Fig. 2.1 shows the coOrdinate system, its Origin coin-ciding with propellér center. Field point cöòrdinates areS , n,

, or, in cylindrical coordi.ntes ,

p,

. Source point

co-ordinates (on the propellèr blade) are x, y, z, or X, r, e. The coordinate system is assumed to be fixed in spacé.

The blade tijickness is represented by continuous sheet of souröes and sinks along the mean line of the profile, .ihich is

assumed to be symmeÌical. Source strength pr. unit area is

a(r,$), where s is the chordwise coordinate, Fig. 2.2. At the sur-face of thin profiles the velocity component jricluced by blade

thickness paralleli to the mean line may bé neglected. The volume of fluid emitted from the source element pr.. unit time is then

(Fig. 2.2):

2vsr

= 4s.r

(Al.1)

where Vn is thè ve1ocity component normal to the section. The

boundary condition of no flow through the surface of the profilè

yields

1 at _(Al.2)

2 as

where T. 1 the thici

Ss

of the profile. From (Al..1) and (Al.2)

one obtains

a (r,s

.S _a

3S

In the case of a ship propeller the chordwise velocity U is

approximately

U .2irnr

s

cos8

where n tiuiber of propeller revolutions pr. econd., 8 pitch angle.

2i

Thus

nr

a(r,$)

2 cos B

The velocity potential at a field point (,p,c) due to the source

distribution over one propeller blade is

center-line (Fig. 2.1),

K _{= rake angle.}

Substitution of Dsf from (Ai.5) and a(r,$) from (Al.3) into (Al.'4) yields, when summing up the contributions from each blade,

z rt er(r)

=Nl

j, rh e1(r) r2 n 2 cos B

((c

ef-

-Kr)2+r2+p2- 2rp

where

(,p,c)

velocity potential due to blade thickness of

propeller with Z blades,

y _{angular position of blade number one (Fig. 2.1),} 61(r) value of' e at leading edge,

(Al. 3) (Al. 6) where where Dsf rh rt si D5f P e =

7t

5(t = rh si distance between hub radius, tip radius chordwise coordinate chordwise coordinate ((-x)2 + r2 P

t(r,$) dsdr

D sf

source point and field point,

of leading edge, of trailing edge.

+ p2' 2rp cos

Kr)2 + r '1- 2 _{2rp cos (c_e))} (Al.5)

of source point referred to blade

((e

pitch,

or

er(r) value of o at trailing edge.

The instantaneous free-space pressure signal

at the field point cn be obtained from Bernouilis equation:

-

i!

+ +

UT2 = C.(t)

-t 2

p - UT2 + c(t))

t 2

where density of water,

UT = total velocity at field point.

The integration constant C(t) may be a function of time, but,

assuming irrotational flow, it is invariant with respect to.space

coordinates. We want to caiculatethe induced pressure signai

at a certain field, point at various times. Since this is

equi-valent to calculating the instantaneous pressure at field points

of various values of , we may still neglect C(t). We shall in-vestigate the case where UT is the vector sum of a parallell stream

of velocity U in e-direction, and velocity components u, u, and u inducd by the pröpelier in -, p-, and e-direction respectively.

2

(U + u)2 + 2 +

= U2 + 2Uu +

By linearisation, i.e. assuming that the induced velocity compo-nents are small compared to U, the above equation may be

approxi-mated by

u2 +21Ju

Since u

-we get by substitutioi into (Al.7)

p _P

_(:E

-2 +

The term gives only a constant pressure and may therêfore be

UT

26

-2

+ _uc2

negleçted here,

p = + u

Introducing the non-dirnnsional pressure coefficient Kt due to blade thickness defined by

-pt

p n

w

p

where D propeller diameter, one obtains

K (ac + Pt - n2D2 u K = pt z i nD M=l r

e(r)

ft

ft

rh 01(r) i

,2ar

as {rrn rp sin(_o_y_(M_l)-I)+ _{(_ pf. - Kr)}}

_dOdr

(Al.9)

For ogival blade sections the numerical calculation _of Kt from (Ai.9) may be carried out in a straight-forward _manner by application of Simpsons formula _{or a similar method of}

numeri-cal integration.. _{Se&tions with a rounded nose, however, require} special attention as -to the numerical integration at the leading

(Ai.8)

The velocity potential in (Al.6) depends on time only by the time variation of y. Thus

a

a-y

at - ay at

-

2irn-ay

By carrying out the differentiations of (Al.6) with _{t'espect to y} and , arid substituting the result into (A1,8)q _{one obtains,}

edge where becomes infinite. The computer program developed

at the Norwegian. Ship Model Experiment Tank handles this problem in the following way: The sourçe distribution over the first 10% of the chord near tlie leading edge is approd..rnated by a line

source at a point 3 of chord c from the leading edge. (Fig.

2.2). Ïntegration of (Al.9) .in O-direction over the last 90% of the chord is carried out by a. 10 intervals Simpson procedure, the

lope }. being calc4lated frÒm values of r taken from propélier drawing.

The strength m of the line source is

s0.lc + s1

m

f

a(r,$)ds

ssl

where the integra..l Is taken. over the first 0% of the chord. Substitution of (Ï',$) from (Al.3) yields

s=.0.ic + s1

f

nr 2 cos

ss1

nr 2 cos 0.1 28 -{rrnrp sIi(c- _(M_l).2L)+

(_Pf-_Kr)}dOdr

. Z r

0t(1)

1

r

( I

2 22

-

T K pt -I (P

+4ir

r ) L.Tr2nD2

M1 rh 001(r)

where T is the profilé thickness at 10% chord station.

Thé pressure due to the line source is derived in the same

way (.A1.9). Writing 2rrr/V'P2+14n2r2 for cos , we have the final

formula för numericál claculatiôn of K_pt

.

z

M1

_rh

((c?

;03

Kr)2+r +p2-2rp cos(

_e0

03....1_(N_i).!L)y3 2

2 U

e003

{lrnrp

SIfl(-O003-y-(M-i)-)+-271

Kr)}dr

(ALlO)

271T01/P2+LT2r2

- 30

APPENDIX 2

Free-space pressure field due to stationary blade loading,.

Each propeller blade is here represented by i infinitely thin lifting surface coinciding with the rotating surface

H = x -

(e+2,nt)L

- Kr O (A2.1)

27F

Later on we shall inke use of the unit normal of this surface. It has the components

2'rrr

2r2+P2+2r2K2}_

-27FrK .7F2r2+P2+7F2r2K2}_

{;.22+p244;.222}2

in x-, r-, and e-direction respectively.

Using the same notatIon as in Appendix i the linearised equations of motion are

u +

--_L

-

L

. a at a au

u -

+

at

au

au

TI ₊ C 'J a; at 1 Pw 1 pw

a(pu)

3p By calculating the trms p a

where p is now the pressure at the field point induced by

blade loading. The continuity equation is

T2 '

(P .

4Ø),

and au i p a -(A2.2) (A2 .3) (A2 L;.) O (A2. 5a)

from (A2.2), (A2.3), and (A2.Ll) respectively,, and by using. (A2.5a)

we find that the pressure field

p(,p,et)

satisfies :th potential equation

+ . . ..L. . + .

L

.

a2

p2 2

Within the frame of linearized, theory we have th? following

boun-dary conditions which have to be satisfied L

p = at préssure side of blade surface., L

p - at suction side of blade surface,

p 0 at infinity,

where La is lift force pr. unit area, in thé opposite direction of the unit normal i of H = O _{Particular solutions of (A2 5b) are}

pressure poles and dipoles distributed over the propeller blade. According to boundary conditions 1) and 2). above, a di'stributon

of pressure dipoles in the direction of _{is suf,fiient in the}

present case. . V V

The pressure p _{at the field point (,p,c) due to a pres-.} sure pole of strength S at (x,r,e) is by definition

pp

(-x)2'+p2+r2-2pro(ce)}

V.(A2 .6)

The corresponding pressure due. to 'a pressure dipole in the direction of is obtained by differentiating _{p in (A2.6) with} respect to (x,r,e) in the direction of i. _Thus

V ap ap ap n + . ___ + . V Vd X _r

_.e

-ar rae Sd (27rr(_x) _{+ 2lrrK(r...pcos(c_e)')_ppsin(ce)i} VV

4(2r2+P2+42K2)*

. {(

)2+p2+r22prcs(_e)

(A2.7) where the pole strength S _{in (A2.6) has now been replaced by}

the dipole strength _Sd.

V

V

By integrating the pressure field of asiñg-I _{dipole over}

{(-x)2 +

2 2

+r

32

-two infinite plane surfaces, normal to the dipole direction, and one on each side of the dipole, we find that the: dipole exerts a force equal. to its strength in, the, direction of its axis. The

dipole strength pr. unit area of propeller blade therefore has to

be equal to La; By integration and sumrna-Uon over the blades we get the pressure field due to stationary bl4de loäding. Division

by pn2D2 yields the corresponding non-dimensional coefficient,

K

2 2'

J f

L(0)

(22(l+2)+P2)

D.

_Ml

(2rrr(-x) +2TrrK(r-pcos(c-e)) - Ppsin(c-e))' df (A2.8)

where the integral to be taken over the area of a blade.

We assume the, lift distribution over the chord at any radius to be a constant over the first 80% of the chord, and. from

there a linear decrease to zero at the trailing edge. (This is consistent with a NACA.a 0.8 mean line, at ideal angle of attack and' ideäl fluid. This profile is 'frequently used iti propeller design.) We further assume an elliptic circulation distribUtion

e)/(e.e0.8) for 008 <

O <

e 0 8value of e at 80% of chord measured from leading edge.

T propeller thrust.

Substitution of La(0) from

(A2.9) and transformation of the double integral in (A2.8)' to an integral over e ánd r yield the final fornula for K5

-3/2 2rpcos(E-e) }

in radial direction. lift distribution

'These two assumptions lead to

, 2r-r -r 2 8'T(l (.. t) ) F(e) the following -L '(r,O) a t- IL , (A2.9)

0.9.(8_.)(r2_rh2)Z.

where 3. fore1 < e _< F(e) _(A2.lO)

1< = PS z T 2 2- 2

2)

.13

0.9

COSK ff

DZ(r

_rh

rh

01(r)

-

2r-r -r

h

t)

r_rh

2.

.

F(0)

{(_Pf- -Kr)2+p2+r2-2rpcos(-O-y-(M-1))}3"2

(2iîr(_Pf- -Kr) + 2,TrK(r_pcos(c_e_y_(M_1).1L))

- Pp

sin(E-O-y-(M-1)--)) 'dÓdr.

(A2.11)

eJr)

3L

-APPENDIX

Free-space pressure1 field due to dynamic blade loading

Tha dynamic component of the lift on the propeller blades can be represented by a lifting sürface of pressire dipoles. The

difference from the stationary case is that the dipole strength is flow not

Qfly

a fcion of r änd e but alsO of tne (or

angu-lar position iÎi the wake field). (A2.8) is thus still valid if

La(r,e) is replàced by L(r,e,yM> where

(A3.l)

By unsteady, three-dimenional lifting-surface theory it is possible, in principle at least, to calculate theoretically L(r,O,yM). However, the numerical solution of this problem is

very complic.ted an time-consuming, even with the.aid of tO-day's fastest computers. For lOw "reduced frequencies" it might be a realistic appoLthaioti to apply the chord-wise lift distribution

of a two-dimensional, infinitely thin wing of zero camber. In

this case the lift pr. nit area would be

L(r,O,yM)'

f(r,yM)I(ët_e)/(ë_),

(A3.2)

where f(r,YM) is some function of r and Aflo,ther alternative is to represent the dynamic lift

by a line distribution of

pres-sure dipoles at the quarter-chord. (The quarter-chord is hosen

because it represSnts the point of application of a lift force

distributed accOrding to (A3.2)). Even the first alternative is

only a rough approxitiatiön for prcticãi ranges of reduced

frs-quencies. On the other hand, from the viewpoint of computer-time

economy the second altenative is to be prefetred because it

eliminates the chordwise integration. Based on these

consider-ations the lifting lne representation has been chosen here. This approximation is expected to leaq to an overestimation of the

pressure flctuation due to dynamic lift, the overestintation

being most pronounce1 at field points close to the propeller and

for propellers with particularly broad blades

distribution of dynamic lift can probably be obtained from an

investigation of the actual wake pattern. In this way it is

possible to apply different radial distributions at the various

angular positions of the propeller. However, in order to simplify

calculations we shall here apply the same (constant) form of

radial lift distribution as in the stationary case, i.e. the one corresponding to elliptic circulation distribution.

With these-approximations for the lift distribution we

can now obtain from (A2.11.) an expression for the non-dimensional

coefficient Kâ of the instantaneous pressure induced by the

dyna-inic component of lift. Since the chordwise dipole distribution of (A2.11) is now replaced by a concentrated lifting line at

quarter-chord, we can carry out the integration over O and get

K = pd Z rt 1 3 2 2 2 2) cosK D (r -rh _M=l rh

2r-r-r

2 h t) J

r_rh

O -3(2 O;25 Kr)2 + p2 + r2

_{- 2prcos(c_OO2S_YM)}}

00 ₂₅ (27rr(-P 2; r + 2TrrK(r-pcos(c.-00 25 -dr. (A3.3)

where td(YM) is thé instantaneous value of the dynamic component of thrust on one blade at angular ppsition y. given by

36

-Table 14.1. Solid boundary factors.

continued Transducer position ID p/D e (degrees) dif f (degrees)

cl

-0.1147

0.757

0.0

1.85±0.08

30 ± 6

C2

-0.328

0.710.

0.0

1.89±0. 08.

31

i 10

P .1

0.635

1.020

11.3

₀₉₁₀₇

(+2. 0

-14 ± 50

P2

0.2414

0.930

12.4

2.401.5

11. ±10

P3.

-0.1147

0.8141

13 8

2.48±0.13

13 ± 7

P4

-0.328

0.800

14. 5

2 .47O.15

13

i

5

PS

-0. 513

0.732

15.9

1. 640 .23

52

i

7

P6

0.244

1.720

20.8

_3.212

1ft.0

0

13 ± 15

P7

-0.147

0.985

22.7

2

30 0,2 Û

9

i 10

PB

-0.147

1.150

29.3

2. 060. 60

-7 i

10 P .9 -0.114 7

1.326

33.7

_lOQ5

f+1. 5

-1 i 10

P 10

-0.293

0.510

8.8

1.830.b6

28 ± 8

P 11

-0.312

0.3614

11.6

1.74 O

07 26 ± ₅

P 12

-9.145 9 0.3 86.

22.3

1. 700. 17

.35

8

P 13

-0. 26 9

0.227

18 8

1.340. 04.

34 10

P 14

-0.239.

0.223

1614.7

1. 10 0. 03

-4 i 10

P 15

-0.325

0.380

171. 9

1 3'4b. 05

-14

i

6

P 16

-0.474

0.387

166.1

1.640 .27

-22

± 6 P 7 -0. 1411

0.540

174. 8

1. 3'4 ± 0.10

-17 i

6

s'

0.635

1.020

-11.3

1.651.0

21 ±25

S2

0.244

0.930

-12.14

2 .14Ó0.40

-9

± ₁₀

S3

-0.147

0.841

-13.8

2 .55i0.10

8 ± ₅

S4

-0.328

0.800

-14. 5

2. 54O .13

'4

i

5

S5

-0.513

0.732

-15.9

1»470.45

2

i

10

Table '4.1, continued Transducer position pfD (degrees) S _diff (degrees) S 7 -0.147 0.985 -22.7 1.34*0.30 31 ± 15 S 8 -0.1'47 1.150 -29.3

3.70l.00

9 * ₂₀ S lo -0.293 0.510 - 8.8

2.260.06

-17 ± 5 S li -0.312 0.364 -11.6 2.2'4O.O7 -20 ± 5 S 12 -0.459 0.386 -22.3 '4.2OO.5O -15 ± 5 S 13 -0.269 0.227 -18.8

l.O20.lO

-14 5 S 14 -0.239 0.223 -164.7

1.O5O.O7

'43 ± 5 S 15 LO. 325 0.380 -171.9

1.O8O.O7

38 ± 7 5 16 -0.474 0.387 -166.1

1.26O.20

'47 ± 7 5 17 -0.411 0.540 -17'4.8

0.67O.08

31 * 7

port

field point (

source point (xyz)

Direction of rotation.

Fig. 2.1.a. Defmntion of coordinates.

a - -'

y;n

starboard

140

60

e

Fig. 2.2. _{Source-sink representation of blade thickness.}

Fig. 2.3. _{Instantaneous thrust on one blade, in percent of} mean thrust on the blade.

vn _sourcé

_sheet

120

100

U

I IL

₁₁₁

n_______

Uil_____

MII

III

H.

:Jj

. i

ii

fTWfl

R

C

f,ml

I-,

1.1 0 _1.59 2.62 3.46

_{4. g}

5.52 6.37 7.30 8.23

tmax

[mm] Fig. 2.'4.

Propeller, Tropst B series, number of blades Z pitch P

ll'4.52 mm (constant), expanded blade area

ratio

0.6

o

0.4

0.2

o

- 0.1

0.3 _Kt. .1OK,, 0.2 90

. K

..-'10

Kt.

Fig. 2.5.

Open water performance curves.

o 01 0.2

03

0.4

0.5

300

250

200

liso

loo

[dei

f18

DA

'P4d 0.8 i .2 1.6

Upstream

downstream

Cp4totl

1.2 0.8 0/. o 310 'p/.tot.i 270 230 190 1.50 110

[d!g]

_'6'--Q33

0.5 1.0 1.5

VD

i 01.6

1.2 0.8 0.4 252.5

225

197.5 170 p4 Jeg] Cp4s '4 t o t 'P45

Fig. 2.8. Blade thickness, stationary and dynamic lift contri-butioñs vs. radius.

0.5 1.0 1.5

o

4tot1 290 280 270 '26 O 25O 0.15

0.35

0.55

0.75

Fig.

2.. 9.

Effect of axial c1eance for various r.adii.

o

2 50

0.15

Fig. 2.10. Blade thickness, stationary and dyn&iiic lift contri-bùtiorisvs. axial clearance. .

0.55

O.75

-X.:

300 s 290 280 70 260

c=0

CpLtot

Lp4

LS

Fig.

11

Blade thickness, stationary and dynamic lift contri-butions vs. radius. o

300

[deg]f

290 280 '270 260

250'

0.2 . 0.3 QL 0.5 - 0.6

i

2 o

280

270 260 25 q

Fig. 2.12.

Blade thickness, stationary and dynamic lift

contri-butions vs. radius.

C. P4 d 0.2 0.3 0.5 o. s i T 4

I

3 300 [de9]Ì 290

P4 d 8 0 180.

4d

[deg] 0

--90

-180

-180 90 O 90 180

£ cieg]

Fig. 2.13. _{Pressure fluctuations due to dynamic lift vs. field} point angular position c.

2

cp4tot

i

o

360

180 90 o -180 -90 0 90 ₁₈₀

c[deg

Fig. 2.14. Total pressure fluctuation vs. field point angular position .

Fig. 3.1.

o

o

5 DWL

copperberyllium corrugated membrane

!

water-tight

seaL

, -

-ferrite core

coil windings

wiring

pressure

transducers

propeller

position

i ndi cat or

Fig. 3.2.

Pressure transducer.

Hottinger

KWS/ 6T-5

amplifiers

130 cps

Low-pass

filters

Fig. 3.3.

Instrumentation setup.

CEC

Fig. 3.5. Some of the pressure transducers fitted flush with model surface.

05 DWL

Fig. 3 .6...:.. Transducer positions on skegand transom, projected

in vettiòal and horizontal plane respeätively. Numbers in circlès are position identifications. Distances are in mil]4meters on model.

5 h 3 2 10 'Pc o -0.25 - 0.20 - 0.15

010

- 0.05 0 0.05 ai o

Fig. 3.7. _{Experimental pressure fluctuation} vs. axial pòsition of propeller.

V S8 V

- PB

N1

VI

-.

/

7

.--.-

'SI.-... 310 250 190 130 70 10

o'--

v_____.-.,

-.--.-.--.

-f.

i O 12 8 L 340 310 280 250 220 i g0 160

/

/

e

S3

V SL

0 P3

V P4

-0.5

-0h

-0.3

-02

-0.1 0 0.1

Fig. 3.8.

Experimental pressure fluctuation

vs. axial position of propel1er

12 9 6 3 o 70 'Pc lo s

-ìg-.i1-

.. I JD 0.6 - 0.5 - 0.4 - 0.3 -0.2 0.1 0 0.1 [%]

Fig. 3.9. Experinenta1 pressure fluctuation vs. axial position of propeller.

//

j7

YI[

/

..[

//

/

'\

r

/

/

_D[

L

31 0 250 190 130 70

N.

Io V ----o r

Fig. 3.10. Distribútion over one frame of experimental pressure amplitudes, one curve for each axial position

103.0 2.5 2.0 1.5 1.0 0.5 o 330 300 270 2/.0

JW

.1/

/

N

VI-ai 0.2 0.3 0.1. 0.5 [YD]

Fig. 3.11. Experimental pressure.fluctuation

on starboard side of skeg vs. clearance.

F

V 'Pc

Ep4 102.2.5 1.5 1.0 0.5 280 265 250 235

opio ---AP12

-Dp13

\

\ L.

"A N

Pig. 3.12. Experimèñtal ressure fluctuation on port side of skeg. vs.

clearance. 0.5 04 0.3

W

'Pc 295

1.2 OE9 0.6 0.3 O

\

\

A

/

Q2 0.3 0.4 OES /b1

-Fig. 3.13. -Experimental pressure fluctuation

on starboard-side of skeg vs.

clearance.

300 Pc 290 280 270 260 250 240 230 220 .210

1.2 0.9 0.6 0.3 o 320 290 260 230

oP1 --P16

--VP15 DP17

\\

\

E:

0V;H

o

--

vi

-.

a

.

i;

N

-. .

'-I

L.

v'/

0.1 0.2 0.3 0.4 0.5 _[Yo]

-Fig. 3.14. Experimental pressure fluctuation on port side of skeg vs.

clearance.

'Pc

9 5 1 0i 2 9 3 1.2 13 1.4 1.5 16 VM [r%] 13.5 14.8 16.0 17.2 '18.5 19.7 VS Cnot

Fig. 3.15. Experimental pressure amplitude

'vs mode spéed. . . . ..-.'

..c. i

.---

V P 6

C2 - P9

i-Fig. .3.16. Experimertal pressure amplitude vs. rnoe1 speed. . 0' .11

-135 12 1.3' 14 15 1.6 _VM[m.4] 14.8 16.0 17.2 18.5 19.7 V5 knotsJ Cp4 1 0 1 2

Cp4 ¡

iO4

3 2 o 2.5 w 1.5 1.0 0.5 VM [m/] V5 [knots]

Fig. 3.17. Experimental pressure amplitude vs. model speed. O 11 13.5 1.2 1.3 14 15 16 w [m,4

]

14.8 16.0 17.2 18.5 19.7 VS [knots]

Fig. 3.18. Experimental pressure amplitude vs. model speed. 8

Opi

0 P8

VP2

si

--VS2

_{--AS7 ..P7}

-- U S

/

.._U

N

J

Vpii

_P12

0P13

sio

--ysli -

_{- OPiO}

-

.----::: .9_

IA...

1.1 1.2 1.3 1.4 1.5 1.6 13.5 14.8 16.0 1 7.2 18.5 19.7 Cp4 i o2ao

JA

o

vP11

APiS

Jo.

V11

_j

V

'-T

1*

_'

'sil

*

S15

0.1

0.2

0.3

0.4

0.5

0.6

_b,t3

0.1

0.2

0.3

0.4

0.5

0.6

_b/D

Fig. Li.

Solid boundary factor vs. propeller-skeg clearance.