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(1)

INVESTIGATION ON THE EFFECT

OF THE PROPELLER RACE

UPON

THE PERFORMANCE OF RUDDER

( Hydrodynamical Research of Ship's Rudder )

The Second Report

By Shojiro OKADA, Ba. Eng. (Kogakushi)

The Technical Research Laboratory

Hitachi Shipbuilding & Engineering Co., Ltd.

January,

1959

Lab.

...,CiseetrA

c i

Techv

g°4JSCiirlt,Or

(2)

INVESTIGATION aN. THE EFFECT OF THE PROPELLER RACE

UPON THE PERFORMANCE OF RUDDER

(Hydrodynamical Research of Ships Rudder .

= The Second Report

By Shojiro OKADA, Ba.Eng.

The Technical Research Laboratory

Hitachi Shipbuilding & Engineering Ca. ,Ltd.

Table of Contents

Abstract

1. Introduction

2.. Theoretical Consideration 4

2.1 The force acting upon the rudder placed in the propeller race

2-2 Calculation of propeller race 13

2.3 Results of the numerical calculation of force acting upon the rudder

a

Experiments on models

3.1 Experimental apparatus and method

3.2 Results of the measuring of propeller race

and the consideration for it. . 44

3,3, The Result of measuring of the force acting

upon the rudder in the propeller race and

its consideration -H.- -7- 6-0 Conclusion --- 95

K

1 3 5 33 40 40

(3)

ABSTRACT

In the previous report (the First Report), the

per-formance of open rudder, particularly the effect of the

velocity of steering, was dealt with, clarifying its

various specific features. In this Second Report,

description will be made on the effect of the propeller

race bearing influences upon the rudder performance,

which is also the principal object of the present

inves-tigation, from both angles of theoretical calculation

and model experiments.

The points made clear as the result of theoretical

calculations are as follows:

The most important factor governing the force

acting upon the rudder in the propeller race is the

velocity increasing coefficient of the propeller race,

succeeding to the advance speed of the rudder and helm

angle, in the scope of the present series of theoretical

calculations. On the rudder, there is acting about 70

80% of the total energy of water which has been

accelerated.

The effect of flow angle of the propeller race

against rudder is insignificant in an instance such as

the present calculation where the rudder has a

symmetrical section.

Even when the longitudinal position of the rudder

(4)

may shift backwards from the position of the propeller

disc by 0.5-1.50R (R:radius of the propeller), no

significant effect will be produced on the force acting

upon the rudder.

In consequence of the experiments on models conducted

in the circulating channel, the following points have

been made clear:

The coefficient of increase of the normal force

of the rudder caused by the propeller race multiplies

in proportion to the 1.5 power of the slip-ratio of

of the propeller.

By the propeller race, the position of the

center of pressure travels forewards, and the amount

of travel will be the larger, as the slip-ratio is

the greater.

Even though the longitudinal position of rudder

may shift backwards from the propeller by 1.00-2.00R

(R:radius of the propeller), it may well be presumed

that the performance of the rudder will not be changed

to any significant degree.

When the position of the rudder is transversely

(port or starboard) biased against the center of the

propeller, or when there exists a marked difference

between the diameter of the propeller and the height

of the rudder, it is enabled to know about the

(2)

(c4

.01

"I - (a) (b)

(5)

-quantitative tendency by making examination separating

the area upon which the propeller race directly affects, from the area other than the above.

(e) The effect of angular velocity of steering is

noted to be acting even in a rudder placed in the

propeller race.

1. INTRODUCTION

In the First Report of the present investigation,

the author described about the performance of open

rudder, and various factors affecting such performance.

Particularly in connection with the effect of angular

velocity of steering, he carried out theoretical

calculation along with the experiments on models in the

circulating channel, and by bringing both data into

comparison, the relevant characteristics have been

made clear.

In the present report, the problems pertaining to

the performance of the rudder located behind the

pro-peller, namely, the main subject of the present

investi-gation will be treated from both angles of theoretical

(6)

2. THEORETICAL CONSIDERATION

In discussing the performance of rudder behind the

propeller, it is ideal to consider on the basis of the

effects on the rudder produced by the resultant velocity

potential of the propeller and the rudder. However, a

problem of such mutual interference cannot be treated as

a stationary theory, hence in the present instance, the

effect which the propeller brings upon the rudder alone

will be considered on the assumption that the propeller

is not to be affected by the rudder in whatsoever way.

In case the rudder is located behind the propeller,

the velocity and flow angle acting upon respective parts

of the rudder may diversify in accordance with the

pro-peller race respectively. For the reason, this cannot

be handled as a problem of two-dimension in which the

rudder is assumed to have an infinite depth. On the

contrary, it should be treated as a three-dimeaional

problem that needs due consideration for the depth of

the rudder, and naturally, the trailing vortices should

also be given consideration as they are generated from

respective parts of the rudder.

Prandtl developed his "lifting line theory" in which

he presumed that trailing vortices flow infinitely

backward from both edges of bound vortex which is fixed

to the aerofoil, thereby to satisfy Helmholz Law that a

(7)

vortex does not terminate in the fluid, and that

series of vortices are consistent in their circulation

at any point of their section.

It is also a known fact that Prandtl theory holds

true no longer, when the aspect ratio of the aerofoil

gets smaller (generally below 3). To replace this

theory, the "lifting surface theory" made advancement,

and more recently, Mr.JonesMand Mr.Lawrence(2/developed

the theory applicable to the instances where the

aspect-ratio is small. When the aspect-ratio is so very small

as an instance of a rudder, it is suggested that these

improved theories should be applied. But these theories

accompany difficulties to strictly satisfy various

conditions attached to them, if ever complex propeller

race is to be treated. Hence "lifting line theory"

comes to be considered sufficiently simple and effective

to know the tendency of the force acting upon the rudder.

This is why the theory was adopted in starting with the

calculation in the present case.

2.1 The force acting upon the rudder placed in the

propeller race

In case of an aerofoil put in a uniform flow, the

"lifting line theory" assumes that the trailing vortex

(8)

disregarding the mutual effects of the trailing vortices

themselves. As shown in Fig.1, in the reverse direction

to the uniform flow u0, the axis x is asslimed, and let

the ads

be drawn in the direction of the height of

the rudder, and the axis y,perpendicular to the x-z

plane. When the rudder is replaced with a vortex line

conforming to the axis z, the trailing vortices will

extend infinitely backward in the negative direction of

the axis x.

When FR represents the circulation of the rudder,

the intensity of the trailing vortices flowing out of

dF

the point z' is

dz,ozc

R and the velocity of the direction

of y induced at the point z by the trailing vortices of

semi-infinite length is given by Biot-Savart theory as

follows:

1 dER

4K dz' z - z

Hence, by integrating this to the range of the whole

height of the rudder, the induced velocity at the point

z is found. dz' 1 b CITA Vi= ( 1 )

47rJrb

dz' z -

z'

2

where, b is the height of the rudder.

In general, when the velocity uo flows in against

the rudder, the velocity at z will be u, by the

afore-(6)

(9)

-mentioned induced velocity vi, whereas the angle of

incidence a decreases by al(Refer to Fig.2) In

respec-tive positions of the rudder, the force acting upon the

rudder will be PuirR, when the theorem of Kuttd-Joukowski

in 2-dimentional aerofoil theory is applied on the

assumption that the stream is 2-dimentional, and its'

direction is perpendicular to ul. On the 'other hand

When the tangent of the lift coefficient CL against the

angle of incidence a' is assumed to be -cAL

k, the

circulation round the rudder

rx

will be expressed by the

following equation:

rR__,i-uock( a

=a1)

where* is the chord length of the rudder.

Hence,

i)

= iu ck ( ty - (2)

vo

Now, in

order to find the rudder performance of

3-Fig.1

Fig .2

c

(10)

dimentions, it is necessary to find the FR for

respec-tive points of the rudder in the direction of the height,

or the distribution of circulation. For the purpose,

the integral equation consisting of (1) and (2) above

must be solved.

As this equation is rather hard to solve analytically,

the method to approximately solve the equation by

developing it into Fourier series is employed, although

there are in practice such methods as Mr.Glauert's(3),

Dr.Lotz's4), Prof.Tani's(5), etc. These methods are almost

similar to one another, and are considered that either

one will meet the requirement, but for the present, Mr.

Glauert's method is utilized:

Firstly, z and z' are set as follows:

Z = -

COS 0 ( o 0 )

(3)

z'=---cose' ( <01<n.

)

2

z represents the coordinates as measured from the

center of the rudder upwards. They vary from 0 to it

the whole height of the rudder from the bottom

edge to the top, and so the circulation around the

rudder can be expressed in the Fourier series as

shown in the following equation:

FR = 2bUIAnsin nO (4)

The values of coefficient An are determined by the

(8)

III

(11)

two fundamental equations connecting rR and vl, and the

series should be chosen for the circulation TRso that

it satisfies the condition that the circulation falls to zero at the bottom and top edges of the rudder.

U in the equation (4) is the uniform flow assumed so as

to make it possible to calculate the induced velocity,

and then the induced velocity at the z point of the

rudder, that is, the point 01, is given by the following

equation:

U InAn cos n O' sin nO,

v1( 01) d 0 U InAn

7r4iu cos 6e-cos 0, sin 0

cos nO'd0' sin

no

dr7 =

cosOLcosp

sin 0

Accordingly, the equation below is established for

the Aeneral point z on the rudder:

visin 0 -U InAnsin n (6)

In this case, the direction of vi gives rise to

question, but it is considered most adequate to assume

that the direction of the vortex flowing out from the

rudder corresponds to that of the stream line.

However, on this assumption, the direction of the

vortex flowing out is caused to vary at respective

points along the direction of the height of the rudder,

whereby it is made entirely difficult to solve the

(5)

(12)

-above integral equation. In the present case, as it

seemed not to be deviating much from being adequate to

presume that the vortex flowing out from the rudder

would coincide with the mean of the directions of

incidence into the whole height of the rudder after

the vortex has flown pasta certain distance rearwards,

that is, the direction of the longitudinal center line

of the vessel, such a condition was decided to be

assumed for the present calculation. Then the direction

of vi is perpendicular to the longitudinal center line

of the vessel and also along the horizontal direction.

In this case, the components of the flow of the z

direction are of the values mutually offsetting between

the upper and lower halves of the rudder, and are

considered not to be affecting the normal force and

moment of the rudder, hence they are disregarded here.

In consequence, Fig.3 below shall replace the

afore-mentioned Fig.2:

Fig.3 (10)

(13)

u = mU + UsinwInAn

sine

(8)

In this case, rR is FRiuick(cy-al), therefore, the

equations (7) and (8) are substituted, and so,

where, co =the angle that is constituted by the

longi-tudinal center line of the vessel and the

proprller race.

ai =the amount of diminution of the angle of

incidence by vi.

aR =the angle formed by the zero lift line to the longitudinal center line of the vessel, that is the helm angle.

Now, assuming that for the angles aR and co, the

anti-clockwise direction as measured from the longitudinal

center line of the vessel is positive, and also vi is to

show positive or negative depending upon the value of

(aR-co), the amount of diminution of the angle of

inci-dence by vi is expressed by the following equation:

v

cos w (7)

u1

If uo=mU,(m is to represent the velocity increasing

coefficient by the propeller race), the following will

be given:

u =u,

vi sin so

sin no

(14)

-ck

sin n 0

ckmU FR

UcoscpI RA.

2

sin 0

2

ck

sin n 0

±-UincoXnA

2

sin 0

On the other hand, by the equation

( 4 ) ,

FR=2bU ZA,, sin nO

and writing that the right side of both equations are

equal to each other, the following equation is obtained:

MAnsin nO (nit( cos yo - asinw) + sin

0 ) =m iice sin

(9)

ck

where,

=

4b

By substituting the known. values for respective

points on the rudder, for example, p-number of points

on the rudder in the equation

( 9 ) ,

as obtained above,

p sets of simultaneous equations are constructed, and

by solving them, the coefficient An can be determined

to

n =1p .

When Anis decided by this procedure, the lift aL and

the drag OD can be expressed by the following

equation:

o L =p b2 U2 { insin 11-41 sin nO

singo( Ink sin no ) (

Asin no )

(10)

OD= pb2U2 cos( lik,sin no )(

nAn sin nO ) dO

Then, ON and OT will be worked out by the

equation

below:

(12)

(15)

-aN= aLcosce + eDsin a 1

(11) aT = aLsina - oDcoscy

When the distribution of the propeller race, or the

velocity increasing coefficient and flow angle

obtained by the above procedure is available, the force

acting upon the rudder in the propeller race can be

found.

2.2 Calculation of propeller race

There have been many publication as to theories

applicable to the calculation of the propeller.

Mr.Pollock(6), Dr.Horn(7), Dr.Kawada0, Dr.Kondoh(9), and many

other authorities, proposed their methods in which they

used the "simple blade element theory", or the Yvortex

theory", and as to whether or not the results of these

theoretical calculation represent the actual propeller

race of the vessel accurately, Dr.Yamagata(10, Mrs. Taniguchi and Watanabe", Mr.Leathard00, and others

carried out the measurement of the propeller race

partially, but there is still no data available which

are based on the results of systematic measurement in a

broad range, therefore it is rather difficult to make

presumption.

To ascertain the effect of the propeller race on the

(16)

number of blades, the mean value of the induced velocity

generated in the course of one revolution is worked out

by making use of the equation for the induced velocity

for a propeller with an infinite number of blades. It

is generally known that when the induced velocity at

the propeller disc is 1, the induced velocity at a

point infinitely backward will be 2 times as large.

The rudder is assumed to be located midway between the

propeller disc and the infinite rear, hence in order to

know what value of the induced velocity will be shown

at the rudder position in relation to the above value,

the following simple calculation was performed to

ascertain the velocity field in its rear as generated

by the bound vortex and trailing vortices of the

propeller as follows:

(1) Induced velocity due to the bound vortex

The induced velocity dy generated by the element of

line vortex dS which has the intensity of Fp at the

unit angle, will be as under-mentioned in accordance

with Biot-Savart theorem:

pp

J.S.r)

dv-47L- R'3

whereR'---\/(x-x' )2 + (y-y')2

+ ( z-z )2

r

is the vector from the position of bound vortex

z) to the optional position(x,y,z,).

(14)

(17)

Accordingly, as expressed in terms of respective

When the number of blades of the screw propeller is

set as infinite, the bound vortices are distributed

over the disc centering the axis of the revolution of

the propeller, where the vortex has a radial axis . The

induced velocity by this vortex element is divided into

components of cylindrical coordinates as shown in Fig.4.

= -dv3 sin 0 +d-4 cos 0 dv!. = dq- cos 0 +d-v- sin 0

dx = 0 dyi = dr' cos 0' dz' = dr/ sin0' (14)

= cos'O zi = sin 0/

y= rcos0

z= rsin 0 hence, Fpdr/ d-cr; - rsin(0- ) 47r1V3 - F dr/ dV4 COS(0-- 0/ ) 471"R'3 (15) component,

dv=

d..q dv; = it will PP be: dy' (

z-e )

dz' ( ) dx1 3r-Yf ) - ( y-37' ) - dx' ( z-z/ ) - dyi ( x-x/ ) } (13) 477R'3 P 47uRf' 4-7L-R 3 dzi

(18)

Fig.4

P

dr'

x-x') sin( 0-0') 4nRi3

When this is integrated in terms of the whole surface

of the vortex, the induced velocity due to the bound

vortex is obtained as follows:

1

R-

cos 0'

ITO - -f

rp(X-X1 )f2 dO' dr'

47E-

0R' 3

In this case, ['is independent to 0, and is the

function of //only, the field of flow is symmetrical to

the axis of the revolution, and since

sin O'

r27r dO' = 0

Jo R/3

v, and vr are zero. In the equation (16), at x-x'

and (x-x' )_ -oc , namely, at the position of the

pro-peller disc and at the infinite rear,

v;

will also

tend to zero. In short, the induced velocity of bound

vortex is generated in the direction of circumference

(16) Fig.5 (16) ( IT A A \ x r )

(19)

alone, and this also disappears at the position of the

propeller disc and in the infinite rear.

When the equation (16) is rewritten in

non-dimen-tional form so as to facilitate the calculation, the

following will be given:

J1

2x cos 0, de'

rirp(

f

d5'

8 7r ° (

(C -

)2+ e2+ e2-2

ee'cos01

(

17 )

where, V-

(n:revolutions, R:radius of propeller, nR

U:advancing velocity )

pp

TP=

h U

(I-11=g, Q:velocity of revolution )

x'

C =

ci=

(2) Induced velocity due to the trailing vortices

Since the trailing vortices are the vortex layer

distributed in the locus where the blades of propeller

have passed, they form a group of vortices with spiral

surface. Henceforth, in case of an infinite number of

propeller blades, a group of cylindrical surface of the

vortex may be assumed. Now, let dS stand for the

element of spiral line vortex, and ds its absolute

value(Refer to Fig.5).

dS =ids sine - ids cos e sin Y+kds cosecos 0' (18)

where,

1, j, k

are the unit vectors of the coordinates

, r

(20)

-(x,y,z) respectively, and

e = tan-1-Dr'

Assuming that the circulation of trailing vortex at

the unit angle is di-, the components of the induced

velocity due to the element of line vortex will be given

by substituting the equation (18) in the equation (12).

dr; {-cosesinb(z-2 )-cosecos Hds -4nR13

dy =

dr cosecoso(x-X )-sine(z- )}ds 47c1R ' 3 dirz -dr {sinE(y-y ) ±cosE.,sino(x-5, );ds 47:R' 3

where (2,9',2) and represent the orthogonal

coordinates of the trailing vortices and the cylindrical

coordinates, respectively. When this is rewritten in

the components of the cylindrical coordinate system, the

following equations will be given:

dv

-dr, cos e

r-rcos ( 64-) }ds

47rR'3

dip

dv =

x-i) cos ecos( 04,)-i'sinEsin( 04')

}ds

42-cR'

drp

= e-(x-i)cosesin(

04)

+sine{r-i4cos(0-o)})ds

42-cli'3

When this is integrated all over the trailing

(18)

( 19)

(20) 3

(21)

-vortices, the induced velocity due to the trailing

vortices will be obtained.

Namely, for ?, it suffices,

dr;

to integrate from 0 to R as drp

dr, for e from 0 to

fd

dr

27T as =

and for

from ,00 tO XI

case

However, in integrating

the diStribution density of

line vortex shOuld be taken into consideration.

The propeller advances for 27ch1 .during the time it

makes one revolution.,

The circulation of

trailing

vortex

generated in the meantime is 271----e- dr, which is

dr

left behind in a straight line in the directiofi of the

axis x. Hence, the distribution density of line vortex

drp

in the direction of the axis x is

dr per unit

length.

Therefore, the induced velocity due to the

whole trailing vortex will be;

.

1 dfp -t"COS

VX

4rhlo

0

dr -00

R1.3

1 drp

,A(x-idicosecose+issin'esin0

rRr27-c

r

rdxdOd

47rh1 cosejo

di4 *I-00 R'3

dr

k

-X) sine case ±(r-i.cos id}sine)

=

-P

47rh, cose o

dr

x' IdddI

(21)

By use of the following formula,

=

ds ,

rdxdOdr

(22)

-sin()

,

dO = 0

R 3

tx

dx

( )(2+ a2 )3

xdx

JO Vi( x2±a2 )3 2 12+ a2

a

Th/x2±a2

the equation (21) is simplified to be,

1,

i''.,(-rcos)

-

x-Y

R,

]

d'ddi"

/R

- n

/.

Vx1-

47rhisio

di'

.13

r2+ r

-

2ri'cos'o

n2 1

ad rp

1>'cosli

f-

, 47ch3710 dr o R 1 a dri; ( r - r c o s 0 ) X-x"

47rJ0 dr ,10

r2+ i- 2`ri'c os 'ON 1 R/

Further, making, use of

the formula,i

1

r

reosO

0,

r<r

2ir 0

r2+

-2r?'cos

r

(20),

d

(22),

and the consideration

is given td the .fact

that rp

disappears at the tip of

aerofoil, and so:

F Q

1 d Fp

i.(r -

rcosro) '

x-x'

P ,de al- ( 25 ),

i.,

r_._ _ ____

2 ^2

^ 2U

47ch1,0 dr 0

r2+ r -2rrcos0

R' 1 drp

r - TCOS

'e X-XP

,

,. rp

vo.

-

CO' di-

(26)

2r

& 47c 0

dr 0

r2+^r2 -2ri-cos 0

Rf 1

dr') ?C0S3

,, r 47rhjlo R

di' jo

R'

( 27)

101 It ( 23)

(24)

-1

(23)

-This is rewritten in a non-dimentional form, as

follows:

= =

r2Ir

rp I drip

Fa

_ecos)

47'10 di -fo e2± 'e COS

c

- c'

x , dd'e (28)

(C-C' )2+ e2± e2

-2e e

cos

rp drp - cos g

U 87r2-*/o dF Jr77

e2+

e "E cos

io = c

- c'

d'eod ( 29 )

Vcccf-f- e 2-42

-

2 e cos '61 1 ti drp r27, 'e cos g

j

j.

dde

= U 47r 0

de

v (C -C'

e2 +

-2 e cos (3o) At the position of the propeller disc,

rp

qx

-(31)

J'rp

q0

47ff

(3) Boundary condition and the characteristics of

the propeller

Regarding the boundary condition, similar to the

assumption as made at the time of the rudder, the

flow is approximated as two-dimentional for every

aerofoil element, and the lift acting upon it, is

assumed to have the lift coefficient CL which is

2-2

47re 2 2

'e

(24)

corresponding to the instance of the attack angle a

where e=arctg

S2/4 and the induced angle

au.

of the aerofoil are substracted from the pitch angle ap, namely, as given by the formula:

a=ap

e

---v, aip= u24_02 r2 e X =X

When the equation (31) is utilized to compute alp , it

is given by:

Ji7p

crip- ( 32)

4;7e

When the solidity of blade element

T

( Number \x(chord

kof blade) llength)) is used, the lift at the 2nr

position of the radius r is:

P 2 9 9

L = --(U

) CL 2r.r dr

2

On the other hand, from the Kutta-Joukowski theorem,

the following is obtained:

L=p(U2+Q2r2) Fp.2ndr

Therefore, assuming that both are equal, the following

equation will be given:

1VU2+ D2r2 r.T.0 =pp 2 L dCL . If is constant, da (22) (33) '1

(25)

As plotted in non-dimentional form, it is expressed as follows: 2 ''' 2

r =

P 2

u

r dCL ( ap-e - ail) )

da

1 )2

-

1 TP

= -2-o- ap -ar c s in

\/1-1- ( L-7re-)2 Vrp dCL 47re / da

/1 42

7te It, /al) arcsin

\ J'

r

(27r,e J ) 1±-1 4 (27re )2 dC, d

By the above procedure, the distribution of the

circulation of the propeller can be computed. (4) Numerical calculation

As already mentioned by the equation (17), the

induced velocity due to the bound vortex has been

found, and the induced velocity due to the trailing

vortex has also been obtained by the equations

(28)-(30), while as shown in the preceding paragraph, the calculating formula for the distribution of the

circulation of the propeller has further been

ascertained. By these data, the induced velocity at

(34)

(35) \

(26)

any optional point of 'the velocity field behind the

propeller can be calculated, but the integration by,-0'

aS given in the equation (17), (28)(300,is rather

difficult to perform in the form as they are given,

therefore in order to alter to a calculable form, the

elliptic function Of Legendre-Jacobi is used for

representation. If

1

_.I-ecos"6

C'

e

M( e , =

°dO

4n-'0 e24\2-2e ecos 0 (c

e2±*2_

2.e-e cos-d

4x 0 ( 36 ) 1 Nce cos d0 ( 371 \/(C-C' )4f2W-2eicos0 cos 0ce 1 jr27r

( 38)r

4n

° \AC

)2+e242-2aCOS

3

then the equations to express the induced velocity can

be transformed as follows:

f-rp(c

o(

2n .0 -

,1rp

d , -j °M( e , 2 0 de' Jen. drp E\L

mc F, e)cil

47re

27cilo d. d-rp

e ?),

o de 0

In the equation, (36)m writing as 'cos the

( 24 )

-= -

-

)

= d

(27)

following is obtained by transformation: C- C' 1 du 27re .V/( e f v/(1.--u2 ) (1-k2 u2 ) ( 1 du 1.

(e±e )2 V( c_co2+ e+-E)2.1,

(1_,u2

where, k2 4eF cc

)2±(F±)2

±e)2 4e1

(e+e)2

c

-NI( e,e) = k[(k)+ 2e V (C

c' )2+

(e4

)2 e_f,

where, K:complete elliptic integral of the first

kind.

II: elliptic integral of the third kind.

k:modulus

c:parameter.

Of the equation (43), the elliptic integral of the

third kind is not calculable, therefore it is

trans-formed as follows:

7rdz

11 ( ,c,k)= f'

(i_cz2 )

i_z2 )(i_k2z2 )

In the equation (44), if z= snp , then

dp 11 =

f

2 K ° 1 - c snp and further, .7., ( -(43) (44)

(28)

K c srfp dp

- K

Jr

2

°

- C

sn -Q

Here, if c = k2sn'a ( a = const) , and is called

cn a dna go(K,a,k) ( H - K) sn a sn2p dp k2 sn a- cn a dn a 2 2 2 ° 1-k sn a sa p

then it will be known that there is the relative

equation:

hro(K,a,k) - H0 (a,K,k) K-E(a,k) - a-E(K,k)

where, E:elliptic integral of the End kind.

However, 2 sn p d-c II (a,K,k)--- sn K cn K-dn 2 2 2 - 0 0 1-k sn K snp and so sn a 11= en a dn a (K-E(a-k) - a-E)+K (45)

In this manner, the equation (44) is transformed as the

equation (45) , but a is an imaginary number, so it is

written as a K 1A(where A is a real number) . Thus,

-1>

sn a> 1.

k=

1

sn a - sn(K+ iA) =

dn(A, )

where k' = complementary modulus .

By the above equation, A is rendered calculable,

( 2 6 ) K

(29)

and consequently a can be found.. 'From this, E( a ,k)

mill be worked out., but the coefficient of II of the

equation sn a cna .dn a. beforehand.. Writing then, sn a - = dn(Ak' cn a odn a Id2 -1) ( 1-d2 k2 ) sn iA - en IA E(a,k)= E(K+iA,k) =E(iA)+E- k2

:dn IA

This is transformed by Jacobi 's imaginary trans-formatiom;

ic-127-a

sn a

E( a )= A-E(A,kf

will have to be calculated

d N/l-d2k

} -FE

-\/(1-ek-( 46)

(4'7)

By substituting the ,equatiOnS, (46) .and (47) Th the.

equation K45) ji

will be

found as follows:

a.

,

{K-EAW'

A(K-E))

d2 -1) (1-d2k2 )

Substituting the equation (48) in the equation (43), 1 \2 where e

/

k C- )A--(e-4 )2 d ,\/d2 If= )

-(48)

1 1

(30)

N(e3)

where .1c2=

N(

(c - )2--F (e±? 7rV( C- C

In the same way

1 0( , e) = 7,r/(c_c, )2,4.:(e+e)2);:y 1

4ee

(28)

-

c' e±,e)2:K(kri:{KE(A m

e,F

(49)

where, in consideration of (C-

n

being negative, { is

with /

positive

to

comply

negative

Next, in order to work out N(e, ) in the equation

(37), write. 2= cost similarly to the previous

instance and transform as follows:

(2z2-1)(az

(c,c1)2+(e-i=e)

2j°

ril-Z2

1-k2Z2

2(K-V_ e-F)2 [ k2

,-A(K-E)d

(50) where D.= 2D } 7C{(C )2 ± +.,e )2} kf2 K-E

4ee

k 2 c')2 e-'-e' )12

Accordingly, by substituting, respective equations (491

(.51) in the equations. (39)-1(42), induced velocity is

converted calculable.. ( 51) 111 II k2k/2 E-k'21C -2 1

of

1 1 , (e

-

} 2

K

(31)

The above calculation is applied to the propeller

of two kind. By way of example, an instance where

a propeller of Troost B°4-40 type with the pitch

ratio of 0.800 is working at the advance coefficient

_ U

of

=0.60, is given in Fig.6, illustrating the

propeller race at respective points at 0.7R in the

rear of the propeller disc. In Fig 6, c =

'

namely R

the non-dimentional value representing the distance

rearwards from the propeller disc in the direction

of x is plotted along the axis of abscissa, and the

ratio of the induced velocity at respective point to

the uniform flow (the velocity of flow as before the

incidence against the propeller), gx, go, gio and Cll.,

on the axis of ordinates.

According to the figure, the induced velocity qx

of the direction of x as caused by the trailing

vortex is the greatest, go of the direction of 0

comes next, and other two components are rather ^

small. Among them, gr gives a negative value,

indi-cating the presence of the flow of contraction.

Next, when the values as on the propeller disc

(ix and go, which are directly related if the

perfor-mance of the rudder is to be considered, should be

given as 1 respectively, the values of gx and go at

(32)

in Fig.?, in which these ratios, namely,

ix

x )at C ----0

( 61 )

at C

are plotted along the axis of ordinates. It is

generally known that the induced velocity at the

infinite rear is two times of that at the position of

the propeller disc, but the status of variations of the

induced velocity between the two points is as

illust-rated in Fig.?.

In the same figure, besides the results of

calcula-ion for the Troost-type propeller (increasing type),

those of the decreasing type propeller of the vessel

wY-Maru', who is actually in service, is also indicated

as obtained in the practical working condition (advance

coefficient:0.69). Even in the case of mutually

different distribution of the propeller pitch, no

significant difference is noticed in connection with

the status of variations in Eix and ie. Therefore, the

induced velocity of the propeller in an ordinary single

screw ship at the position of the rudder (ordinary,

c

--0.5--0.7) may well be considered as io to be about

1.7-1.85 times and ix to be about 1.60-1.75 times of

the induced velocity as at the disc of the propeller.

Now, with a view to finding the distribution of the

incidence velocity against the rudder, the propeller

(33)

10

/Or

Fig.6

The curves of induced velocity,

x

Fig.7

-15 .TROOST 54-40 TYPE P/D- 0.800 3-0.60 'Y"MARU J- 0 6q 0 0

(34)

g%L.) - 0.800

J060

J.

3 2 Lt. g

00

TROOST B 4-40 TYPE

DISTRIBUTION OF PROPELLER RACE

IC 0.5

00

0 05

RADIUS OF PROPELLER

OR

race in respective positions of longitudinal center

plane of vessel are worked out as shown in Fig.8 and 9.

Fig.8 represents the instance of the Troost type B-4-40

with an advance coefficient of J=0.60, and Fig.9, the

instance of the aforementioned "Y-Maru", where the

advance coefficient was J=0.69. In both figure, the

velocity increase coefficient in of the propeller race

at 4 sections behind the propeller, that is, sections

behind the propeller at the distances of 0.25R, 0.50R,

(32) < 3 1.5 2O0

/

. . Y MARU ---J- 069 m IT 10 5" 05 0 0.5 IOR RADIUS OF PROPELLER Fig.8 Fig.9 0.25 -- -- 50 --150

(35)

1.00R, and 1.50R and the flow angle 49 to the center

line of the vessel are used to express the data.

According to them, the increase coefficient is the

greater as the position is further backward, whereas

the flow angle does not show any heavy variation.

2.3 Results of the numerical calculation of force

acting upon the rudder

In the preceding paragraph, the distribution of the

propeller race at respective points on the rudder as

in various rudder positions, have been made known. These values are to be substituted in the equation (9)

to set up the simultaneous equations. Now, in order

to know the tendency of the force acting upon the

rudder placed in the propeller race, calculations were

carried out with the following five instances. In

the rear of the Troost B-4-40 type propeller with 200

mm diameter and P/D= 0.800, operating at the advance

coefficient of J=0.60, the same rudder as used in the

experiments described in the First Report (N.A.C.A,

aerofoil

NO.0018

symmetrical rudder, with a height of

b=200 mm, and a chord length c==120 mm) was set with

its center of height conforming to the position of

the propeller shaft center, and when the gap between

(36)

at 15', 25°, and 35° respectively, and when the gap was

1.00R and 1050R, the rudder angle to be at

35'respec-tively.

In this calculation, the rudder was replaced with

the lifting line located at the point behind i of the

chord length from its leading edge, hence this position

of lifting line was to represent the position of the

rudder. As to the points to be selected on the rudder

in establishing the simultaneous equations, 9 points

were presumed to be sufficient including the top and

bottom edges. In respect of the tangent value k of

lift against the attack angle, it was decided to use

k=5.24 (See foot-note), which was obtained from the

results of the open rudder tests, about which report

was made in the First Report. By solving the

simulta-neous equations thus constructed, the coefficients A,

A2, 'A7 were found as shown in Table 1.

When An is obtained by the above procedure, oL,

op,

oN, and OT can be calculated by the use of the equations

(10) and (11), and then by applying these to the whole

height of the rudder, L, D, N, and T for respective

instances are found. As tabulated in the form of

Note: This value is slightly smaller than the value

found by K.E.Schoenherr (k= 5.46).

(37)

Table 2. C , a Ar'LN\

c=-0.5

a= 15'

a= 25

c=-05

a= 35

c=-0.5

c=-1.0 a= 35° C=-1.5

a= 35

Ai 0.1654 0.2756 0.3898 0.3927 0.3996 A2 0.0673 0.0627 0.0394 0.0549 0.0539 A, 0.0209 0.0348 0.0654 0.0518 0.0534 A4 -0.0148 -0.0142 -0.0279 -0.0147 -0.0129 A5 -0.0023 -0.0076 -0.0059 -0.0060 -0.0063 A6 0.0018 0.0020 0.0053 0.0026 0.0023 A7 0.0013 0.0023 0.0067 0.0027 0.0036 \C\\(\y C \, CL CD 1 CN CT -0.5 15 1.139 0.199 1.135 0.174 -0.5 25 1.895 0.441 1.890 0.507 -0.5 35 2.675 0.826 2.665 0.891 -1.0 35 2.760 0.869 2.753 0.934 -1.5 35 2.875 0.907 2.870 0.962

respective coefficients (

C=

, where A: the

p A

-area of rudder, U:uniform velocity, which flows into

the propeller, and so on), they are as indicated in

Table 2.

Table 1. Aii

(38)

By plotting ON and CD for respective rudder angles in

the

case of c=-0.5

along the axis of ordinate, the curve

shown by the solid line in Fig.10 will be obtained. In

the same figure the experimental results in the case of

open rudder which was previously reported in the First

Report(rudder stock position ratio: h=0.25, and

V-0.99 m/s) are also given. By bringing them into

comparison, the condition of the increase of Cx and CD

by the effect of propeller race will clearly be seen.

In the experimental results, as it began to stall

around 20% CN decreased at any greater angle than this.

Meanwhile,

a stall is not

taken into consideration in the

calculation of ON in the propeller race, so ON is shown

as though continuing to increase even at greater angles

than above, making it no longer adequate to be the

object of comparison. Accordingly, ON was similarly

calculated for instances of open rudder without giving

consideration for a stall (The empirical value of 5.24

was used as tangent value k of lift against attack

angle), which result is given in the same figure in the

dotted line.

The tangent of CN against a in the case of open

dC

rudder --14(aspect ratio A =1.67) obtained by the above

da

calculation, almost coincides with the experimental

result, being 2.49. On the other hand, the tangent of

(39)

CN against a of the rudder as located at the point

C= -0.5 in the propeller race g.c, (aspect ratio: A==

1.67) is 4.34. Therefore, it will be learnt that the

normal force increases for about 74% by the effect of

the propeller race. This is, however, smaller than

the increasing coefficient for the force

It=

2.01

which was calculated from the mean value of velocity

increase coefficients E(1.42) of the propeller race

at the position of the rudder as found in Fig.8. In

other words, according to this experimental result,

of the energy imparted to water by the propeller, the

energy at the position of the rudder is not

effec-tively utilized as a whole, but about 75% of the

whole energy is working effectively to the rudder.

On the other hand, comparison is made by use of

Table 2 in the changes of CN and CD between the

instances where the longitudinal position of the

rudder is changed as -0.5, -1.0 and -1.5. The further

the position of the rudder departs from the propeller,

the more both CN and CD would increase though slightly,

which fact agrees with the tendency of increase in the

velocity increasing coefficient m of the propeller

race.

(40)

RUDDER ANGLE

Fig.10

CN

C* iPA(TILT

f

the values for the cases where c are -0.5, -1.0 and

-1.5 respectively are plotted in Fig.11, which gives

almost constant values, showing that 0, is proportional

to the square of the increasing coefficient Fn. of the

propeller race.

Presuming that 0N in the propeller race increases in

almost linear proportion to the increase of the rudder

angle as shown in Fig.10, the effects of the angle of

(38) 14 13 1.2 1.1 10 -05 1.5 30

BEHIND PROPELLER (CAL.)

OPEN ( E X P ) (CAL.) 20 10 Cr

z

OPEN 20 30

(41)

incidence c9 appear at respective points of the rudder

to ascertain degree when calculating for a rudder

having such a symmetrical section. But in case the

rudder is considered as a whole, these effects are

reverse mutually between respective points of upper

and lower portions, whereby only slight effects alone

are thought to manifest.

Generalizing the above results of calculation, the

following can be mentioned.

The most important factors governing the force

acting upon the rudder placed in the propeller race

are the advancing velocity, and rudder angle, then

follows the velocity increasing coefficient m in

the propeller race within the scope of this

calcu-lation. On the rudder, about 70-80% of the

acce-lerated energy of water is working effectively.

In case of a rudder having a symmetrical

section as used in this calculation, the effect of the angle of incidence is slight.

The condition of axial changes of the induced

velocity of the propeller taking place in its rear,

does not vary significantly even when the pitch of

propeller may gradually decrease or increase. Fig.7

may serve to represent this axial changes for the

propeller of an ordinary vessel.

(42)

(4) Even when the longitudinal position of the

rudder may shift for 0.50-1.5R(R:radius of propeller)

behind the propeller, the normal force and drag will

not be much affected.

3. EXPERIMENTS ON MODELS

3.1 Experimental apparatus and method

;rust as in the case of First Report, the present

series of experiments were conducted at the circulating

channel at the technical research laboratory of the

Hitachi Shipbuilding and Engineering Co.,Ltd.

Regarding the dynamometer for the measurement of the

forces and moments acting upon the rudder, detailed

description was made in the First Report, therefore it

will not be repeated here, and instead explanations

will be given here principally about the driving

apparatus for propeller and the 5-furcated Pitotis tube

for measuring the propeller race.

For the purpose of driving the propeller, a

column-shaped displacing body having a stream-lined section

as given in Fig.12 was fixed at the center of the

measuring section in the circulating channel. The

propeller was fitted at the rear end of the propeller

shaft projecting horizontally rearward from this

displacing body for about 0.8m. (40)

(43)

The fore-end of this propeller shaft was engaged with

a bevel gear through a thrust block. On the other

hand, inside the column along the vertical direction,

another shaft was provided, the lower end of which was

engaged with the propeller shaft by the medium of a

set of bevel gear, while its upper end was connected

with a ill) driving motor by the medium of pully and

belting. The revolutions of the propeller were

counted by the revolutions of the another shaft which

were reduced to 1/20 of the revolutions of the

pro-peller shaft by means of a worm and a worm gear,

utilizing the system of making and breaking electric

circuit which was related to each revolution of the

another shaft, so as to give the revolutions of the

propeller.

In order to examine the race of a propeller working

at definite revolutions (definite advance coefficients)

by a driving apparatus such as above-mentioned, the

5-furcated Pitotis tube as illustrated in Fig.1 was

used. This had an analogous form to that which was

used by Mrs.Taniguchi and Watanabe(11) to measure the

race of the propeller fitted at the stern of a model

ship. It is provided with a static pressure tube in

the center and 4 pieces of total pressure tubes were

(44)

right and left sides of the central tube, each of which

was inclining towards the center at an angle of 45%

Namely, this 5-furcated Pitot's tube consisted of 5

pressure measuring tubes combined in one.

The calibration of the 5-furcated tube was carried

out at the towing tank of the Naval Architecture

Depart-ment of the Faculty of Engineering, Osaka University.

The 5-furcated tube was mounted at an inclination of

given angle against the advancing direction of the

towing carriage, and the indication of respective

pressure tubes were read as the carriage advanced at a

given speed. In this case, the angle of inclination

was set at every 5' up and downwards and towards the right and left for the range from C--30 'including the

angle at which they were combined to the both

direc-tions. If the differences in pressure between the

upper, lower, right and left total pressure tubes, and

the static pressure tube are written as

4, y_, z

z_, and the sum total of all the four as M, then m

is considered to be greatly affected by angular

deviation in vertical direction, and y+-Y- by angular

deviation in horizontal direction. Hence, the

cali-bration curves obtained were as shown in Fig.14, as

these data were plotted along the axes of ordinates and

abscissa, by which it was possible to learn the angular

(42)

(45)

deviation of both directions. Regarding the velocity

of incidence, since the value of M(M0) at the time

when the two angular deviations are both 0% is

pro-portional to the square of the velocity, so the ratio

of M at respective deviations to Mo were put together

into the calibration curves as shown in Fig.15, from

which the velocity of incidence was to be calculated.

In the actual survey of the propeller race, the

5-furcated Pitot's tube was mounted on a base which

could be moved up and down, and also to the right and

left at a low pitch so as to facilitate the 5-furcated

tube to shift to respective measuring position as

might be required, and every consideration was given

so that the velocity of flow within any required section which was at a right angle to the mean flow

could be measured speedily and handily. The status

of actual measuring operation of the propeller race was as shown in the photograph of Fig.16.

Prior to the measuring of the propeller race, with

a view to checking whether the effect of wake

generated by the pillar-shaped displacing body mounted

in front of the propeller so as to drive it, was

extending to the propeller position, the distribution

of the flow velocity was determined with the section

at a distance of 100mm behind the propeller, by

(46)

-conducting the measurement with the propeller removed.

From the results of these measurements, it was affirmed

that there would be no objection to regard the flow

distribution as uniform, except that at localized points

the flow was observed to be inclined to the center line

of the channel at an angle of about If- 2°. Therefore

necessary correction was made for such inclination with

the results of measurement of the propeller race

enforced thereafter.

3.2 Result of the measuring of propeller race and

the consideration for it

In order to facilitate the comparison with the

results of the numerical calculation as carried out in

the previous Chapter, the measuring of the propeller

race was also carried out by use of a propeller of

Troost B.4-40 type, having a diameter of D= 200=4).

Including the instance of the pitch ratio of P/D=0.800

and the advance coefficient of J=0.60, of which

calculation was made previously, with the propeller

having a pitch ratio of P/D= 0.800, the 3 advance

coefficients of ,T= 0.40, 0.60 and 0.80 were chosen,

and with the propeller having a pitch ratio of 1.000,

0.40, 0.60, 0.80 and 1.00 were selected, and the

measuring was performed with 4 sections as at 0.25R,

(47)

0.50R, 1.00R and 1.5R (R= radius of the propeller)

behind the propeller. However, inasmuch as the Troost

type propeller used in this case had a blade rake

angle, the propeller surface adopted was the section

that was at a right angle to the propeller shaft, and

which also included the intersecting point of the

maxi-mum thickness line of the aerofoil section at 0.7R

with the pitch surface.

The measured points included in a section were 97

in number within a square of 250mmx 250mm as shown in

Fig.17. Fig.17-19 indicate examples of the results

of measuring. These figures were prepared in the form

of resultant vector of vertical and transverse

compo-nents, vz and vy respectively of the velocity at

respective measuring points at the time when the

propeller was viewed from its rear side. Fig.20 is

intended to show the status of distribution of the

ratio of the resultant velocity composed not only of

the vy and vz 9 but also of the axial component vx to

the general velocity of incidence. (The same method

of representation was employed as in the case of m

utilized in the numerical calculation of the propeller

race in the preceding paragraph.) By the figure, it

is clearly known that the velocity around 0.5--0.6R

is most accelerated.

(48)

-Fig.17--19 are indicative of the measuring results

when a propeller with pitch ratio of 0.800 was working

at the advance coefficient of J=0.60, over the

sections at 0.25R, 0.50R and 1.00R behind the propeller

respectively. According to these figures, the

circum-ferenceially accelerated condition of water can well be

observed. The direction of flow at every point outside

the projected area of the propeller's actuator disc, is

pointing more towards the center of the propeller than

the direction of the tangent of the cocentric circle

which passes the point. This fact gives exact idea

that the propeller race is a flow of contraction. At

the section at 0.25R directly behind the propeller, the

centripedal velocity is at the largest, and it decreases

in intensity as it gets further backward. This may be

deemed to show that the intensity of the flow of

contraction is at its maximum at the point most nearly

behind the propeller.

With a view to ascertaining what changes the induced

velocity generated by the propeller will undergo, as it

departs farther from the propeller disc, the induced

velocity at 0.7R of respective measured sections were

computed from the results of these experimental surveys,

-and represented in the form of

4.,

go and q, according

to the similar method of representation as used in the

(49)

numerical calculation in the preceding Chapter, that

, in the form of the ratio of respective induced

velocity to general uniform flow, as shown in Fig.21.

In the figure, the results of numerical calculations

as mentioned in the preceding Chapter, are given as

well. The comparison of these data reveals that the

empirical values of qx, ir are presented larger

than calculated values near the propeller. This may

be due to the lifting line theory us ei in the

calcu-lation. As it moves farther backward, the values of

,

q0+% and qr shows a tendency of approaching the

(50)

Fig.13 The apparatus to measure the race

of Propeller.

^

values of numerical calculation, but qx alone is large

in its value when it is close to the propeller, and

diminishes as it goes farther back behind the

pro-peller, which is a trend quite different from the

calculated value. However, within the extent of the

present experiments, respective induced velocity

settles down to a constant value at positions farther

(51)

-70

11411

MMEOMOMMIN mommMiumm HOOMMUNIO IIMINOMMumen IIMOMMOOOON UUOMMOOm

/Ailiktik

2' 2 0 g-to

Fig.14 Calibration diagram for the

5-furcated Pitot's tube.

backward than 1.00R.

On the other hand, in order to ascertain what

distribution the propeller race makes in its radial

direction , or in other words, what changes it shows

for the direction of the height of the rudder, the

radius of propeller was plotted by the axis of

abscissa, and the velocity increasing coefficient m

and the deviated angle of the flow (the deviated

(52)

Fig.15 Calibration diagram for the

5-furcated Pitot's tube.

Fig. 16

.7-514°-.

The Model Propeller and its race

Measuring apparatus.,..

(:50

OA 111ALSM

I

II

rdi

1111

oo

lb

r,Pli

Mieladd

Sill

(53)

TiOOST B.4-40 TYPE

D-200'1 moo

GO-2514, 7a60

NOTE:FIGLIRE SHOWS THE RESULTANT VELOCITY VECTOR°

OF AND 1 DIRECTION, VIEWED FORE WARDS

FROM THE AFT. AND THE SCALE OF VECTORS AT THE MEASURED POINTS ARE SHOWN IN PER-CENTAGE OF THE RESULTANT VELOCITY OF THREE

DIRECTIONS, OF WHICH SCALE SHOWN IN FiGua

SCALE OF THE RESULTANT VECTOR

Fig.19 The result of propeller race measurement(1)1

F/D=0.800. GAP- 50mm, 3=0.60

( 51 )

Yig.18 The result o propeller race measurement(2)

rim

itirgistimbs

wins_ainnom

.

i

Ei_

mrAwaringiiiiim

NEVINAMMEIBIN

inemigioninn

11111MLIMENVA

18111.*S11,

Mi

,1

(54)

Fig.19 The result of propeller race measurement(3). GAP i002'1, 3=0.60

nummoimmi

eiraiteraim

111110111M161111111

arMill1111110111

minimmanri

sammonams

IRILIEMIP/11=1111

111111111111101111111111

II= NM=

,

gdrA

ZILI\

1112H1111111M11111

NNW 1,111ri

rjFr

-iirtramorow

Fig.20 The distribution diagram of in

(P/D= 0.800, J=0.60, GAP= 25rnm)

(55)

1-<craz,

<DT

<cr-4

Fig.21 Experimental results of induced velocity

compared with theoretical ones..

FA,-=0.800. J-CALCULATION 'EXPER PENT CO- 0.25R 05OR' 1.00R 5 0 R 05 'RADIUS OF ,PROPELLER

Fig.22: The measured results of propeller race.

(56)

00

EXPERIMENTED CALCULATED 05 DR RADIUS OF PROPELLER (54) 10 10 5 0.5

00

D- 0 800, 3- 0.65 RADIUS OF PROPELLER GAP' -0 .3t/P 86R ---- I.36R 5, e 15 LS %- 0.800. J- 0 60, GAP - 0 50 R M

/..---i/

i

/ `.... ..."...N 1 10 1.0i

\

\\

5 0.5

Fig.23 Measured results of Fig.24 The diagram of the

propeller race com- propeller race

pared with calcu- according to ,T.F.

lated race. Leathard's experiments.

and respective measuring points) by the axis of

ordi-nates, and so Fig.22 was obtained. As shown in Fig.20,

it is clear that the velocity around 0.6R is most

accelerated. With a view to making contrast with the

result of calculation, the data in the case of J=0.60,

at gap= 0.50R were written in the same figure which

(57)

the velocity increasing coefficient in and the

variation in the angle of incidence co are fairly

con-forming to each other, while in their absolute values,

in is smaller in the empirical value, but co is larger

in the same value.

With regard to the result of measuring of propeller

race of this Troost B4-40 type propeller with a pitch

ratio of 0.800, Mr.J.F.Leathard published his data as

he obtained by use of hot wire with a propeller of 8"

(208mm) dia in a state of J=0.65. As rewritten in

the similar form to Fig.22, it is as given in Fig.24.

However, the author's experiment was conducted at

J-0.60, while Mr.Leathard's at T 0.65. Moreover, there

is a difference in the position of the section used

for measuring, thereby making it impracticable to make

comparison in a strict sense. Even then the author

dared to bring the two data into comparison inasmuch

as there had been no available data of this kind so

far as he knew.

Since the author's data show a smaller advance

coefficient, that is, its slip ratio is larger, it is

quite natural that the velocity increase coefficient in

is larger. With respect to the position where the

maximum value of in is shown, it is also considered

(58)

were measured at the nearly point to the optimum

efficiency, shows its presence in the proximity of 0.7R,

whereas the author's data gives it to be located at a

position nearer to the propeller boss than the former.

Furthermore, both data are found well conforming to

each other in that the velocity increase coefficient

directly behind the propeller is larger than those

which are farther in the rear of the propeller.

In the author's experiment, J was made to change in

the same propeller, therefore the mean value of

velocity increase coefficient Ft of the section at

1.00R in the rear of propeller was plotted along the V

axis of ordinates and the slip ratio s ( s=

1-nP

V:advancing velocity of the propeller, n:revolution,

P:pitch) along the axis of abscissa, whereby Fig.25

was obtained. In this case, in consideration of its

effect on the rudder, the mean velocity increasing

coefficient of the radial direction is used as M.

When the aforementioned Mr.Leathard's data as of the

section at 0.86R behind the propeller are marked in the

same figure with

"x"

mark, they give values a little

smaller than the results of the present experiment, yet

it will be noted that they can be said to be in good

conformity.

In Fig.25, the experimental results in the case of

(56)

(59)

1.0

.3

A ( SUP RATIO OF THE PROPELLER)

Fig.25

s-7, diagram.

15 I.0 .7 .5 .2

0 RESULTS OF THIS EXP.

x RESUCT OF IF LEATHARD'S ExP. -2 .3 .4 -5 .CEXPERIMENTED THEORETICAL 02 3 .5 .7 1-0

(60)

the pitch ratio of 0.800 and that of 1.000 are

con-jointly given. This reveals that in an instance of the

Pitch ratio difference at this degree, the mean of

velocity increase coefficient may be regarded to be

almost on a common level when it is viewed in terms of

slip ratio s.

Besides, when the increased portion of the mean of

velocity increase coefficient M, that is

m= -1

is

considered and plotted against the slip-ratio s on

logarithmic cross-section paper and so Fig.26 will be

obtained. On the other hand, when Am is calculated

in accordance with the "simple momentum theory",

dis-regarding the radial velocity and its changes,

- (52)

1 - s

When this is rewritten in Fig.26, it comes forth as the

theoretical curve in the figure with 'which the

empiri-cal values compare a little smaller, but they may be

said to be in a very good agreement with each other in

their general trend. Within the scope of s= 0.2-0.6

which is also within the range of the practical

condition in ordinary vessels, the curve in Fig.26

indicates an almost straight line, hence the following

can be established as empirical formula:

(58)

(52)1

I

Lm= 2.4,6 s1-5

(61)

Summarizing the above results of the comparison

between the results of measurement of propeller race

and the simple approximate calculation in accordance

with the "lifting line theory" , the following

con-clusion will be reached.

Among various induced velocities generated by

the propeller, the general trend of the results of

measurement of the radial component CI', and the

cir-cumferential components ol-4-q coincide well with the

results of calculation, but the axial component

ix

does not correspond, 'where the velocity directly

behind the Propeller is greater than that at position

farther backward.

The status of the radial distribution the

velocity increasing coefficients in and the angle of

incidence 92 show a tendency of conformity between

the experimental data and the results of calculation,

but in in the calculated values are found a little

larger than experimental value, while in So, the latter values are a little larger.

Regarding the propeller race, at the point

backward for about the propeller radius from the

propeller disc, it reaches an approximately constant

value, and for the extent farther backward by another

propeller radius, the value may well be assumed to

(62)

remain constant.

(4) The velocity increasing coefficient will be the

higher, as the slip ratio is the greater. The increased

portion of the velocity well corresponds to the result

of the calculation by the "simple momentum theory".

3.3 The result of measuring of the force acting upon

the rudder in the propeller race and its

consideration

The measuring conducted for the purpose of knowing

the effects of propeller race on the rudder performance

was carried out on the undermentioned conditions, so

that the calculation of the propeller race and the

results of actual measuring can be used for reference

The data of the rudder and propeller used are as

mentioned in Table 3 below.

Table 3

(60) Speed of flow at the time of

experiment about 0.8 m/sec

Propeller used Troost B.4-40 type

Rudder used

Form of section NACA-0018 type

aspect ratio 1.67

Position of axis

of rudder

0.30 the chord length from the

leading edge

-I

(63)

-Note: As to the size of the propeller and

rudder, a propeller with a diameter of

200mm and a rudder with a height of 200mm

were used in usual experiments with

exception of special cases.

Experimental Conditions

Firstly, in order to know about the effect of

variations in the advance coefficient J, for the

propeller with 200=0 and P/D= 0.800, three conditions

were chosen, namely, J=0.40, 0.60 and 0.80, and for

the propeller with the same diameter but P/D 1.000,

4 conditions, namely, J= 0.40, 0.60, 0.80 and 1.00.

So as to ascertain the effects of changes of

the rudder position along the longitudinal direction,

( the rudder position is represented by the point

behind i the chord length from its leading edge.),

or the effects of changes of the gap between the

propeller and the rudder, the gap/R (R: the propeller

radius) was changed in three ways, that is, 0.60,

1.00 and 1.50 with the each of the 7 instances

above-mentioned, respectively, that is 21 conditions in

total, to which one more condition was added, namely,

an instances where the propeller had only the boss

(64)

With each of these 22 conditions, the rudder angle was

set at every 5' for the range from CPto 400 onboth port

and starboard sides, and the measuring was carried out

as usual.

With a view to affirm the effects of bias

conditioryof the rudder towards transversal direction,

that is the instance where the rudder is not right in

the rear of the Propeller, the experiments were

con-ducted with the instances in which the rudder position

a

varied for (R:the radius of

3 ' 3

Propeller) in both Port and starboard directions

respectively.

In the abo'Ve experiments of (i), (ii) and (iii),

a model propeller with 200mm diameter, and a model

rudder of the light(h) 200mm xthe chord length(c) 120mm

x the thickness(t) 21.6mm vere used. Lastly, in order

to know about the effects of the propeller on rudders

with different sizes, three instances of different

rudder heights, that is, 200, 150 and 100mm, were

experimented with the 200mmo propeller, and another 3

instances of different rudder heights, that is, 200,

150 and 100mm, with the 100mmcb propeller, 6 different conditions in all.

For the purpose of investigating whether the

effect of the angular velocity of steering, about

(62)

(iii)

R and R

(i70

(65)

which full description was given in the First Report,

could also be witnessed with a rudder in the propeller

race, a several experiment was conducted with the

instance of P/D =0.800.

Concerning the non-dimentional form of

represen-tation of these experimental result, there was employed

where,

V : Uniform velocity which flows into the

propeller (m/s)

p : Density of water, A

: Area of rudder(M),

: Chord length of rudder (m).

the similar procedure to the instances appearing

the First Report, that is, as used in the

Resultant force coefficient

Cp-Non_al force coefficient

CN-Tangential force coefficient CT

Lift coefficient

Drag coefficient CD

Moment coefficient CIE

-equation in (53).

(53)

2 1-2 N. A v2A PV2 A iPV2A 2 P1T2 A MH i-PV2Ac CL=

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