Investigation on the effect of the angular velocity of steering upon the performance of rudder

LA,

_{v. ...Sch,?:,Tr!nu,vAnde}

,

1

JCi1001

INVESTIGATION ON THE EFFECT

OF THE ANGULAR VELOCITY

OF STEERING UPON

THE PERFORMANCE OF RUDDER

(

_{Hydrodynamical Research of Ship's Rudder )}

The First

Report--By S. OKADA, Ba. Eng. (Kogakushi)

The Technical Research Laboratory

Hitachi Shipbuilding & Engineering Co., Ltd.

INVESTIM.TION ON THE EFFECT OF THE ANGULAR VELOCITY OF STEERING UPON THE PERFORMANCE OF RUDDER

/ Hydrodynamical Research of Ship's Rudder \ The First Report

By Shojiro OKADA, Ba.Eng.

The Technical Research Laboratory

Hitachi Shipbuilding & Engineering Co.,Ltd.

Table of Contents

Abstract 1

Introduction 3

Theoretical Consideration 5

2.1 Circulation round the non-stationary aerofoil 7

2.2 Force and moment acting upon rudder 13

2.3 Method for solution of the integral equation

to determine the distribution of free vortex 16

2.4 Example of numerical calculation 22

Experiments on models 30

3.1 Experimental apparatus and method 30

3.2 On the non-dimensional expression of forces

acting upon the rudder 35

3.3 Law of similitude and effect of scale 37

3.4 Experimental results, and comparison with

the results of theoretical calculations 43

3.5 Other effects on the performance of rudder 49

Conclusion 58

---ABSTRACT

In the field of ship-hydrodynamics in which resistance, propulsion and steering are principal objects of study, it

seemed that so many problems related to these subject matters, particularly those connected with rudder, were

remaining unsolved. Therefore the author conducted a

survey on the twisting moment acting upon the rudder-stock taking advantage of the opportunities of sea-trials of several large cargo vessels in the 1951-1952 period and

made report on the results. The phenomena as observed

with actual vessels were very much complicated, and in

many points difficult to comprehend. In this respect, the

author stated in the conclusion of his report that further theoretical study and investigations were required to be made on the basis of experiments on accurate models.

Fortunately such chance was made available for the author. So he contemplated to make an all-over investigation in

forces acting upon the rudder-stock. In this paper (Report

I), the performance of open rudder, the related theoretical calculation and the results of experiments on models on the effect of angular velocity of steering are dealt with as the preliminary problems to enter into the question of the effect of propeller race.

From the results of theoretical calculations in con-nection with the effect of angular velocity of steering as obtained, in accordance with the vortex theory with due consideration of dynamic effects, the following facts have

been made clear:

N.P

The twisting moment acting upon the rudder-stock

develops to be greater than that in the stationary condition owing to the effect of the angular velocity of steering.

In this case, the lift acting upon the rudder-stock can be divided into two, namely, the one acting upon the point behind * of the chord-length from its leading edge, and the other acting upon the middle point of rudder.

The lift acting upon the point behind of the chord

length is the function of the angular velocity of steering and the rudder angle, while the lift acting upon the point behind .J2- of the chord length is the function of the angular

velocity of steering alone.

The twisting moment acting upon the rudder-stock increases by the effect of angular velocity of steering, but the nature and the status of such an increment differ very much depending upon the location of the center of

rudder-stock. The amount of such an increase grows larger

as the position of the center of rudder-stock shifts

rear-wards, provided that the location of the center of

rudder-stock is within the range of the chord-length.

In addition to the fact that the results of model tests

confirmed the above points, it was also ascertained that

when such experiments are made in a circulating channel,

reasonable results can be obtained even in case Reynold's

number is so small as

0.1x

106.

(1)

1. INTRODUCTION

Within the scope of ship's hydrodynamics, of which

resistance, propulsion and steering are the main object of study, there still remains many problems pending for

solution. Problems related to the hull, propeller and

rudder also belong to such important items outstanding. On the basis of the hull, the phenomena related to these three subject matters are covered by the conception of the so-called "hull efficiency", which takes principally wake

and thrust deduction into consideration. When these very

same subject matters are considered on the basis of the

propeller, they are termed the "relative rotative efficiency", in which investigations have been conducted in the past to

a fairly minute degree. In the mean time, studies relating

to these subjects on the basis of the rudder seem to be

limited in their number as compared with the above-mentioned two branches of studies with exception of several reports

(1),(2),(3),(4) and (5), in the author's

knowledge.

The

author desired to obtain some clues to solve these out-standing questions by the investigation centering around the rudder, and conducted the spot-surveys on the twisting moment acting upon the rudder-stock on the occasion of the

sea-trials of several large freighters in the course of 1951..1952, publishing their experimental results in an article titled "On the Twisting Moment Acting upon the

Ship's Rudder-Stock". The phenomena as observed with

actual vessels were both complicated and difficult to

-understand in many points, and the author stated in the conclusion of his paper that he contemplated to conduct

further theoretical study and experiments on models. This

time it had been made available for him to utilize a

circulating channel for such experiments, hence he planned an all-over study to confirm the results of the afore-mentioned spot surveys, in addition to the reexamination

of force acting upon the rudder.

In considering the force acting upon the rudder, even putting aside the effects of various aerofoil section which affect the rudder performance in the case of open

rudder, there are involved such related problems as the relative size of rudder and propeller, air draw, and many other questions in addition to the effect of the propeller race, and the dynamic effects of the angular velocity of steering.

The present study deals with these problems, particularly

forces acting upon the rudder centering around the effect

of propeller race.

This first paper principally involves the theoretical

calculations and the results of experiments on models

carried out in connection with the performance of the open

rudder and the effects of angular velocity of steering acting upon the open rudder as the preliminary problems

prior to entering into the question of the effect of

propeller race and other questions, description will be

2. THEORETICAL CONSIDERATIONS

In the case of a rudder under water, when an angle of incidence against the direction of the flow, or a rudder angle, is applied to the rudder, normal force is generated with the rudder, which develops into the turning moment of ship. In this case, the normal force caused to the rudder

has substantial effect not only on the ship's turning performance but proves an important factor to decide the

capacity of the steering engine. Hence attempts were made

to assume this normal force by Mr. JOssel(6) abroad and

Dr. Aka#aki(7) in'apan, and also by many other people(8)

(9)(1o) Besides, the rudder may well be regarded as an

aerofoil with a low aspect-ratio, and relating to aerofoil sections, experiments were made with a great many of models

(12)

in Gbttingen(11) and N.A.C.A. . Regarding the effects

of aspect-ratio, rudder section and rudder outline, Dr.

Akaraki(7 Dr.

Lammeren(lN

Mr. Gawn(14)and many other

people have previously conducted fairely detailed investi-gations.

However, in these investigations irrespective as to whether they were theoretical or experimental, the problem

was mostly handled on the stationary" basis, and it seems

that almost none of them discussed as to the effect of angular velocity of steering.

The angular velocity of steering in the present day ships

are, of course, not so great as may need handling with

For instance, compared with the magnitude of the normal force incidental in dealing with the ship's turning

performance, the effect of the angular velocity of steering

may be of a negligible degree. However, in considering the

twisting moment acting upon the rudder-stock which will directly be necessary in dealing with the capacity of the steering engine, it appears that even a slight effect can bear much influence upon the rudder performance, because the center of the rudder axis and that of water pressure are

located so much adjacent to each other. Hence the present

study was decided to be made.

The question of non-stationary aerofoil as viewed from the standpoint of the vortex theory has already been dealt with on the occasion of investigations relating principally

to the non-stationary motion of aircrafts by Birnbaum(15),

Wagner(16), Glapert(17), Theodorsen(18) von KArmAn(19)

and others, and solution has already been reached as to

simple motion. In discussing the question of non-stationary

phase of the rudder, the author also intends to proceed with the question in accordance with the circulation theory.

(Other methods, for instance, such as that based on the

acceleration potential, may also be suggested, but it has

alredy been affirmed by Prof.Kondoh(20)that the same result

may be obtained from either of the methods. Therefore, the

author is going to follow the circulation theory.) In

dealing with an object shaped like a rudder, it should,

sense. However, it has already been ascertained by Mr.

Tohnes(21) that the effect of a finite wing on the

non-stationary lift is insignificant. Henceforth, for the sake

of simplicity, the rudder was replaced with a two-dimensional

thin aerofoil is advancing through fluid which expands to an

infinite extent.

2.1 Circulation round the Non-stationary Aerofoil

According to the aerofoil theory, when an aerofoil

under-goes a change in its condition of motion, it induces a change

in the circulation

r

round the aerofoil section, and a free

vortex of the opposing direction of a corresponding magnitude

leaves the trailing edge and flows down stream. In

observance with the theory of thin aerofoil, and assuming

that the attack angle and the camber of the aerofoil section

are insignificant, it can be concluded that the change in

the position by the mutual effects of free vortex may be

omissible. Assumed that the aerofoil starts motion from a

stationary condition at time t=0, and proceeds in the

negative direction of the axis x at a uniform rate, and that

the free vortex remains at the position where it was

generated, the free vortices will form a vortex layer along

the axis x.

Now, let the advancing velocity of the rudder be V, the

circulation round the rudder

r,

and the intensity of the

free vortex at the distance s down stream from the trailing

1 di'

k(s)

_{= --}

(1)

V dt

However, it is required that the right side of (1) is calculated on the basis of the time when the trailing edge of the rudder passed the point s.

When this is expressed as: ,t

V = / V dt (2)

0

the following equation is given:

k(s)=- [dr

(3)

d t _

where k(s) is the function of(r-s)alone, and should be expressed as k(r-s).

V

Fig.1

When the equation (3) is integrated, a equation

expressing that there is not present any circulation round the closing curve involving the aerofoil and free vortex was obtained.

rk( s)ds = 0 ( 4)

Now, assuming that the rudder is a plate rudder for the sake of simplicity, let a represent the rudder angle, c the

chord length of rudder, and x= 0 the center of the chord

length. V is to denote the velocity of motion advancing in the negative direction of x, and suppose the center of the rudder-stock is located at the point H which is at the

distance he from the leading edge of rudder. Normally,

there is no objection to assume this to be within the extent

of 0<(h=1. Suppose the rudder is steered centering about

the center of rudder-stock H at the angular velocity of (the clockwise direction is positive), each points of the rudder is presumed to hold the following velocity in the direction of y as appearing in Fig.1, if the boundary condition on the rudder is taken into consideration.

da

- 2h)

+x}-) dt

This may be rewritten as

da da

Va+

T(1 -

2h) + x

dt _dt

The second term of this equation represents the velocity of the y direction which can be determined when the central point of the rudder stock and the steering velocity are

given. Since a common velocity is appricable to each

points of the rudder, let Wi stand for it, and then the following equation will be given:

da Va+ W1+ x

-dt

da

where W1= (1 - 2h)

2 dt

Thus the equation (5) expresses the velocity of y direction at every point between the leading edge and the trailing

edge of the rudder. On the other hand, the first

approxi-mation is performed in accordance with the theory of thin aerofoil omitting the thickness of aerofoil, in which the

circulation

r

round the unit length of aerofoil is expressed

by the following equation, when x---2-coso

00 1- cos n 0

= 2V a tan - 2 nan

2 1 _sin

Accordingly, the induced velocity w(x) of the y direction at the co-ordinates x is,

w(x) =V -a + 2 ci nan

sin

and so the boundary condition on the rudder surface is

filled. Therefore, assuming that the velocity of minus

direction of y at every point of the rudder as expressed in

the equation (5) is equal to -w(x) of the equation (7),

al-

wl

2V,

a2=-sin ng (7) C da\ 16V c dt ) (0_0 <rc ) (6)

is obtained. By substituting al and a2 in the equation (6),

the distribution of circulation of vortex layer on the rudder, that is,

0 da

ro(x)= 2(Va+ _w1) tan +c - sine

2 dt

will be given. This vortex layer is fixed to the rudder,

(8)

and is termed as the bound vortex, relative to the free

vortex which flows away down stream. The circulation r

round the rudder by this bound vortex will be given as

ro

- Je

f

ro(x)dx

(2(Va÷vytan+c

sin 0}dx

-7

_"T

r rP-7 0 da c c 2 dt 2 da (v) ---itc(V a+ w1) 1 _{4 dt}

Next, the circulation by the effect of the free vortex

shall be found. For the purpose, let us assume a plate

rudder with the rudder angle 0, and also a vortex with a

strength K located at the position s in the downstream from

the trolling edge of the plate rudder. With this vortex,

its resultant effect on the rudder is to be found.

Now, the z-plane which is regarded as the plate rudder is to be represented to the c-plane by the transform-equation of Joukowski, c2 C

16C

C -2- --I---z=X+ty, Fig.2

-Then the plate-rudder on the z-plane, , is

represented to a circle having a radius of *c on the

c-plane, and the vortex K moves to the point c-i+bi-Vcs+s2-r. To fulfil the boundary condition of the vortex K for the circle, it only needs to place a vortex having the strength

of -K at the position of the image r, for the circle of r.

[c\ p,

c s 1

ke

r,-

- +--- cs

s2

422

To satisfy the condition within the circle of the vortex having the strength -K, it only requires to place a vortex

with a strength of K at the origin. When the circulation

caused along the rudder is 1-0, the complex velocity potential 0 will easily be found to be as the following equation: iK C (c- Jr' log + log C 27c

C -

27r, K (10) (12)

In the above calculation, only one vortex was assumed in the

rear of the rudder. However, since there is no change even

By adopting the Joukowskies hypothesis, that is, water flows away in the direction of its tangent line from the trailing edge, K needs to be decided so that infinitely great velocity will not occur, namely, assuming that

d c c

=0

the following equation is given,

z= Ecos

1

if it is replaced with a vortex layer k(s)ds as shown in Fig.1, and the same result will be obtained therefrom, the

following equation will be got for the circulation round the rudder under the effect of free vortices:

Thus the circulation round the rudder is ultimately given as;

c2 da /c +ss

r = Po+ ze(Va +1111) +7r _dt -1-_-1

1 k(s)ds

4 ₀ s

(14) This may be simplified by the equation (4),

wc2 daroo

c s

n 1ATc(Va+ )+

1 _{dt s} k(s)ds - 0 (15)

in which the integral equation is obtained to determine the

distribution of free vortex k(s) when respective conditions are given.

2.2 Force and Moment Acting upon Rudder

To obtain the force acting upon the rudder, pressure is

to be integrated along its surface. Applying the

Bernoulli's equation,

p = Const - v2 - P

2 Ot

on both port side (suffix:p) and starboard side (suffix:s),

then the difference in pressure between the two sides is

computed, a pp p )(v -v ) 4_ (0 - )

Ps=- 2spsp, , at

Sp

rl

.,,,c)c) s 1 k(s)ds (13) -c(Va + -- (16) (17)

However, since vs+vp- 2V, vs-v= r(x), by integrating the equation (17) from the leading edge to the trailing edge, the force L, which is perpendicular to the flow against the rudder, is found.

L =f2(p

_{c p -s}-D 21p-v-r(x 2 dl*

=pvr+

dt where I* = P.i; kss-c5D)dx. 2

The first term of L is the lift by the Kutta-Joukowski theorem, which is equal to the quasi-stationary lift, to which the time differential of the integral of the pressure product on the rudder surface is added as a non-stationary term. Just the same as circulation, the velocity potential

0 is divided into two and calculated as follows: 1* x c2

+

P 2 _-0

and then substituting the equation (14) ,

gc2p d a dV dWi 1-_-_. rcpV(Val-Wi)+ ( 2V dt +a +dt at ) + V P ldx 0 + P -43 )dx at 2 (18) \iCS 9 k(S)dS (19) 2 k(s)ds cz2p da dV dWi pVe k(s)ds = 7repV(Vai-Vi, )+ ( 2V+ a +

j

-1- 4 dt dt at 2 0 v es±s2 (20) 7rc2p da dV ail c +2s ( (V --I- a dt .4. L= Pvr+ 4 at +dt )+V P - , 2 V cs 4- s2

]1CS)dS

)dx c +s

Next, the moment Mil (the anticlockwise direction is deemed positive.) around the center of the rudder-stock H is found,

2

c di* dN*

+ dt =VPI' o (1-h) - VI- +(-he)

at 2 Vpc ,,x, k(s)ds ... MHz= po____ _ 2h)1 ( zcvp(va+w1 )__,_

j

,/ es +c,3,c,

)

_2 2 _o Y 4 ,2 7re2P , 2.vda dV +d at dt dt ill 1 )1+ 2-1'p o u a + [-9--(1-2h)]( j _{128 dt2} (21) 2 4 '

Relating the aforementioned equation (20) to the equation (21), which denotes the moment about the center of the

rudder-stock, the following equation is built up:

L L1+

_L2 pVC k( S)dS = 71"CPV(Va+ Wi) + 2 0 VCS S2 n.c2p da dV dWi (2V

+a

+ )

4 at at at

and by so doing, the equation (21) can be rewritten as follows:

(22)

where N*- pj

(os - op) x dx

N* c2 _{zo4 da} + o cs+s2 S ± 2 CS-1-S2 k(s)ds P 10 _{128 dt} 16 2 = P P 2 ,,c - Psit-T- - ho + x)dx 2v4 (cs

)1

(P- -hc + x)dx _ 2 x

-p

-(1) t s P -00 -000-C2

- 2hd Li 4- 1 - 2h)]L2+1,1m

-c

In the equation (23), [i(-32-- 2h)] indicates the distance

between the center of rudder-stock H nnd the *c point from

the leading edge of the rudder. Besides, since [I, _ 2h,1

expresses the distance between the center of rudder-stock H and the center of rudder, it may be known that Li is the lift acting on the ic point from the leading edge of rudder, and L2 is the lift acting upon the center of rudder.

It is not found that ME is the product obtained by multiplying the lift L1 and L2 by the lever of couple

respectively, to which Mm is joined as a term related to angular acceleration.

By the above procedure, the general equation for the

moment and lifts acting upon the rudder-stock with necessary modification for the effects of angular velocity of steering is obtained.

2.3 Method for Solution of the Integral Equation to Determine the Distribution of Free Vortex

When the conditions as to the variations by time are

given for V,wi, h and a, the integral equation to determine

the distribution of free vortex k(s) is given by the

above-mentioned equation (15) , that is,

ree2 da

f

gc(Va w1) + 4 dt C S k(s)ds = 0 where 7rp 4 A 2 u a (23)

Mm

128 dt2 MH = [1(

da where -c(1 - 2h) Co dt 7rcVa+ nc2(--3- - h) da +

f

4 dt 0 2 / 3 ncVa+ Th.c j - h)

+f

r

-0

fl

+s

Now, it is needed to introduce various condition for the steering of ship into the equation (24) by substitution, and it is considered approximately right to assume that the following conditions prevail directly after the steering motion:

V - Const

da _{-Const =J}

dt

k( s ) is to be found on condition that a - 0 for t - 0,and

a is given for t >0. Assuming further that -s, the

equation (24) will be as follows:

Further, r

_f

tV dt and V ._,Const, r

0 da d a and a dt dt' dt Const, a _jt a -

r ,

3 7r cj 74-7rc2 j(T - h) + is thus obtained.

In order to change the above to a non-dimentional form,

the following values are introduced. Putting that

-k( )

et

X ,

c

= e

r(e)

.

the equation (25) is transformed.

+ r k( )de' = 0 (25) V

-

e

k(s)ds - 0 (24) k(ei)del

-

0 -t C

-x

V1+ X

-(X+ (1- h)}=f

i( e) de ( 26 )

and the equation (26) can be rewritten as follows:

-x

± X - e

Jr

r(e)def(x)

(26')

o vx - 6

however, since f(0)---- constant -A(A:constant h), it has

1 A

the solution + 211

X }

at x< 1, and

r

holds singularity

vx

(22)

at x-0

. Hence, assuming that

r(e)

+

211-0- U(e)

(27)

substitute in the equation (26),

j..x

V1+X - e

-

u(e)de g(x) 0 x e where g(x) (A + x)(ga.(x) g2(x)) - g3(x)

g1(x)=(

+x - 1) 1 2e- x (28) x sin--g2(x)=

(x)._f

0 V 1+ x - e

x

e(x -g3( x)

f

d e o 1'1+

X

-by which the integral equation relating to u( is obtained.

This has no singularity even at x0, and numerical integration can be made.

It is then assumed that the following equation can be

formed for suitable interval.

eV1+x-e

eV - e

f

u( e)de

)f

de -=U(i-e)Fol(X) v X- e ₀ V X -=

-1 ,.,e \/1+ x - e

where F0

(x)

-

1 de

'o - e

Accordingly, when it is to be generally written as follows, u ((m -2 me V 1-4- X C

f,

_{m lie} de = Fmm-1(x)

\-

x - e

by putting e , 2e, ne, the following simultaneous

equations are given,

g(e)

u1F10(e) g( 28) - 111F ( 2e ) + u.,

10( 28 )

o

g(ne)-uiF1(ne)+u

_{o 1+2 1}iF2(ne)+ However, Ille \/ 1+ ne - e e Fla

(ne)-

de 111-1

_'(m-1)e

_\ine - e V(k+1)e+(k+1)2 +(k+1)e+-7--ff log k e

+k2

e2 +k +12

+(k+1 )

+ ( C2 -

ke+

k2

e 2 where k n

-As it will be seen with the equation (30), F3211 (ne) is

m-the function of n-m and e, so it is to be written as

Gn-m' and when it is required to be particularly mentioned,

it is written as Gn-m(e). In this case, the equation (29),

which are simultaneous equations become

gkne)-n-m m-1 -1

(29)

4_ un..,Fi( ne) where n _1 , 2 , ..n (30) (31)

-E

-By solving these equations respectively, 11 can be found.

However, in this case, if e is assumed excessively large, the kernel of the integral equation will be infinitely large at x= e,and in the integral for the final section,

n \/1± fl - e

u(e)cie

un_L_ Go ( 32)

-5(n-1)8

ne

e

where Go

the error is feared to become excessively large. Therefore,

even when e is assumed fairly large

in

carrying out the

calculation, there will be the need of a corrective measure to minimize the interference of error, namely, supposing

u(e)

is a straight line in the section of ne->e(n-l)e,

the following equation is given,

u(e )

=u* 1+x(

- L-)6

n-T-3

and substituting in the equation (32) ,

IG

- 11.4

eGo -

N,J* ne 1 + n.e -where, T

=jr

(n, e )

de

(n-l)s

n-

e

2e

+ 1 \/ 1 e+e - log( 2 +1 +2 e2 + 4

*e

Hence, I = A. G0 - 2 ( e Got - Got1+ 2 G0 11 )) =_Go 1 _* A At ) u' _{n- g}

On the other hand, by the equation (31),

n-1 g(ne)= _{um-,1 Gh-m} I Is-.1 2 (33)

-e) ( t

1.0 05 1 n-1 g(ne) -LLH G 11,1 -0 n-m 1 un_A_ u G

GI I

f

_l+*

n-rs

In this way, by finding Gto and u, I can be obtained

from the equations (33) and (34) with the interference of

error diminished.

In practice, assumption is made to the extent that e=1,

and calculation is carried out by the

u(e)

up to n=4.

Any further calculations are made by the method of

multi-nomial expression just equal to the equation (29) by

directly using i(e). In the meantime, n=10 and onwards,

Gn-m -*

0.30

3 5 _{SCALE FOR Ulg )} 20

SCALE FOR _dil-Lg)

30

(34)

0 10

+

the slope of

rw

becomes extremely slaw and presents

convergence. Therefore, e is set still larger to facilitate

calculation.

For example, in case h= 0.3, that is, when the center of the rudder axis is located at the point 30% of the chord

length from the leading edge, u() and

r(e)

against e are

plotted as shown in Fig.3.

2.4 Example of Numerical Calculation

By the solution of the integral equations of the

preceding paragraph, r(e), that is, k(s), or the distribution of free vortices, is found, and so it is made possible to

calculate lift and moment. In this case, in order to make

clear the non-stationary effect, the lift and moment in the stationary condition are expressed by Lo and MH0,

Lo_ mcPV2a

(35)

-h) c2 p V2a

MHO=

This Lo and MHO, as well as Ll, L2 and MH, are expressed in the non-dimentional form for the convenience of calcu-lation as follows: Lo CL _ 2 n: a o pv2 c CL1 1L - s (1-2h - A ) 2 PV c 0L2- L2 - 2 71. c2, P V2 c

cir(e)

de a where

a=

A 4

x

-V '

_{1/(x-e) +(x--)2,}

a

( 3 6 )

-clvdr

MR2 - h) (CLo -1- CL1) + - h)c1,2 (37)

1--PV

c-and the difference between the stationary condition and the

non-stationary condition is expressed by ACL, and LNCLAH as follows:

ACL=CLi+CL2

- h)CL1+ - h)CL2

One of the points on which the lift acts are h- 0.25,

and the other is h- 0.50. Therefore in numerical

calculation, both extremities, that is, h= 0.25 and h= 0.50

are selected, and as intermediate points, h= 0.30 and 0.40,

4 points in all, are chosen for calculation. In ordinary

ships, about h =0.27 -0.32 are selected in most cases.

Assuming that the maximum rudder angle is 35°, x at this

time is expressed by xmax,

annx a_ithax V

XITLaX

cj

When V and c are definite, xma, is the parameter which

is inversely proportional to the angular velocity of

steering j.

For example, in case h=0.30, and xmax =20, CL and CmH

are calculated, and plotted in the same chart as shown in

Fig.4 and Fig.5. The difference between

Clmoof

stationary

condition and Chin of non-stationary condition as shown in

Fig.5, corresponds to the increased amount of the

coefficients of moment, AC, arising from the effect of

the angular velocity of steering.

40 30 c;..1 2.0 /0 = 0.30 X max= 20 STATIONARY

NNON

STATIONARY 10 20 30 40° RUDDER ANGLE 10 20 30 40° RUDDER ANGLE Fig.0 Increment of CmH against respective rudder angle. (in case h. 0.25) 10 05 -02 A=OM 20 ;<6 Cfivi DUE TO CO

A CA1H DUE TO Cce

STATIONARY

NON-STATIONARY

10 20 30 40'

RUDDER ANGLE

Fig.? Increment of Givgi

against respective rudder angle. (in case 0.30) 6 Cmn 05 10 20 30 50 70 40' /0 20 30 RUDDER ANGLE Fig.5

Note:CmH of the anticlock-wise direction is noted

as positive in Fig.l. Fig.4

0.15 0 (0 0.05 20° 30° 40 - RUDDER ANGLE Fig.8 Increment of CmH against respective rudder angle. ( in case 0.40) Fig.9 Increment of CmH against respective rudder angle. (in case h= 0.50)

In order to know the effect of angular velocity of

steering on ACmH, by changing velocity in various ways,

that is, within the range of xmax

=5-70, then ACmH was

calculated against the rudder angle a at h=0.25, 0.30,

0.40 and 0.50, which results are as plotted in Fig.6--Fig.9.

In these charts, dotted lines indicate the increment of

moment due to Cu, and the full lines show the total of

increment in the case where there is an increase in moment

owing to CL1 in addition to the increase due to Cu,

namely, in the case of 0.25, there exists the increase

10 20 30 40° RUDDER AM-LE ° -rmax= m0X=20 =30 ..--Xmax =50 xinax =70 20 50 A

-due to _0L2 alone, and that due

to Cia

does not come up,

as it is a matter of course. Accordingly ACmH is the

function of the angular velocity of steering alone, and even though the rudder anglea may undergo a change, LCmil

stays unchanged, as illustrated in Fig.6. Compared with

this, in the case of h- 0.30, both increases due to CIA.

and CL2 are present, of which the former is the function

of not only the angular velocity of steering but also that

of the rudder angle a. The latter has no relation to the

rudder angle just the same as in the instance of h.= 0.25,

but the sum of the former and the latter, that is, ACivai

will form a group of curves as indicated in Fig.?.

At h==0.30, the increase due to Cu is larger than that

arising from CIA_ in its amount, but at h= 0.4, the increase

due to _0L1 grows larger reversewise, and at h =0.50, the

increase

will

be only that which arises from CIA., and a

group of curves radiate from the origin.

All though Fig.6-9, it is always observed that the

smaller

xnax

is, that is, the faster the angular velocity

of steering is, the greater t_scuila will come up. With a

view to making this relation clearer, the groups of curves

are cut at a-15°, and plotted AC IE along the axis of

abscissae and fl(when the rudder angle a and the advancing

velocity V are constant, the non-dimenconal value

propor-tionate to the angular velocity of steering.) along the

axis of ordinates, and AC MH at a=150 are laid down as

A = 025

Civil/ DUE TO CL2

I I 1

05 10 15

Fig.10 Effect of Angular

Velocity of Steering on the Increase of CLui

(at h =0.25)

Fig.12 Effect of Angular

Velocity of Steering on the Increase of C. (at h= 0.40) r-0 10 A =030 A.= 0.50 NEIC/414 DUE TO CH

" " CL?

.05 .10 15

r

2

-Fig.11 Effect of Angular

Velocity of Steering on the Increase of CLEI.

(at h =0.30)

CMH DUE TO CL1

.05 .10 .15

cvd:.

Fig.13 Effect of Angular

Velocity of Steering on the Increase of Old'. (at h 0.50) /5 .10 005 0/0 a 110 . ,05 DUE DUE C 0

According to these charts, the increase due to 0L2 is perfectly proportionate to the increase of the angular velocity of steering as in the case of h= 0.2,5 in Fig. 10,

and such increase grows in a straight line. On the

contrary, the amount of increase due to 'CIA. does not draw a straight line, but, as indicated in Fig.13, shows a tendency of gradually approaching the extremity value.

On the other hand, in order to clarity the effect of the change in the position of the rudder axis on the amount of

increase in the coefficient of moment, the rudder axis

chord ratio h is plotted as the ordinates, and ACNE ata

are laid down with

n

as the parameter, and thus Fig.14 is

obtained. Since the

n

is 0.02-0.03 at the time of steering

in ordinary ships, it tan clearly be observed that the larger the h-value is, that is, the farther backward the rudder-stock is located, the greater the LCmH value will

be, within the above range of 2.

15

w

6 D5

02

Fig.14 Effect of the Position of

Rudder Axis on the Increase of C.

Summarizing various points discussed above, the following conclusion will be reached.

Within the above extent of calculation, the twisting

moment acting upon the rudder-stock becomes creater by the

effect of the angular velocity of steering as compared with that in the stationary condition.

In this case, the lift acting upon the rudder may be

divided into two. The one acts upon the point at the

chord length from the leading edge of the rudder, and the

other acts upon the center of the rudder.

The lift acting upon the point at 37.- the chord length

is the function of both angular velocity of steering and the

rudder angle, but the lift acting upon the point at the

chord length is the function of the angular velocity of

steering alone.

The twisting moment acting upon the rudder-stock

increases by the effect of the angular velocity of steering,

but the nature and status of increase greatly differ

depending upon the position of the center of the rudder.

Within the range of the angular velocity of steering in

ordinary ships, and 'when the position of the center of the

rudder-stock is located within the scope -1- the chord

length, this increase grows larger as the position of the

rudder center shifts farther from the leading edge rearwards.

3. EXPERIMENTS ON MODELS 3.1 Experimental Apparatus and Method

In the preceding chapter, various conclusions were led from the results of theoretical calculations in relation to

the effect of angular velocity of steering on the twisting

moment of the rudder-stock. For the main object of making

comparison of these results with that of experiments, the present experiments on models were conducted.

This series of experiments on rudder models were carried out by use of the circulating channel constructed in the

compound of the Technical Research Laboratory of the Hitachi

Shipbuilding and Engineering Co., Ltd. This circulating

channel was completed in the 1956 spring for the purpose of experimenting on ships, propellers and other hydrodynamic subject matters, being 11.500m in length, 3.400m in width

and 1.200m in depth. As a circulating channel, this ranks

among the largest as well as best in Japan. Its general

arrangement is briefly as illustrated in Fig.15. The water

in the channel is actuated by a -bladed propeller with a

diameter of lm driven by 15HP 3-phase A.C. Comutator Motor.

The water is thus made to circulate in the horizontal

direction at the rate of 0.6-1.8 m/s. There is free

surface at the measuring section alone. All other parts

form a closed tunnel. The measuring section is provided

with windows for observation of water on the side in the

frontal part and at the bottom. The distribution of the

indicated that the degree of irregularity was within 5% except the zone in the close vicinity of the side walls and

the bottom. The ripples on the free surface were also

insignificant, and particularly low at the time of a slow speed. Thus it had been proved that the channel was

sufficiently eligible for utilization in hydrodynamic experiments.

PLAN

ELEVATION

11500

Fig.15 General Arrangement of the Water Circulating _Channel.

(31) -1 _EE 1 li i----P 1 0

0

N --, k 4 II 1 II II '; 1. I o o v-,

-I

,

0

0

_

VV a e ! , 1 , 1 o o

,

-1 ! ! ;

IiI

I

In order to measure the forces and moments acting upon the rudder, a rudder dynamometer as illustrated in Fig.16

was designed and prepared. This dynamometer was planned to

measure the three factors, that is, the moment about the rudder-stock(MH), the force acting perpendicular to the

chord of rudder, that is, normal force N, and the force acting in the same direction as the chord (Tangential

force: T) in continuation not only in a stationary condition but even in a non-stationary condition during the process of steering.

In measuring the above mentioned three factors, hollow

steel tube with a thin wall thickness, in which a electric

wire strain-guage was pasted, was inserted into a part of

the rudder-stock, so as to transduce the mechanical unit

into electric units. The details of these mechanisms are

as shown in Fig.16. The rudder-stock was held in a vertical

position by a pair of bearings at the point higher than the

hollow steel tube, and was connected with the rudder at the

lower end, while the upper end with the worm and the rudder

angle indicator, and the variohm for recording the rudder

angle. A 1/32HP commutator motor was utilized as the

driving motor for steering, and the two-step reduction system including the worm and worm-gear was so designed that the rudder-stock would be turned round at a proper

velocity. Fig.17 is a photograph to illustrate the

condition in which the aforementioned dynamometer was

rudder angle was available by the indicator moving around the dial having the angle graduations, and also by an

electric bridge formed by the variohm directly coupled with the rudder-stock, which converted the changes in the rudder-angle into the changes of electric resistance, so that continuous measuring could be made by use of an oscillograph.

HELM ANGLE INDICATOR

11,,.11=1: .

III.

Agit*

milwropmnimmin

=IF

iktPMOTOR FOR STEERING

IT/ N OF THE

PASTED STRAIN_CAUGE

CUT WATER

RUDDER

The unbalanced voltage induced by the electric wire

strain guage were also amplified and rectified respectively

and led to the oscillograph. Fig.18 shows the amplifier,

oscillator, for the electric wire strain guage, bridge-box, magnetic oscillograph, as installed near the circulating

channel. Besides, before and after experiments, force and

moment were applied to the rudder-stock by means of the weight hung through a pulley, and by changing the amount of weight, calibration was made so as to confirm the accuracy

of measuring apparatus for conducting experiments.

Fig.17 General View of the Rudder Dynamometer.

Fig.18 Arrangement of the Experimental Apparatus.

3.2 On the Non-dimensional Expression of Forces Acting upon the Rudder

The forces acting upon the rudder are generally like

those shown in Fig.19. When the resultant force acting on

the rudder placed in the water at the rudder angle of a is is denoted by P, and the distance from the leading edge to the point acted upon by f, the component at perpendicular to the chord of the rudder is the normal force N, and

another component along the chord is the tangential force T.

This resultant force P may be considered dividing it into a component acting at perpendicular to the flow, and another

acting in the direction of the flow. In this case, the

former is the lift L and the latter is expressed by the drag D. As previously stated, in this experiment, the

normal force N, the tangential force T, and the moment

about the center of the rudder-stock MN alone were measured. So other factors were required to be computed from the data

5

The non-dimen,tional notation of these forces were made in

accordance with general method as follows:

obtained by what were actually measured. the following equations are cited:

For the purpose,

p

\i N2 4_

T2 -

\I L2 4. D2 ₍₄₀₎ L= N cosa - T sina (41) D = N sina+ T cosa (42) ME hC + (43)

-Resultant force coefficient

Normal force coefficient

CN-P V2A

Tangential force coefficient CT=

Lift coefficient

Drag coefficient

Moment coefficient

same manner as follows:

I 2 = 2 2 CP= \I GIT ± / CT CL CD CL- CNcos - CTsin a CD- CNsin a ÷ CTcos a C h CN CD= P V2A PV2A PV2A PV2 A 2 o vI P V2 AC (44) (45)

where V: Velocity of the flow (Ws),

kg 82 p: Density of water

in

A: Area of the rudder (m.2)

The equations between one another coefficients are in the

L

=

+

-Fig.19 The Forces acting upon the Rudder,

3.3 Law of Similitude and Effect of Scale

To presume the performance of the full size rudder on the basis of the experimental results on models, it is, needless to say, necessary to satisfy the law ocVsimilitude

between the model and the full size. It Is generally known

that when the phenomena relate to inertia force, it needs to follow Newton's law of similitude; in the case of

phenomena relating to gravity, Fronde's law, and the

phenomena governed by viscous force of the medium, Reynolds'

law. Since the present experiments on models aim at

surveying the effect of the angular velocity of steering on the twisting moment of the rudder-stock, it suffices to make comparison between the values as measured in a stationary condition and those in a condition where the dynamic effects of the non-stationary condition have joined, and there may be no necessity for law of similitude in the strict sense. However, the results of the present series of experiments

will be brought into comparison or referred to in the contemplated various experiments hereafter to be carried out, or it may be necessitated to make assumption as to the condition of actual vessels on the basis of the results of the present experiments, it was decided to make investi-gation into law of similitude at the start of this series of experiments.

Suppose the rudder is in an open condition where it is completely submerged deep into the water and there is

assumed no wave on the surface. Then there will not be

involved anything related to gravity, and instead it needs to consider about both inertia force and viscous force of

the medium. Even when these two forces are acting, it is

sufficient to follow Reynolds' law as introduced by the

general principle od dynamic similitude(23). In order to

perfectly satisfy Reynolds' law of similitude when the

model is

4-

of the full size, and the same kind of fluid as

in the case of the full size is used, the experiments on model must be conducted under the velocity a times the full

size, which makes it unsuitable for the practice of

experiment. Accordingly this law is usually ignored, and

experiments are made with a velocity as fast as possible,

and on a model as large as possible. This gives rise to

the problem of the scale effect, and lays down the condition

relating to Reynolds' number for the contemplated experiments

on models so as to be able to precisely estimate the

(23)

publicly announced in 1939 that Reynolds' number was

required to be

1.5x

106 at the minimum in order to obtain

a satisfactory agreement in every case. _{Later in 1948,}

Dr. van Lammeren(8) stated that for a rudder of ordinary shape, a reliable results can be obtained even at a

Reynolds' number of around 0.20x 106, basing upon his tank experiments on open rudders for Reynolds' numbers ranging

from 1.66x

106-0.05x

106. In the present experiments,

prior to various experiments on models, tests were conducted

with geometrically similar models in two sizes so as to

ascertain the scale effects which might arise in the

circulating channel. The shape and size of the model

rudders used were as shown in Fig.20 (Model No.1), and another one which was similar in its shape to Model No.1

but twice as large in its size (termed Model No.2). _Fig.21

is the photograph showing these two models. As to the

rudder section, N.A.C.A.0018 was selected in the _{sense that}

it is equal in the thickness-chord ratio to _{those commonly}

used among ordinary ships, having a symmetrical section, and its characteristics are already known.

The exioeriments were conducted with Model Nos.1 and 2, in which the rudders were held in a condition that the

upper end was submerged under water for one-half its height. Besides, with a view to preventing the rudder-stock from disturbing the water surface by making waves, or causing air-draw, a cutwater with an ordival section was prepared, and made the rudder-stock pass through the inside of the

2/6

NUMERALS:IN Myi.

Fig.20 Rudder No.1 as

tested in the

circulating channel.

Fig.21 Similar Model of Rudder.

Note:Models No.1 and No.2 are arranged from the right to the left.

cutwater, so that the force acting on the cutwater would

not be transmitted to the rudder-stock. Thus efforts were

made to satisfy the condition that on the assumed water surface, no wave is considered to be raised.

With Model No.1, experiments were conducted by changing the velocity of the flow in three ways, and with Model No.2,

in two ways. The experimental results were as shown in

Fig.22 and 23, where the rudder angle a is plotted along the

ordinates and CN, Cp, and E/C values were laid down. The

comparison between the two Figures affirms the agreement in

the tangent of the coefficient of normal force ((qT'') the

stall angle, etc., with an exception of part of t/C, and there was not noticed any substantial difference between

the two. Although slight difference was seen with the

maximum normal force coefficient, etc., this was due to the

appearing even when Reynolds' number is as large as 5.5 x

106. Therefore, this may be considered not to relate to

the difference in Reynolds? number. Beyond the critical

angle which causes a stall, the phenomenon became unstable

with differences arising in 0N, however this is inevitable

since it is a matter of natural _{cause. Table 1 shows the}

Reynolds' number at the time of experiments. It must be

noticed that they cover a range of 0.085 x106

0.22x 106.

As above-mentioned, the maximum Reynolds' number at the time of experiment was over 0.2 x106 as advocated by Dr. van Lammeren, and the minimum number is so small as one

place lower than the aforementioned. Yet these results

showed a good agreement. The principal reason why these

experimental results gave reasonable values for important items such as the slope of normal force, the coefficients of maximum normal force, and the rudder angle at which the stall occured as compared with other experimental results (to be described in detail later), is that in the present experiment, a circulating channel was used, which condition was quite different from other cases of experiments.

Namely, an experiment by use of a circulating channel differs from an ordinary towing tank experiment with the model advancing through the stationary water, in that the water itself is activated into circulation by means of the

impeller, and the water flaw had come to include its own

inherent turbulence. By this reason, the so-called

when a turbulence stimulation grating

(24)

is inserted, a larger Reynolds' number was brought into action in the present experiments as compared with ordinary towing tank

experiment, and diminished the portion of laminar flow in

the proximity of the leading edge of the rudder, and in spite that the apparent Reynolds' number was below 0.2x 106, the experimental results showed agreement as good as

any experiments conducted at a larger Reynolds' number.

RUDDER Ng I RUDDER AL? 2

v =0 62 m/s

--- V

064m/s , 0.85 0.93 1.04 /0 1

0

20 30° 10 20 30°

RUDDER ANdLE RUDDER ANGLE

Fig.22 Results of Rudder Fig.23 Results of Rudder

Model No.l. Model No.2.

In this way, it had been affirmed that reasonable values

could be obtained even at an apparent Reynolds number of

around

0.1x

106 in experiments by use of circulating

channel. Accordingly, in the experiments hereafter to be

-conducted, the model rudder having a height of 200mm, and

a chord length of 120mra, and the velocity of flow of about

1 m/s or below which raises but least ripples on the free

surface, were decided to be used, except special instances,

with due consideration on the convenience of experiments and, also on the fact that the propeller model to be used on the occasion of the experiments on the rudder fitted behind the propeller, Was 200mm in diameter, about which descri'ption will be made in Report 2 of the present series of experiments.

3.4 Experimental Results, and Comparison with the Results

of Theoretical Calculations

To facilitate the comparison with the results of

theoretical calculations as carried out in the preceding Chapter, the experitent was made with the instances where

the center of rudder-stock was located at tile points 25%,.

30% and 50% of the chord length from the leading edge, and

!Rudder Dimensions of Rudder

(Height ChordxThickness -4* I Velocity of Water Flow Reynolds' Number 0,82 0.085 X 10 'No.1 200

x 120 x 21.6,

.

0.85

0.117 x 106

0.143. x 106' 0.64

.

, 0.153 x; 10 No.,2 400

x 240 x 43.2

0.93 0,..,221 x

106

---Table 1 Reynolds' Number at the time of Experiment

for the instances in which the rudder was at the stationary rudder angle and also in the non-stationary condition in

the process of steering. In case of the stationary rudder

angle, 0-350 of both port and starboard sides were

measured at an interval of 5°. As to the non-stationary

condition in the course of steering, expstriments were mnde by changing the maximum rudder angles in a number of ways, and at 3 different kinds of steering speeds for both port and starboard sides.

Figs.24, 22 (previously shown), and 25 illustrate the results of experiments in the form of coefficients, as used

in the case of the equation (44), for the instances of

0.25, 0.30 and 0.50, plotting the rudder angle along the

axis of abscissa. The dimension of rudder model used at

the experiments were 200mmx 120rimx 21.6mm, height >< chord

length x thickness, respectively. As they are given as CN,

CD, E/C, they should have shown the same values regardless

of the h value, but owing to the finish given to the model

surface, particularly the roughness of the surface near the

leading edge, it influenced sensitively upon the critical

angle for stalling or the coefficient of maximum normal

force at the time of experiment, and some slight differences

were noticed in these respects, which are all due to the

effect of inequality of the three models. However, so far

dCN

as the slope of the coefficients of noimal force da , and

was observed. Besides, at the rudder angle beyond the critical angle for stalling some deviations were noticed, because the phenomenon was unstable as alredy explained. This is, however, thought inevitable under the present condition.

As expressed in the form of the coefficient of the moment about the center of rudder-stock CmH, these

experi-mental results will be as shown in Fig.26 In this case,

the effect of the location of the center of the rudder stock is clearly observed as it should be so.

1.0 8 6

z

A= 0.25 v = 0.5 9 in/s , 0.76 -0.99 20 30° RUDDER ANGLE 1.0 -8 2 A = 0.50 0.63 m/s , [04 -10 20 30° RUDDER ANGLE

Fig.24 Results of rudder Fig.25 Results of rudder

test for the case of test for the case of

.05

05

10

20 30' RUDDER ANGLE

Fig.26 Moment coefficient

curves at stationary and non-stationary condition

Fig.27 An example of the oscillogram 20 - STATIONARY I --- NON-STATIONARY! 15 i -0.25 =1.5

-Regarding the Threes acting upon the rudder- in a

non-stationary condition while in the course of steering motion,

Fig.27 shows an example as measured by oscillograph.

By analyzing such an oscillogram, the coefficient of

moment Cial is found against the rudder angle just in the

same manner as in the case of the stationary condition.

Tig.26 gives a typical example of such analytical results

for instances of h==0.25, 0.30 and 0.50. These curves

Indicate greater values of the coefficients of moment as compared with the instances of stationary condition owing

to the effects of angular velocity of steering, Generally, .

in such a non-stationary condition, thefl critical angle

'where stalling occurs, usually lags by 5° 8° to be about

27'30° by the effect of the angular velocity of steering, as compared with the critical angle for the stationary

condition which is about 21.50 This is thought to be the

reason why this effect was observed with the coefficient of

moment for the rudder angle of over 25° in the, instance of

h==0,50, as appearing in Fig,26, whereas the Civil'. in the :non

stationary condition was given a smaller value than that in the stationary condition.

The ONE values in the nonstationary condition thus

measured was brought into comparison with the Cll values

obtained at the stationary rudder angle, so as to find

AC, thereby to make

comparison with the results of

calculations. However, in the theoretical calculation, the

calculated on the assumption that it keeps on increasing

until the maximum angle of 350 Therefore, it was necessary

to select a rudder angle below the stalling angle in making comparison with the results of experiments on models.

Accordingly, it was decided to make comparison in the difference of the coefficients of moment at the rudder

angle of 15°. Thus AC

IE

in the case of Fig.26 was found

to be 0.016 at h=0.25, 0.034 at h=0.30, and 0.062 at h= 0.50. The ACIvill values calculated in this manner, were

also led from other results of measuring, and were laid

down with the non-dimensional values

a

along the axis of

abscissae, obtaining Figs.28 30 for respective h values. For the sake of convenience in making comparison, the curve based on the results of theoretical calculations which were previously shown in Figs-10-13, is put down concurrently.

Since the conditions at the time of experiments do not

fully satisfy the assumed conditions for calculations, that

is, the condition that the angular velocity of steering is

to be constant, was not perfectly fulfilled, or the

calcu-lation was made on the two dimensional basis, or due to

other measuring error, the experimental results showed

deviations, and cannot be said to be quantitatively in

agreement with the results of calculations, yet they can be

considered to be close to the results of calculations in

their general tendency.

In Paragraph 3.3, under the title of "Law of Similitude", it was mentioned that the experiments were made on the

assumption that the rudder submerged deep in the water, and there was no wave on the water surface, whereby the effect

of gravity may well be neglected. In order to satisfy this

condition and to prevent the air draw as induced by the rudder-stock, the cutwater was provided around the water surface. This was also considered to be a kind of aerofoil,

and so it was feared to give rise to interference with the

body of the rudder. Thus it was thought important to make

this point reaffirmed. So by changing the length of the

rudder-stock so as to give 5 different depths down to the top of the rudder, a series of experiment were carried out. The result was that in the case of Model No.1 the changes in the water-depth down to the top of the rudder within the range of 6-15cm, would have no effect on the performance

of the rudder. Therefore, it should be added that this

series of experiments were decided to be carried out under

the condition that - the height of the rudder was submerged

in the water.

3.5 Other Effects on the Performance of the Rudder

The above finishes up the discussion in connection with the comparison between the theoretical calculations and the

results of experiments on the effect of angular velocity of

steering on the performance of the rudder. In order to

examine the propriety of the experimental results as viewed from other phases, let us deal with the effects of other

06

ci .02

01

.02 .03 .04 .05 .06 to,

=

9 c

_S-KtmL,C.

Fig.28 Experimental results

of Cjv compared with

theoretical ones.

(for the case of hr=0.25)

..0/ = 0.25 .01 02 03 04 05 06 Cd. Si= Fig.30 Experimental results of ,f_ CLE compared with theoretical ones.

(for the case of h =0.50)

01 .02 03 .04 05 .06

Fig.29 Experimental results

of LC compared with

theoretical ones.

(for the case of h 0.30)

Fig.31 Rudder models for

5 different draft. A-050 /0 08 1. 06 z 04 02 _c5)

making comparative study on the results of other experiments. Among many factors acting upon the performance of the

rudder in the case of open rudder, the principal ones are summarized to be the under-mentioned four items:

The effects of aspect-ratio. The effects of rudder section. The effects of rudder outline. The effects of Reynolds' number.

Of the above 4 items, (4)Reynolds' number has already been dealt with under the title of "Law of Similitude", henceforth, other three item will be discussed below.

(1) The effects of Aspect-ratio

When the dimension of the rudder is reduced more in the direction of the height, the stream which flows into the back of the rudder from the upper and lower edges gradually grows larger in proportion, and the force generated by

negative pressure acting on the back of the rudder, that is,

the lift (Normnl force), is reduced for the same rudder

angle. Namely, the smaller the aspect-ratio (in the case

of rudder, the height/chord length ratio) is, the more the lift will diminish, and accordingly the critical angle for

stalling will lag and grow larger. In order to generalize

the effect of aspect-ratio, the following relative formula is led from Prandtl's theory of aerofoil.

eLb CLa 2 CL Ab Aa CD1) = CDa+ 2r ( hb2 ha2 ) CL Ab Aa ab= aa +

573

7-,T ( ) nb na

where A: Area of the rudder

h: Dimension of the rudder of the direction of height

a: Rudder angle

The attached letters "a" and "b" signify the "rudder a" and the "rudder b" respectively.

This relative equation was confirmed, to agree compara-tively well with the results of experiments when the

circulation round the aerofoil is making elliptical

distri-bution(25). Basing upon this equation, Glauert, Betz, and

others introduced corrective coefficients r and

a,

and

assuming them as the function of aspect-ratio x, gave the induced angle otand induced drag CDL as the corrective

(46)

As Wieselsberger(25) stated, these equations cannot be

applied to a wide range of aspect-ratio , and the

coeffi-cient of maximum lift remains unchanged irrespective of

of aspect-ratio. Thus they cannot be said to be perfect,

terms against the rudder of infinite height.

r

77- X CL ( 47 ) + a 2 Cpi= _n7x 0L -+ 4

but even then they express such effects quite well.

Mr. Fisher(27) published in the "Werft, Reederei Hafen' the result of the application of the equation (46) to the

experimental results obtained at nttingen(26). This has

recently been referred to by Mr. Jaeger(28). Therefore, it

is again mentioned in Fig.32 for the sake of reference. It

may be observed that as the aspect-ratio gets smaller, CL becomes smaller for the same rudder angle, while the

critical angle for stalling grows more larger. The rudder

on which the author experimented had an aspect-ratio of 1.67, and it is laid down in dotted line in the same chart. It can now be noticed that the coefficient of maximum lift does not correspond by various reasons aforementioned, but the shape of lift is found to position about the half-way between those for the aspect-ratios of 2 and 1.33, which is

considered to be a reasonable result.

Next, among other experimental results, of which the coefficients of normal force Cm and the position of the center of pressure were given, and having about the same values of aspect-ratio as the present experiment, were selected and correctively plotted in a chart for the sake of comparison with the present experimental result as per

Fig.33. The chart includes the result of experiment

conducted at the water tank in Washington with symmetrical hydrofoil having an aspect-ratio of 1.0(23), the experi-tal result obtained by Dr. Akazaki with a model having an

. J00 t 12 080 og 0.3 .97 - .20 , RUDDER ANGLE 30' AspECT RATIO THIS EXPERIMENT 167 NACA 167(CORRECTEp WAGEN,N6EN 200 A C.A ISO WASHINGTON' 400 Dr.AKAZAKI, 4 1. 0.0

Fig.32 Effect of aspect-ratio of rudder

on CL according to Mr. Fi er.

r

Fig.33 Results of comparison with. other

r

: experiments: _z . :,

Kfi,c 1

,

111

litrjri

irr irVPI/TAr

₄₁₄

II(

11

"

A '

All/Prfr4V1*-

41°

Ail

/

or

i 1 ,

M.

le"

RESUL TOT THE AUTHOR'S

(ASPEC7;RATIO=167) J _ ASPECT RATIO 5 -7 - ° 133 -.20° 25° 30' 43 40* --- RUDDER ANGLE 060 10.40 020 08 067 /0° /5°

by Cowley, Simmons and Coales at the Advisary Committee for Aeronautics with plate rudder(29), and the test results that Dr. Van Lammeren obtained with a model of aspect-ratio

2.0 at the towing tank of Wagenigen(3), and that of the

experiments on N.A.C.A.0018, having the aspect-ratio of 6, the same aerofoil as used in the present experiment, which was corrected to the aspect-ratio of 1.67 by the equation

(47). According to those data, although the result of the

present experiments somewhat differs from that of N.A.C.A. as corrected, it is considered to give reasonable values as compared with other results so far as the coefficient of maximum lift and the critical angle causing stall are

concerned, when it is taken into consideration that the aspect-ratio used was 1.67.

(2) On the Effect of Rudder Section (Thickness-Chord Ratio)

A marked advancement has been witnessed in the field of studies principally on aircraft wings by experimenting their performances through comparisons between aerofoils

having different sections. Many such experiments were msde

by the wind tunnels at Gbttingen, N.A.C.A., and others. However, the aerofoils used in these experiments are mostly

unsymmetrical, and data on symmetrical sections which can

be used for ordinary rudders are limited to a small portion thereof.

The section generally used for rudders, with expection of special cases, are nowadays those of aerofoil types

having symmetrical sections, hence we will principally deal

on such rudder sections. As indicated by the results of

experiments on a series of symmetrical aerofoils (7 kinds from aerofoil 0006 to 0025) as carried out by N.A.C.A., the

slope of lift does not change even though the thickness/ chord length ratio varies within a range of 0.06-0.25, but the coefficient of maximum lift and the stalling angle

undergo changes as illustrated in Table 2.

This result suggests that the most desirable thickness/

chord length ratio is found within the range of 12-18%.

In this case, CD does not change up to the rudder angle of

about 300, even when the thickness/chord length ratio may

change far about 0.12-0.25. Among the Flugel's

experi-mental results obtained with different rudder sections,

those which have possibility for practical use as ordinary

ship's rudder, are cited in Fig.34. As illustrated by the

results II, III and IV of Fig.34, when the Joukowski's

symmetrical aerofoil is used, there will be no particular

change in the rudder performance even when the thickness/

chord length ratio may change within the range of

0.16

0.25. On the other hand, the plate rudder I which was used

in the past, and the rudder V of an existing ship with

which almost no consideration was given as to its section

RUDDER I RUDDER 5-7V (.70UKOWCKI PROFILE) RUDDER

240

--06 10 02 5° /0 /5 20° 2 RUDDER ANGLE

Fig.34 _{Results of experiment according to Mr. Flagel.}

Table 2, Experimental Data of N.A.C.A. Symmetrical Aerofoil. 0 30 /C 20

4

iimaly E

prr,

Am.

Ill

Fr

IPA

//

, damp,

P

47 Thickness/Chord Length Ratio Coefficient of Maximum Lift (GLmax) Stall Angle (Degree) 0.06 0.88 16.0 0.09 1.28 18.0 0.12 1.54 22.5 0.15 1.52 22.5 0.18 1.50 22.5 0.21 _{1.38 22.5} 0.25 1.20 20.0 V I 30°

(3) Effects of Rudder Outline

Generally speaking, The effect of rudder outline may be

said insignificant. Putting aside the R.W.L.Gawn's(9)

results of experiments on special forms of rudders used for warships, according to the results of experiments conducted

by R.C.Darnell(31) on rudders by changing the rudder

out-lines within the range of availability for ordinary ships, there is observed almost no effect in this respect.

Accordingly, for ordinary merchant ships, this effect will not be needed to be considered, so far as an ordinary type

of rudder is used.

4. CONCLUSION

Of various factors acting upon the rudder performance, principally the effect of angular velocity of steering has been discussed from both viewpoints of theoretical

calcu-lations and results of experiments on models, and could make their physical properties clearer.

It is contemplated to conduct further study on various problems such as quantitative estimation of the twisting moment acting upon the rudder-stock with due modification

for the effect of the propeller, which is considered

essential for ship-building technicians.

In conclusion, the author wishes to acknowledge the guidance and encouragement rendered by Dr. M. Kinoshita, ever since the construction of the circulating channel, and

also for cooperation of Mr.M. Takagi, who participated in