Measurements of side forces and moments on a ship model and a comparison with some simplified theories

17 ME 1979

ARCHIEF

AUG 1978

Lab. y. Scheepshouwbnde

Tcchnisdìe HogechooI

Deift

KUNGL. TEKNISKA HOGSKOLAN

I STOCKHOLM

HYD ROM E KA NIK

MEASUREMENTS OF SIDE FORCES AND MOMENTS

ON A SHIP MODEL AND A COMPARISON WITH

SOME SIMPLIFIED THEORIES

OVE SUNDSTRÖM

THE ROYAL INSTITUTE OF TECHNOLOGY IN STOCKHOLM

DEPARTMENT OF HYDROMECHANICS

SU1IiAPY

TRITA-HYD-78-O 3

MEASUREMENTS OF SlOE FORCES AÑO MOMENTS ON A SHiP MOVEL ANO A COMPARISON WiTH SOME SIMPL1FIEV THEORIES

by

O. Sundström

Using an oscillating mechanism hydrodynamic coefficients for a ship model are determined. Yaw angle, Freude number, keel clearance and reduced frequency are varied.

Treating the free surface as a rigid plane a basic equation for the influence of keel clearance is derived. It is

simplified for small and large aspect ratios. An alternative approach to the effects induced from the free surface is presented.

TABLE OF CONTENTS

EXPERIMENT

Pase LIST OF SYMBOLS 2 INTRODUCTION 3 DESCRIPTION OF MODEL 3 TEST APPARATUS 4 TEST PROCEDURE 4 INSTABILITY 5

REDUCTION AND PRESENTATION OF DATA 6

FIGURES

10-24

THEORY

BASIC EQUATION 25

SMALL ASPECT RATIO 29

LARGE ASPECT RATIO 33

DISCUSSION ON THE SIDE FORCE 37

AN ALTERNATIVE MATHEMATICAL MODEL 39

NUMERICAL VALUES 47

ACKNOWLEDGEMENT 50

LIST OF SYMBOLS

Symbol Definition

f _Frequency

g Acceleration of gravity

t Time

CL Side force coefficient

Moment coefficient F Froude number L Length of ship N _Moment T Draft U Velocity Y _{Side force}

A _{Reduced frequency parameter = irfL/U}

p _{Density of fluid} Yaw angle 410 Amplitude w Angular velocity ( )C

Refers to quantity out of phase with the displacement

Refers to quantity in phase with the displacement

INTRODUCTION

The prediction of the path of a surface-ship when it is

disturbed by currents, its rudder is deflected etc. requires knowledge of the mass of the ship, its center of gravity and its moments of inertia about several axes. Furthermore, the hydrodynamic properties of the ship must be well known. These are only to a limited extent available for theoretical calculations. Therefore, methods have been developed to

conclude from model tests the actual properties of large ships. For this purpose different kinds of test facilities are in use in order to isolate the movements that are most

important.

In this report an oscillating mechanism has been used to obtain experimental values of the hydrodynamic properties of a small ship model. Special attention was called to the

influences of Froude number and restricted waterdepth on side forces and moments. This is an extension of Ref. 13

DESCRIPTION OF MODEL

The tests have been carried out with a 1.1 m model. It was laid up of glass-fibre reinforced polythene plastics and laquered with polyuerethane. Longitudinal and transverse lists were inserted to strengthen the model. Model

parti-culars are shown in Table 1. and Profile Outlines are sketched in Fig 1.

The design draft was 70 mm but the immersion during the tests was 72 mm. Although Fig 1 shows a rudder the tests were run without.

The model was fixed to the balance and the water forces could not modify the model in any degree of freedom. Turbulence stimulation was effected by means of studs

cemented close to the forward perpendicular.

Table 1.

Length Between Perpendiculars 1.100 m

Beam B 0.160 m

Design Draft T 0.070 m

Design Displacement V

_9.918

dm3

Block Coefficient _CB 0.805

Longitudinal center of buoyancy

forward of L /2 LCB 0.028 m

pp

TEST APPARATUS

The test facility is described in more detail in Ref. 13. The towing tank at the Royal Institute of Technology has a rectangular cross-section with a width of 3 m and,in this case, a water depth of

1.2

m, and extends for a length

of 60 m.

The balance had two pairs of aluminium plate springs, which sustained the side force and measured this force by means of cemented strain gauges. An ABEM Ultralette 5651 recorder registered on a photo tape the electrical signals for forces, speed of towing carriage and, at the dynamic tests, yaw

angle.

At the dynamic test the model was allowed to oscillate around two different axes. For these axes dynamic equilibration was made when the towing tank was empty. Lead weights were

inserted into the model so that the integrated signals draw straight lines on the phototape at the most rapid frequency, that would be used in the tests. These places were marked and for each change of the axes the lead weights were re-arranged.

So that varying water depths could be simulated a plate of aluminium of the dimensions

_2000x1000x10

mm was used. The plate was sharpened at the leading edge and was attached to the towing carriage so that its depths of immersion could be shifted. Close to its leading edge, in front of the model, a string was streched to stimulate turbulence.

TEST PROCEDURE

The plate springs were individually calibrated with standar-dized weights.

The model was symmetric about its lateral plane. In order to obtain the yaw angles test series were run at three very

small yaw angles. At the third run the side force had reversed its direction. The measured values were faired using a

least-squares procedure and the zero yaw angle was calculated. Afterwards the yaw angles were determined by measuring the

displacement of two bens with a micrometer.

For the static test, the towing carriage was brought to initial position. Reference signals for the two integrators were recorded. The towing carriage was then brought up to a predetermined speed, and when steady conditions were reached the recorder was started. At the end of a run, the model was towed slowly back to the starting position in the tank.

A waiting period of _15-30 minutes between the beginning of succesive runs was taken to allow the water in the tank to become free of waves and currents.

The sequence of tests is summarized in the following table. Table 2 Yaw angle 593 326 220 Table 3 Reduced frequency

0.15< À <1.5

Keel clearance mm co,

150,

43,

30,

_{24, 12,} 6, 3, 1.1-1.2

Each angle was carefully measured. For the hydrodynamic coefficients used variations of ±00.1 were regarded as negligible in the summary. The Freude number was varied for each angle and depth.

The velocities of the signals from the integrators were measured and the values were converted into forces by means of the calibrations. The gauge length of the

weakest forces was about 30 meters and the integrators were operating for about 90 seconds.

For the dynamic test one Froude number was selected

(

0.141

±0.003 ).

Reduced frequency and simulated water depth were varied. At first the model oscillated around an axis through

Ihen around an axis

269 mm

forward. The frequency range and simulated water depths for each axis are shown in the following table.

Keel clearance mm

co_,

10

_{1 ,}

43,

30,

24,

12,

6,

3,

0.9-1.0

Besides the natural phase shift of 900 between the integrators and the yaw angle there was a phase shift due to the hydrodynamic force. For each integrator the

amplitude (a) and the phase shift of the signals were

measured on the phototape. The frequency (f) was easily

determined. The number 2rraf mm/s could then be converted into a force by means of the calibrations. The force of each plate spring was divided into sine and cosine

components.

INSTABILITY

At the static test, the rear pairs of plate springs

measured a non-stationary force when the water depth was small. This was first observed when the keel clearance

was 6 mm and the yaw angle 2°. Occasionally the distur-bances were nearly sine-shaped. Some values are given below

that show the order of magnitude and period of those forces. The yaw angle is 20 and keel clearance 3 mm.

Table L

Fn Period sec. Amplitude

0.09 6.5 Side force

0.11 " 5 ' Side force

0.22 3 (Side force)/2

Due to the integrator and rather high frequencies these forces were recorded on a reduced scale on the phototape.

At the dynamic test it was possible to find traces of this disturbance when the keel clearance was 12 mm. As the reduced frequency increased the disturbance force decreased. For all depths it was thought of to have disappeared when A " 1.

REDUCTION AND PRESENTATION OF DATA

For the static tests, the total side force (Y) and moment (N) acting on the model were obtaind from the two plate springs as

follows:

+ 'R N

Fi -

_R''

-where the subscripts F and R denote forces measured at forward and rear plate spring, and is the distance from midship to forward plate spring. The space between the plate springs was

1,000 mm.

Using Q pU2TL/2 the nondimensional forms are

CL Y/Q sin jJ _and

C N!QL sin '

The nondimensional derivatan be written

y' z _{T/L and N}

_CM T/L (Ref. 11)

For the dynamic test the yaw angle varied sinusoidally

'1'(t) z _{sin wt}

( L1°97)

The side force was the sum z

F + R where F and

had been divided into different components

- sin wt + Y cos wt

YR: Ysinwt +Ycoswt

The side force could thus be divided in the same way

z _{sin wt} YC _{sin wt}

The moment acting on the model was obtained from

N z

F1 -

-

NS

sn wt

+ NC wt where

NS _Y

_Y(l

-) and NC

Y(l

-he nondimensional forms are

C Y5/Q sin o , Ns/QL sin

C C

Y ¡Q sin q'0 and

C,

NC/QL sin

The diagrams show values when the model oscillated around an

axis through L(ebut the moment is refered to an axis through L /2

pp

In order to estimate the generally used coefficients in ship

hydrodynamics, the model was allowed to oscillate around a

second axis. Then the following assumption was necessary: The measured changes can be explained by the linear

expressions

Y = Y y + Y r + Y. + Y.f

y r y p

N = N y + N r + N.r + N.f' (Ref. 11.)

y r y r

where the dot above the symbol signifies the first derivative

with respect to time and rq', vvelocity in y-direction and

Y =

Y/v

etc.

V

When the model oscillates around an arbitrary axis at a distance s from midship the velocity of the origin, in

y-direction, is

rfhe side force is then

Y (Y.sw - Y.u2 - Y U)P0 sin ut +

y r y

(Y - Y.0 - Y s)'F0 cos ut

r y y

Let index i signify quantities derived from TD oscillation

and 2 the axis forward. Thus

Y1 - Y2 Y.(s1- s2)w2P0 sin ut - Y(s1- s2)w'Yo cos ut

In accordance with the linear assumption

a + hA2 and cA _; where the constants

a, b and c were determined by means of a least-squares procedure. Identification of w2 gives

(b1 - b2)T/14(s1 -

s2)

The in-phase component gives then

Y_r

Ys1/L

_y - b1T/14L and the side force was

deter-mined from Y _{-T(a1 + a2)/2L}

Out of phase component gives

+ (T/2L)(c2s1 - s2c1)/(s1 - s2) The prime symbols are, according to Ref. li,

Y 2Y /pL2U Y 2Y./pL3

V V V V

2Y /pL3U and Y: 2Y./pL

r r r r

The same method was used for N, which has the prime symbols

N (Y/LY

) etc.

V V y y

Table 5 shows a summary of the derivatives obtained by this method. Keel clearance is 6.

The measured out-of-phase values were regarded as linear in the range 0< A <1 . The above mentioned method was not

applied to the keel clearances

_6,

₃ and 1 mm because the linear assumption was then too inaccurate.

-9

Table 5 00 -19.9 -8.0 4.0 -3.4 -14.4 -0.9 -0.6 -0.8 2.09 -22.2 -9.1 3.8 -4.1 -15.4 -1.5 -0.7 -0.9 0.60 -28.9 -14.3 5.6 -3.4 -19.5 -1.8 -0.6 -1.1 0.42 -33.9 -16.8 6.3 -4.2 -23.2 -2.8 -1.2 -1.5 0.33 -37.8 -19.2 3.8 -4.1 -28.8 -2.6 -1.5 -1.7 0.17 -48.9 -26.9 3.4 -5.0 -34.2 -2.7 -2.0 -2.1 cS /T y-V V y-r N r V V y r N r x103

Fig.

J.

Profile Outline

-0.6 0.4 0.2 0.6 4 0.2 0.6 0.4 0.2 0.6 o . 4 0.2 11 Q Sa S Q O

.

C

p

S

S

I

a

OD

_{o p} U C D _o Q C D

Keel c:len rarrr,e YaW i np I u: r : s q) = 3:)) q) »:n O W

DI

D a

.

D C C O C _o o o

Kuci u .1 ui :lruc : IO nia

Yaw ;irilc: q r

q) r D q)

r 20

O

r!0e

«:i0P0ó:0.

D

o 3

J

o

Koc I u 1c)irariuc 14 _nia

Yaw nnh1c: q) r = n J) r ,)() O

"

L U e o

:

Ki3u I u l:ii.iriuv: 111111 Yaw ;irj J) L: :o n

:i

o 0 10 0.15 0 20 0.25 0.10 0.15 0 20 0.25 0.10 0.15 0 20 0.25 L ri 0.10 0.15 0 20

CL 0.8 0.6 0.4 0.2 0.8 0.6 0.4 0.2 1.2 1.0 0.8 0.6 0.4 0.2 0.10 Fige 2 _Contd.

-

12

-D _D S D _o q s u O o o D D _0Ds o o O _o O O O O o

Keel clearance: 2L inn

Yaw angle: _5:3

,

3:6 D = _2:0 o . C s s D

.

s I D o O -O O o o a o o O o o Keel clearance: 12 mm Yaw angle: p 5 3 36 o 20 ° a a w S W s s L D D D o o O _L D O o o

Keel clearance: 6 mmmi Yaw angle: mp = 5.3 s

= 36 D

p = _2:0 ° 0.10 0.15 0 20 0.25 0.10 0.15 0 20 0.25 CL 0.15 0.20 0.25 F n

1.6 1 .4 1.2 LO 0.8 0.6 0.4 0.2

13

-D o n s _s

.

s ) s D _9. D U D o o o D

- -a--o o u Keel clearance: 3 mm Yaw angle: i ₃ s

36 D

O 0.10 0.15 0 20 0.25 . 2 Ccntd. CL 1 .8

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 s D s s

o

s s D 505 D o o a

'D

D C _o e o o Keel clear'ance: i mm Yaw angle: _5:3 s 3.6 o p

20

O 0.10 0.15 0 20 0.25 _n

Fige 2

Contd

0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 O . 2 0 10 0 15 0.20 0.25 n

Fì. 3

Nondimensional moment obtained froon

tatic angle tests

15

-O o o O OD

_.

C

I

V 43. o J3 Keel clearance: Yaw angle: q _{5.5 e} z 3.6 D 2.0 0 a o _o C O O O o o o s s o

,D ,O.0

o

n

Keel clearance: 150 rna

Yaw angle: 5.5 s z 3.6 2.0 0 o o o ₉

C O

c

p.g

Keci clearance: )43 mm Yaw angle: ,Ù _{5.3 e} = 3.6 D 2.0 0

U

o B

c°WUs

111111

i

° e

IN

U

III

uuau

uuua

uuu uu

_UI

u

UUUU lu

_{KeeJ clearance: 50}

rien Yaw angle: _{5.3 e} z 5.6 _n 2.0 0 0 10 0.15 0 20 0.25 n 0 10 0.15 0.20 0.25 n 0 10 0.15 0 20 0.25 n

CMV) 0.4 0.3 0.2 0.1 0.4 0.3 0.2 0.1 0.5 0.4 0.3 0.2 0.1 -

16

-D O t; 4) . LP a

O. D0

-OD ecl D O . O. Keel clearance: »4 mm Yaw angle: l

53

.

3:6 2:0 0 o CO OU s s o q s °a OD D .

O

1 s Keel clearance: 12 mm Yaw angle: _{5 3}

r36

20

s'

8 s a _a a

sassa .a

o O O O o ----o---Do O u e Keel clearance: 6 mm Yaw angle: L ₅ 3

pz36 D

r20

0 0.10 0.15 0 20 0.25 F n O 10 0.15 0.20 0.25 F n 0.10 0.15 _{0 20} 0.25 F

Fig. 3

Cotd,

n

CM 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -

17

-o . _. ri ' Do D o o ° a D o o o D Keel clearance: 3 mm Yaw angle: 5 3 S iJi 36 a o h _t

.,

D

e . S :iP -o o o o C I D o Keel clearance: 1 mm Ynw angle: t S.

'

56 D 20 O 0 10 0.15 0 20 0.25 CM n 0.10 0 15 0 20 0.25

Fige 3

Contd.

1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

05

10

1.5 À

Fig,

4

Nondimensional side force in-phase with displacement

18

-o

j

iumuui

oSlUj.IUU

I

u

o °

inhllil

i:

u..

iu::n:n

_uuuuu

_..uuu

iii

uuuuuu

_A A

iu

u

AA

11111

uil

iii.

_

.11:

_I.

ii

Keel clearance: mm

150 mm £

I

ICI

U

INRI

.

II. III

R...

I

.iiiII.

i

ii

_.I.II

III...

.1.1.1_Iv

L'

.1.1111_I

urnu

UlvIlv

r1i

auuuuuuuu

U.UUI.IuuII

Ivvv

U..

IU1R

uruu

uuii

u_u..

.-1

I

i

FiiïiuIiiI

....I.I

iliPli

i

0.5 1,0 1.5 À Fige 4 Contd.

19

-s Lt 1.6 1.5 1.4 1.3 1 .2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Cs 1.6 1.5 1.4 1.3

12

1.1 1.0

09

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

IIKeel

clearance:1 nrc o

III

mm A 3 mm

i

II. iiiiiIiU

i

uiiiuuiui

IIIIIIIIIIIIIIIN

iuu uu uuuu u u u

"III.'-i "III.'-iu"III.'-iu"III.'-i"III.'-iuIl"III.'-i"III.'-iuuuI

u usauuuiuuuuuu

IINiiiIIiiiiIi

uiiiiU!UiIiii

Iii.

1

1111111

lu. aauuu

-

20 -0.5

10

1.5 X

s MJ 1.6 1.5 1.4

13

12

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

- 2

-Keel clearanCe: 3 nun

::

150 mm L

iuuu urns

uuuuuuuuuuuuusu

uu...__. u_

u.u...

u.u...

I lui

il.

uIuu,uIuuIIIuII

11111 11111111

IIUIIII!iIiiII

.uu.uu-..uu

uuiuuuicuu

o5i

u.u...

V

uusuuu

INii1,IuII"IN

L

.uuuiuuuuuuui

05

10

1.5 A

Fige 5

Contd.

cc L 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 2 mm A

iiuu

mm

u.

uuuuuuu

"l'In"

IININIIIIIIIIII

iiiiiiiiiiii

NI'-"-

u...

u.. u...

_{u. uuuuu}

iI,NIIÌIIiiIII

u.

1II"u'uIIuI'II

_{II uIM.uu}

I uiiiuiiiii

-D A u

22

-1.0 1.5 X

3. 3. 2.5 2.0 1 .5 1.0 0. 5 4.

-

23

-Keel

111111111

_clearance: 3 mm 150 mrnA

.1

I IIlIllllIll

1IIIIIllli

IllIllIllIll

IIIIlIIIIIll

lIlllUIRl.ii

IllIllIllIll

llIlIl

IIIIIIIIIIIIII

IlIIIIIllllIC

IllIllIllIllIllIl

IllIllIllIllUIlIl

IlIllIllIllIllIl

IllIllIllIlMIll.

IlIllIlIl."..'

1111111111

1IlIRI'

IlIllIllUlli

_1.I.1111.

"mii.".

lIIllIlIrnI

iì,iì'I,iiì,

I 11

II

1111

Il

11V

II

G

I

III

_.

all

_Il

I

WIIII

05 1.0 _1.5 À

Fig.

E Contd,

-0.1 -0.2 -0.3 -0.4 -0 1 -0.2 -0.3 -0.4

Fig 7 Nondimensional marnent out-of-phase

with displacement - 2 -!

:'E

o

I'uÑ

o . ¿ a o o o o e: lmxno Keel clearanc 6 mm I 21 mm ¿ 3mm s

p,.

.

VVV

£ £ .

.

V V .

3 mm

Keel clearance: 12 mm 30 mm G l5OmniA 0.5 1.0 1.5 A 05 1.0 1.5 A

THEORY

BASIC EQUATION

The fluid is assumed to be inviscid and of constant density p. The Froude number is low and the free surface is

suppos-ed to be plane arid horizontal. The bottom is also assumsuppos-ed to be a horizontal plane.

Let (x,y,z) be a cartesian coordinate system fixed to a

plane rectangular plate, which moves at a constant speed.

The z-axis points vertically upwards. The angle between

the x-axis and the free-stream velocity, U, is 'V. The

length of the plate is i and its immersion is b. The water depth is d. The clearance underneath the lower edge of the plate is 6, so that db+6 . The planes

zO

and

z-d

now act as mirrors. The real aspect ratio is b/i and the re-flected aspect ratio is 2b/i . The space between the

reflect-ed planes of bottom is

H2d.

y

25

-X

d i

Let the velocity field in the x-direction be

U cos V +

a/ax

is a harmonic function and its singularities are

distributed on the planes yO ; O x .Q _; -b z+nH b

where n= O, ±1, - - -

-To begin with there are a finite number of plates to take the bottom into consideration. Hence nl <

In accordance with a form of Greens theorem the pertubera-tion velocity can be written

b

)/3x

(1/2)

dI

d [ yir3 + 2y/p3 o -b n n (i)

where

_rr(n)

{(x_)2y2(z__2nH)2}

and

pp(n)

{ (x_)2y2+(z+_H_2nH)2}

Then

4Ix

±3M/3x

_{as y ±0}

on the defined cuts. Thus,

2M/Bx

defines the x-velocity jump across the plates. The Kutta condition, which requires finite velocity along the trailing edge, is

M/x

O when x.Q. The velocity jump is

zero along the lower edge of the plate. Hence

M/x

0 when

z±b.

A first integration in the x-direction gives

y + f1(y,z) +

(1/2)JJddc M/[

_{iy2+(z_ç_2nH)2} r R y y2+(z+-H-2nH)2 p

where R is the rectangle O x .Q

; -b z b . The condition

that there is only a uniform stream when x- gives

fi yU sin +

(1/2)JJdd

M/a[

_y2+(z2nH)2

₊

Fi

y

y2+(z+H2nH)2

As a result of this condition the free vortex sheet down-stream of the plate is at once included in the expression for

x ). A partial integration in z-direction gives

yU sin ₊

(1I2)IJdd

[

tan1 z--2nH

i

y R

-1 (z--2nH)(x-)

_tan1

z+-H-2nH

t an y r 2 tan1

(2)()

y p

Differentiation with respect to y gives the y-velocity

component and its limiting value when yO is

26

-y

-/y

+ U sin

(1/2)JJdd

I i

z--2nH

+ R

/(x_2+z__2nH2

i

(x-)(z--2nH)

- 2 c

z+-H-2nH

+

/(x_)2+(z+_H_2nH)2

(x-)(z+-H-2nH)

The integrals here and later are to be taken with their principal value in the sense of Cauchy.

Now we intend to enter into details respecting the sums of the equation. If the notations are changed for this occasion then the troublesome part of the first sum is

(3)

Now the

integrarids are

positive and the sum is convergent.

According to the monotone convergens theorem summation with respect to n can be done when N - . Using n as a

com-plex variable the "cot" method of contour integration gives

27 -s1(N) xiiNç- i ' N +

i

IdOsjn2O z' x1 du -N n-z1 nt-N j Jx+(n+u)2sin2e -z1

nl

zi-n

where

z1<

The plates are put out in pairs so that

S1(N) /x+(zi_n)2

ii

N

_{/x+(n_z1)2}

n z1-n

-' n-N Z1-fl Z1+fl

j

where N< For

z10

this series is absolutely convergent.

If

x10

then S1(N)z1/z1

Using ir

2(x+(z1+n)2)

j

dO(xi+i(n+zi)sin

e)

when x1 >0 the sum can be written

S1(N) 1. 1. 'I n=-N n+z1 n-z1

j

+ N 1T/2 _n+z1

_n-z1

j

aecos2e [ x+(n+z1)2s2e - x+(n-z1)2sin2e 11

n-N

o

ir/2 1Z1 _f

x1

cot

z1 + de Slfl I

cot

7F(u-2i

_-z1

sin

xli

cot

1T(u+

sin e

_]

Evaluation of the

u-integral

gives

11/2

Si()

x1

cot z1

+

desin O ta1

_[

_(tan

_z111)

_coth

X111

se]

Jo

If the plates are put out in pairs then the second sum is

-

&+(zi__n)2

1

/+ (z++n)2

2\ '

- 2

Z1--fl

-

₂

z1--n

+

zi++n

n

n-N-1

where

_zij

< .

If

x1O

then

S2(N)O

Using the same method as above the sum is when

x1>O

N N 71/2

S2(N) -

- [ X Xi + _{1 Y I dO}

xisiri2e

du

-

fl+Z1+

flZl]

71

n-N-1

nz-N-1 i J

x+(n+u)2sii

If now

N -

then

he result of the calculations is

Fr12

X17T

tan z

+

de sin e ta1

_[(tan

_{z171) tanh}

J

For

N

the boundary condition is

/yO on the plate.

Using the original notations in

S1

and

S2

the integral

equation for the distribution

2M/

is

1 f1 _{________}

2M()

7F z+ç U

dd

[ 1+

)( cot

+ tan

=

sin

JJ R 1F j

[(tan

coth

712H sin O 71/2

-

2

_{1 1d0}

sine ta

1(tan

7F)taflh

_{712H sinO}

J j

ri

x-J

O

If H-

then

2 1 fTf/2

i

z-1 11 2M

L4_(1+!I)

₊

- dO sinetari sin

U sin

JJ

dd

o R o

28

-71/2 2 J

de sin e

ta

7F X-o

r +b

=

dJ

dM

I ¡(x_)2(z_)2

-b

(x-)(z-)

The second form in the case H=c. is given in Ref. 12.

On the basis of the complexity of equation

₍₎

the two cases small and large aspect ratios are considered in order to obtain approximative numerical values for side forces and

moments.

SMALL ASPECT RATIO

If xx2 and

then the argument of the hyperbolic functions is

2 H sine

Assuming iril(H sin e

) »1 we put

tanh coth 1 . Then the

simplified equation

()

is 2 '

i1

a2M

1--1+)(cotî-

+

taruiî--)

-U

sin

'Y -jj 2H- x-R (5)

It is clear from this equation that the distribution has the

form which gives

A(x)r+b dE()( z+ U sin'Y 2H d cot + _{j +} r+b

i rZ()d1

I B(ç)dç + d o

B() is an e4n function, therefore, in this case,

tan

(z+)/2H

may be replaced by -tan

(z-)I2H

Hence U sin'Y H 'b z- ç cot ir

Using the transformation equations

tan zn/H -tan(brr/H) cos and

tan çrî/H -tan(bir/H) cos e we have

29

-+ B(ç)dç

d

A(x)

_Irde

(i+tan2y cos e

cos

i

U sin

_{cos e}

_{- cos}

+

NJ

---

¡ _{B d}

H tan y

o

(6)

where

ybir/H. If

dB/de

(i+tan2y cos2e

'cos

e

then

3(e)

_2siny

cos2y

i

+ sin y sin e

_sinysinO

This describes the loading across the span. In the limit H

it is elliptic

B(z)

/(1 - (z/b)2). Using

dB/de

and the

formula

Principal value of

ir

o

equation

(6)

becomes

ir cos2y

U sin

A(x)

_siny

cos nO

de

cose - cos

7r

sin n

sin

2. +

i

dA()__

_I

B() d

d Jo

In order to integrate the loading across the span, the following

integral is needed.

cose

_ta

_{((tany) cose) de}

1(y)

I

i + tan2ycos2e

2

n 2n-I-i

2cos y

_{de ta}

_{((tany) cose)}

_(-i)

_tan(y))

Sifly

J_-ir

fl0

X

cos(2n+i)e

cos2y

sin y

cos2y

_{in cosy}

sin y

(tan2(yfl2fl

2n+i

n0

III

Hence

fBd

-J

.(e)

dB(e)/de de

HI(y)/2ir. Then the equation

o

i

dA d

is

U sin

A(x)

_{H siny}

cos2y + cos2y

_siny

( Th(1/cosy))

_J

(7)

o

This equation may be put in a dimensionless form using

T

(2H12.rr) ln(i/ cos y)

and the transformations

A(x)

0(x) U sin

H sin y /(î cos2y )

x

(i - cos s )/2

- cos

)/2

-Henc e i C(s) +

I fl

dC(o) do (8a)

IJ

do

coso - coss

o

It is not known if this function C(s) is tabulated. To obtain approximative values we proceed as follows. The

func-T fldC0(o) do

i + r(s,T) Co(s) +

J do cos o - cos s

o

The remainder r has a simple physical interpretation. It

is an error from the real angle of attack. Using functions that fulfills the Kutta condition we can write

C(s)

z

_{a1(s + sins) + a2(sins}

_{+ (sin2s )/2 ) +}

..+ an((sin(n_i)s)/(n_i) +(sinns)/n

In order to make the influence of r small, the following

conditions are here used:

r0(,T)

= O and

Jr(s,T)

sin js ds = O

for

j

1,

2 ...(n-i)

These conditions give a system of n linear equations for

the n unknowns a1, a2 ...an .

If

n4

then we obtain the following coefficients

a1(T)

(rr3/32

+ T7r221/5 + T2ii376/25 +

_{T3163814/75)/(T)}

a2(T) =

(ir3/32

+ _Tir216/5 +

T2r736/25)/(T)

a3(T) z

(ïr3/32

+ Th21L1/5 +

T2r32/3)/(T)

ak(T) = _(Tr33/32 + _T7127/5 +

T2r32/5)/(T)

where

(T)

3/32

+

T32i/5

+ T27r2i!36/25 + T3ri89414/75 +

Ti638!/75

31

-(8b) tion is divided thus:

C(s) = C0(s) C1(s) and further we put

C1(s) + T J

rdc1(0)

do - r(s,T) then do COSO o - cos

32 -table.

Table 6

number of functions in Cv(s) is shown in the following

T f. 0 0.25 0.50 0.75 1.0 1.5 2 ₃ 10

n2

1 _{.970 .918} .865 .8i .725 .653 .5142 .2116 ¶a1(T) n3 i .91414 _{.891 .840 .792 .708 .639 .533 .21414} nz4 1 _.956 .900 .8146 _{.797 .711} .641 .534 .21411 nz2 0 .097 .140 .164 .180 .198 .209 .221 .241

n3

o .084 .128 .154 .171 .192 .204 .217 .239 0 _.099 .139 .162 _{.177 .196} .207 .219 .240 çi ç+b 2pU d d Jo _-b

sionless by pU2b sin ' thus

aM( /a

For small yaw angles cos ' 1 and the side force is

This force is made

dimen-rrTHa1(T)/b (9)

The moment coefficient about mid-chord and the center of

pressure are given by

CN, rrTH(a1(T) + a2(T))/11b (10)

x/

(i - a2(T)/a1(T))/4 where X denotes di-stance from leading edge.

IARGE ASPECT RATIO

In this case the chord is small compared with the span.

Therefore, equation (4) is simplified by setting x- O

Henc e

2. r+b

32M îr

Z+)

1f2.aMd

u sin d I d (cot + tan

d'-b

Then the distribution has the form

M(,) = F()G()

If F(x) -

si«

(1-(2x/2.)) + 2/(x/2.)(1-(x/2.))

then

f dF()/(x-)

is a constant. Thus the equation may be written

1+b

dG() cos(7r/H)

U sin'

J_bd d sin(z/H) - sin(/H) + 2G(z)/2.

If H - this equation tends to the lifting-line equation

for a rectangular wing.

Using yb/H and the transformation equations

afldx(o) do asin2y

1100)

sino

da cos a - COS 5

-

l-sin2ycos2s J 1-sin2ycos2o

+

a sin2y cos s 1 1

1-sin2ycos2s o

where ar22.I(4H sin y).

In this case the dimensionless side force is given by

K(o) sino

cos a - Cos S + K(s) (1_sin2ycos2s)

-

33

-(12) sin(iîz/H) -sin sin(ir/H) = -sin U smi' y cas s and y cas a we have (11) í dG(a) /1_sin2ycos2a I da + 2G(s)/2.

2H sin y J do cos a - cos s

o

It is convenient to put

G(s)

K(s)LUsin(4(1_sin2ycos2s))

in order to avoid the square root in the integrand. Then the equation may be written

K(s) sins

FI siny CL _{- b} J ds

_{1-sin2y cos2s}

o

and the moment coefficient is

CM

First suppose that

y0

.

Then

ctîr2/14b

and we put

K(s)

e

sin(2n-1)s

n1

To solve this

K(s)

we use GlaurtTh method, which consists in

neglecting all coefficients

e

of order greater than some

specific n. To determine the oher coefficients

n

distinct

values of

s

are selected to obtain a system of

ri

linear

equations for the unknown

a1, a2,

.

_{. .an}

.

Thus for the case

n14

we use

s

220 .5, 1450, 670.5 and

9Q0

and obtain

e1 (a)

(a2375.8728 + cx21121.6513 + a161.5589 + 7.11) /Ai(a)

e2 (a)

(a2133 7595 + ci314.5285 + 2.1166) /A1(a)

e

(a)

(a210.6664 + a8.9214 + 0.9149) /Ai(a)

ek (a)

(a21.5113 + al.2166 + 0.2812) /A1(a)

where

A1 (cx)

a"2375.8728 + cz3138.9985 + a21091.1111 + a138.2515 + 5.6571

The side force coefficient is here

CL

(15)

Now the case

>0. In the expression for

_0L

there are the

following integrals:

í

sinno sino

(t(1/2))n_l

_for

_n

_odd.

1-52yCoS2

do

- 2cos2(y/2)

o

If

K(s)

sin(2n-1)s

then

2n-2

CL

f1

+ f2(tan(y/2))2 +

. .

.+ f(tan(y/2))

+

and as

y

increases terms with large n do influence

. Even

if

f

is small, this contribution is large if

r

_{n+1 as}

-

ir/2

.

Therefore, we put

K(s)

f1 sin s

+ f2 sin 3s + f3 sin Ss

and use the angles

30°, 60° and

90°

to obtain three linear

equations. This method is limited to the case when

is rather

-

34

small. An estimate is given by f2(tan(y/2))

<<

fi

1-(tan(y/2fl2

For numerical application it is convenient to solve f1, f2 and f3 from the following equations where p cos y

30° 1 + 2 12a(1+7p+7p2+9p3) 2

t132

/(1+3p2) t (1+3p2)(l+p) /(1+3p2) 2a(-1+8p+14p2+2Op3+9p) +f3 + (l+3p2)(1+p)2 /(1+3p2) 600 1 f1 /3 - f2 2a(1-p) + 3+p2 /(3+p2) 1+p - f3 2a(9+8p+16p2+!pB+3pL) /3 ) (3+p2)(1+p)2 /(3+p2) i 1 f1 (ap+l) - f2 ((p+)+1) + f3

[(

13p+2p2+p3) (1+p)2

An alternative procedure, suitable for larger -Y, consists

in setting

G(s) N(s)th sin ₍ 1-sin2y cos2s

+ N(s) (1_sin2ycos2s)

where

2/(lHsiny)

as before. If

N(s)

Asin(2n+1)s

we now obtain

nQ

CL (A0îrHsiny)/2b -

35

-Then equation 1 = (11) ( dN(a) da is (1-sin2ycos2a) ₊

_asjn2yI

11) Sina COSa daN(a) da o

Using this series the equation may be written

ss

a(1- (sin2y)/2) (2n+1) sin(2n+1)s +

- ( asiny)/L (2n+1)(sin2n-1s + sin(2n+3)s) +

nQ

+ ( asin2y)/14 A (sin2ri-1s _{- sin(2n+3)s)} + zQ

+ N(s)( sins) (1_sin2ycos2s) (16)

This equation can be treated by the method suggested by

Carafoli. For this purpose the following series expansion is made. When O s 7t so

( 1- sin2 y COS2S ) sin s c cos ns

no

Neglecting all coefficients of order greater than 2 we put

( 1- sin2y cos2s ) sin s - K cos 2s where

((y! sin y ) + cos y ) / and

K (y(1-cot2y) + cot y ) /( sin y)

If y'îr/2 then and the expression is exact.

A selection of sinns components gives then the following system of infinitely many linear equations for the infinitely many unknown coefficients A

n sin s

2 (a(2- sin2y) +2+K) A0 - ( a sin2y +K) A1

sinns n >1

O a2(2- sin2y)(n+1) A - (a(2- sin2y) -2e) A - a sin2y nA1 +

- KA1 - asin2y (n+2)A1 - (K_a sin2y)

On the assumption A

_{1tF(t) dt}

this system may be solved. A partial integration gives a first order differential equation which is satisfied if J tan2(y/2) and

F(t) constx (CO2(1/2)_t)Q_(cos'(Y/2)/cosY) X

x (tari2(y/2) -t) sin(y/2) ¡cosy) -Q

-where

Q = (a(2-sin2y)-2)/(a4cosy) K(l+cos2y)/(a2sin2ycosy)

Using the substitution t s tan2(y/2) we have

A COfl5tX(t(Y/2))2n

1ls(Ksmn2y)

ds n ((cot(y/2) -s)(1-s)) [ cot(y/2) -s ]R where

R z (K(1+c0S2y)/ sin2-y - )/(a2cosy)

It is clear from the integrand that R < is a needed condi-tion which restricts the availability of this solution. To make the integral suited for numerical integration a new

change of variable gives 2n

An k(tan(y/2)) I where

I (12/(1_2R)

)fl+ K/(Sifl2y)

(cot(ç/2)

1+y2/2 )R_dy

n _J0

The constant k is chosen so that the sin s relation holds.

Thus for the dimensionléss side force we obtain

0L (iîH/b)siny /(ct(2-sin2'y)+2+K-(K+sin2y)(I1/Io)tan2(y/2)) (17)

and here z

(18)

DISCUSSION ON THE SIDE FORCE

-

37

-If the aspect ratio is small and the water surface is treated as a rigid plane then the theory underpredicts the measured

side force by a factor about 1.5. To obtain theoretical values for a restricted water depth a common method is to

correct for this deviation by counting the influence of bottom from an empirical level.

For the reflected aspect ratio 1, according to Ref. 13, the center of pressure is close to the quarter-chord point. For this reason the large aspect ratio case has been regarded in the theory. Each strip of the span is supposed then to be in a pure two-dimensionel stream. This, of cource, is a

The experimental datas in this report show that the side force is nearly constant in the range Fn <0.2 . According to

measur-ements reported by Gerritsma et al Ref. 14 the length/beam

ratio affects this force only to a limited extent.

At F0.21 the side force coefficient of

_{a surface model is}

about a factor 1.5 larger then on a submerged double-body according to measurements by Norrbin Ref. 9 . Therefore, in

the following the free surface condition for an ideal fluid is discussed in view of a surface-piercing flat _plate.

The undisturbed free surface is the plane

_z0

_{and the z-axis} is positive upwards. The vertical elevation of the free sur-face is f(x,y) . The plate has the time independent velocity

U. It is now assumed that the surrounding velocity field is

irrotational and has the potential Ux ₊

The pressure (p) is constant on _{f(x,y), which also is a} stream surface. Hence, on z=f(x,y) ,

Bernoulli's theorem: ( v )2/2 + gf U2/2

Kinematic condition:

_{(f/Bx,af/y,-1)}

o

If V

»U then

_{f _(v)2/2g}

. Thus, a large velocity

gradient makes a depression on the water surface. According to the wing analogy,this is the case at the leading edge. This cavity changes the pressure and disturbes the velocity field. The linerazed surface condition,

(U2/g)2/x2

+

/z

O on

z0,

can_..not describe this effect because

_f-(U/g)a4/x

_and

there is a depression or heigh depending _{on the sign of}

_/x

Using this surface condition Hu (Ref. 6) has studied the

effects of finite Froude numbers on the side force. He obtain-ed a F dependence which is not realized in model tests.

Chapman (Ref. 2) has studied a restricted non-linear descrip-tion of the free surface. Using

jrT

+

(a/y)2/2

+

(/z)2/2

¡tí&

on the free surface the result is that the flow and free sur-face elevation are influenced but the side force and moment are not significantly altered.

These investigations point to the fact that, maybe, only the

total free-surface condition can describe the behaviour of the flow as

F 0

On the basis of the mathematical difficulties to obtain such a _{if it exists, a different approach to the problem} is here made.

-According to Kelviris theorem: If a totally submerged and yawed flat plate starts and moves in an ideal fluid then the circulation is zero all the time. But, if a plate is surface piercing then it creates circulation underneath the water surface as it moves.

If it is assumed now that this is the most important feature of the free surface at small Froude numbers then it is

possible to derive another mathematical model. Such a linear potential model is given in the following section and its properties

i

studied.

AN ALTERNATIVE MATHEMATICAL MODEL

Let the potential be

()

= xU cos ' + yU sin Y +

The derivative

B2c/xz

is harmonic and contirious in the entire space with the exception of

_yO;

O x Q and

-b z O where it has a jump.

Then we may write

JdÍ

d

BM()

Y 2 3/2 (19)

2

-b ((x-)2+y2+(z-) _J

o

A first integration with respect to z gives

r O

d d B2M Y + fi(x,y))

-b (x-)2+y2 r

where r

((x_)2y2+(z_)2)

f1 is harmonic and the condition

/xO when

z=- demands

f1 y/((x-)2+y2) . Integration with respect to x gives

-Jd

r _2M

(t1

(x-g) + (x-)(z-) + f2(y,z)) y y r

The perturbation velocity is zero for

_x-

. Hence

f2

ta1

_(z-ç)/y . Then the potential is

9 o Idç j( d a2M(ç,ç)

(t1

(x-ç) ₊ (z-ç) + 2iï -b

çç

y y + tan1 (x-)(z--ç) y r

The distribution M(ç,ç) must satisfy the following

condi-t ions:

1 The Kutta condition

M/ç0 for

çQ

2

M/ç=0

_for

ç0

Condition 2 describes that /az=0 on the waterline

z0;

0 x and

y±O

. But it can also be interpreted as a

condition that requires that _{no particles of water are allowed} to cross the upper edge of the plate. This separates the

surface-piercing plate from the case when the upper edge of a submerged plate ends arbitrarly close to the water surface. A differentiation of with respect to y gives y-velocity.

For y ±0 the limiting value of this component must satisfy

U sin /By on 0 x Q ; -b z 0 . In this manner

the following integral equation is obtained

Usin

J dçJ dç

i i

, -b

çç

z-ç _{(x-ç)(z--ç)}

The integrals are to be taken with their principal value as usual. In order to get approximative numerical values of the

forces the two cases large and small aspect ratios are

con-sidered. For small aspect ratio we put Hence r r0 Usjn' d dç 21T J o a2M(ç,ç)

1(1x_t)

z-ç x-ç

As a consequence the distribution has the form

LQ

-(20)

e + e

M(ç,ç) A(ç)B(ç) _{and the equation can be written}

A(x)

Usin

í°

(ç) dç B(0) (ç)

-b dç 2 dç x-ç

If z _{-b(1+ cos )/2 and}

Ç ¡ =

-b(1+ cos e)/2

then fdB(ç)/(z-ç) is a constant for B(0)

Henc e

U sin = 2A(x)/b +

1dA() d

2J d

x-o

This equation is converted into the normal form (8a) by the

substitutions:

x (1- cos s)/2 ; (1- cos

A(s) C(s)(bU/2) sins' and T ¶b/29.

Thus

1 C(s) + fldco do

'rrj da coso - coss o

The approximative solution of this equation is (8b) The

side force is

2pU

Jd

1b

3M(,)

and from this it follows that

CL

(32/2)(b/z)a1(T) = 3Ta1(T)

The center of pressure from the leading edge is Xc (z/4)(i - (a2/ai))

and the dimensionless moment about mid-chord

(32I8)(b/)(a1(T) + a2(T))

For the small arguments equation (23) may be modified

(3/2)(b/)

For large aspect ratio we put

Then equation (21) becomes

9 _ro

Usin d I d

aM()

i

(1+-i1)

+

-b

z-This equation is identical to equation (22) provided that

z -- x . The same procedure as above leads to the result

(2/2) ( 3 a1(T) + a2(T))

(23)

where here T

n/2b

. The moment coefficient is 0LP

In Fig. 8 these two limiting cases for the dimensionless side force are compared with the corresponding coefficients when the plate is reflected in the free surface.

Fig. 8 / /

--/'

r /

7/

_,//7

--/

/

/ _r

/

/,

//

/

1/

/

/

I//

/

I/I /,I

I,

/

//

/

/

I' / / /

///

/

I,

_/

I, _/ II / II / /7, ,I, /

//

0.5 1.0

1.5

2 .O Aspect rat io Jones formula Lifting-line EQ

₍₁₅₎

Small aspect ratio EQ (23)

Large aspect ratio EQ (26)

42

-4

3

2

For a restricted depth of water the additional condition is

/z O

when

z-d

A

d

H

A partial integration with respect to x together with the

condition that there is only a uniform stream when

xz-give the velocity potential

d d 32M(,) (x-)(z-) + +

o -b

y r

-1x-- -1(x--)(z++2d) -1z++2d

+ 2tan tari tan

y

j

(28)

Differentiation with respect to y gives y-velocity. As

y -*0 the limiting value of this velocity component makes the

integral equation for the distribution M Thus

Usin -i--j

dj

d

2N(,) 1x_2+z_2

2Tr

b

(x-)(z-)

Z X o

/(x_)2(z2d)2

_î

(x-)(z++2d) z++2d

When the aspect ratio is small and the water depth large the following approximations may be done

Then the simplified equation (29) is

X

(29)

A reflection in the plane

z-d

gives the x-velocity

d r

B2M()

d I y (Z_c z++2d) + 2y 2ir _j -b (x-)2+y2 r (X_)2+Y2) (27)

2. O

r° ₁

d

(-

z++2d

b

w

+ in

The velocity of the free stream is

1

and we use

y(n)

as

a vortex distribution on the cuts

O; Os

n sb;

-b - 255

n

s-2S

.

Then the complex potential (F) may be

written in the form

b

F(w)

+

dty(t) ((w-it)

_{- Th(w+It+216))}

2Tr1 J0

The complex velocity is

b 1 1 dF(w) _z

_{- 1 +}

I

dt 1(t)

_w-jt

_w+it+2i

U iV

-du 2'Trij0

Separating real and imaginary parts and letting

±0

on

the cut

Os n Sb

we obtain

U

1 -

y(t)

and

y

±y(n)

- 144

-The first integral on the right must be made independent of

z. Then we put

(31a)

This equation can be solved using complex variables. Let two

plates be normal to a free stream as the figure shows

n

and

UsinF

M(,)

2

has

D(x) d o

the

J° I d 1 + (30) (31) -b

form

dE

x-D()E()

so

(1

1

z-E(0)

z++2d

fdD()

x-

j d

-b

d

z-

z+ç+2d 2 J d o

x-as

An identification and a change of coordinates give

i I

-(2+2d)

d T _-b2+2db+2+2d

ro

Because E(-b) O it follows that E(0)

= J

dE()

-b Equation (31) is now Usin = D(x)/ E(0) ídD() d 2iî J d x-o

It may be converted into the normal form (8a) using:

x = (1- cos s)/2 ; = (i- ces o)/2

D(s) C(s)îrUsinY and T

E(0)T/

Hence

i C(s) +

I

fldc(o) do

irJ do coso - coss

o

Using the solution (8b) we can write for the coefficient of the side force

O

(T2a1(T)/b) I

E()d

-b

The constants can be rewritten in the formform

T E(0) = ((2d/b)-T) J d (b/T)P0(d/b) ¶J0 (1-T) ((2d/b)-1-T) T E(0) = ((2d/b)-T) J d (b/T)P0(d/b) ¶J0 (1-T) ((2d/b)-1-T) 10 _b211 i ((2d/b)-T)T

E() d

= - T

I d (b2/T)P1(d/b) T _J (1-T) ((2d/b)-1-T) i 10 _b211 i ((2d/b)-T)T

E() d

= - T

I d (b2/T)P1(d/b) T _J (1-T) ((2d/b)-1-T) i

The integrals may be expressed in terms involving complete normal integrals of the first and second kind. Hence

The integrals may be expressed in terms involving complete normal integrals of the first and second kind. Hence

P0(d/b)

(d/b)((k) -E(k) )

and

P1(d/b) _{(dlb)2 (} i) - (1c ₎ - 2(d/b) - i)111

where k z _{(2(d/b) - 1)/(d/b)}

( first kind, second) Then we have T = (b/)P0(d/b) and

z _{(l1iîb/) a1(T) P1(d/b)}

(32)

The center of pressure is given by

x (/'1)(1 - (a2/ai))

where x _{denotes distance from leading edge. The moment}

coefficient is

z

cL( 0.5 -

_{(x/) )}

₍₃₃₎

Because as (dlb) ±1 this approximation does not

hold when the lower edge of the plate is close to the bottom.

When the clearance is small the distribution M is assumed

to become nearly constant across the span. Therefore we may

use the following estimate

r

r°

2M( )(/(x_)2+(z_)2 _{M(,0) o}

Id I d

/x_2+z+2d2)

Using M(,r,)

z F()E()

_{equation (29) becomes} F()E(0) o

E(0) 1çjF() d

+ Usin -

2b(bO)

+ _J d X o +

F() fdE()

2 J d

(-

z++2d o

If x z _{(i- cos s)/2} and z 2(1- cos a)/2

the first integral is a constant for: F(s) z constx(s+ sins)

Using

E()

as above the second integral becomes a constant

and we obtain 8Trb P1(d/b) 2 {O/(b+o)} + (14b/2))Po(d/b) + z

₍₃₆₎

(324) (35) i

(x-)(z-)

(x-)(z++2d) b(b+6)

NUMERICAL VALUES

The following table gives a comparison between theory and experimental datas obtained from Ref. 24 LIB denotes

length/beam ratio. The aspect ratio is

_T/LO.057

Table 7

Nondimensional side force: -Y'x103

- ₎₁₇

-Nondimensional moment:

-Nx1O3

FO.15

6.i 6.7 7.3 7.8 7.0 5.0

FO.2O

6.5 7.2 7.9 8.0 7.0 24.5

In tables 8 and 9 the measured values at the static tests in this investigation are summarized by mean values in the range O.1 F 0.2 in order to facilitate the comparison with theory.

Datas obtained from Ref. 13 refer to mean values and standard deviation values in the range 0.1 F 0.3

LIB 24 _{5.5 7} 10 20 Theory: EQ(23) F 0.15 18.0 17.0 16.0 124.5 i24.o 15.0 n 15.22 F 0.20 18.5 17.6 17.5 15.0 124.0 i6.o n EQ(224) 6.83

Yaw angle

Static test

503

306

2°.0

Dynamic

test

EQ(9)

EQ(32)

EQ(35)

K e e

i

150

0.336

0.383

0.299

0.351

0.266

0.312

0.308

0.341

0.206

0.215

0.3014

0.337

113 0.1431 Q/4/43

0.403

0.441

0.249

0.1400 C 1 e

a

r

a

n c e 30 24 12 6

0.520

0.603

0.681

1.087

0.472

0.550

0.560

0.98'l

0./477

0.1177

0./181

0.783

0.522

o.8o

0.750

0.267

0.281

0.333

0.1101

0.430

0.452

0.535

0.6/42

0.672

0.817

0.991

m m 3 1

1.523

1.71114

1.507

1.721

1.487

1.572

0.480

0.618

0.765

0.987

1.176

1./479

F9 Çi) (D Yaw angle 5°.3 Static test 3°.6 2° .0 Dynamic test EQ(10) EQ(33) EQ(36) K e e i 150 0.123 0.1141! 0.139 0.139 0.149 0.175 0.123 0.140 0.103 0.102 0.135 0.148 143 0.210 0.216 0.220 0.218 0.112 0.173 C 1 e a r a n e e 30 24 12 6 0.247 0.276 0.318 0.1488 0.253 0.298 0.341 0.4714 0.267 0.296 0.342 0.423 0.258 0.294 0.412 0.119 0.124 0.143 0.166 0.184 0.193 0.224 0.263 0.168 0.204 0.248 m m 3 1 0.553 0.615 0.539 0.602 0.515 0.578 0.193 0.238 0.307 0.384 0.294 0.370

Table 10

Side force coefficient:

C

Aspect ratio

0.5

Gap Experiment

mm

Ref. 13

EQ(9) EQ(13) EQ(17) EQ(32) EQ(35)

2.15

±0.11

_1.57

2.04

_1.98

55

2.61

±0.09

1.92

2.74

2.65

2.93

2.84

35

2.99

±0.15

2.08

3.09

2.84

3.20

3.04 15

3.69

±0.09

2.39

3.97

3.20

3.77

3.41

1

4.34

±0.26

3.34

_5.99

4.16

5.55

4.32

Table 11

Moment coefficient:

CM

Aspect ratio

0.5

Gap

Experiment

mm

Ref. 13

EQ(10) EQ(14) EQ(18) EQ(33) EQ(36)

0.53

±0.04

_0.79

_0.51

_0.66

55

0.58

±0.03

0.62

0.69

0.66

0.91

0.71

35

0.63

±0.05

0.65

0.77

0.71

0.98

0.76

15

0.77

±0.07

0.72

0.99

0.80

1.12

0.85

1

1.01

_±0.13

_0.93

_1.50

1.04

_1.55

_1.08

ACKNOWLEDGEMENT

The author wishes to thank the staff of the Department of

Hydromechanics for their invaluable assistance in different

phases of this work.

-REFERENC ES

CARAFOLI, E. "Tragflgel theorie,"

VEB Verlag Technik Berlin (1954)

CHAPMAN, R. B. "Free-Surface Effects for Yawed Surface--Piercing Plates," Journal of Ship Research September 1976

CHISLETT, M. S. and STRM-TEJSEN, J. "Planar Motion

Mechanism Tests and Full-Scale Steering and Manoeuvring Predictions for a MARINER Class Vessel," HyA Report Hy-6 April 1965

GERRITSMA, J. BEUKELMAN, W. GLANSDORP, C. C. "The effects of beam on the hydrodynamic characteristics of ship hulls," Delft Progr. Rep., Series C August 1976

HESS, F. "Rudder effectiveness and course-keeping stability in shallow water: A theoretical model," International Shipbuilding Progress, Vol. 24, No. 276, August 1977

HU; P. N. "Forward speed effect on lateral stability derivates of a ship," Stevens Institute of Technology, Report No. 829, August 1961 JAHNKE, E. and EMDE, F. "Tables of functions,"

2 nd edn. Leipzig (1933)

NEWMAN, J. N. "Lateral motion of a slender body between two paralleli walls," Journal of Fluid Mechanics 39, No. 1, 1969

NORRBIN, N.H. "Forces in Oblique Towing of a Model of a Divided Double-Body Geosim,"

SSPA Pubi. No. 57, 1965

MGRRBIN, N. H. "Theory and observations on the use of a mathematical model for ship manoeuvring in

deep and confined waters," SSPA Publ. No. 68, 1971

il. Principles of Navel Architecture

New York (1967)

ROBINSON, A. and LAURMANN, J. A. "Wing theory," Cambridge (1956)

SUNDSTRM, O. "Experiments with a surface-piercing

flat plate," The Royal Institute of Technology, Stockholm June 1978

THWAITES, B. "Incompressible aerodynamics," Oxford (1960) TITCHMARSH, E. C. "The theory of functions," Oxford (1939)