Experiments with a surface piercing flat plate

17 MEl 1919

ARCHEF

JUNE 1978 Lab.

y. Scheepsbouwkunk

Technische Hogeschool

Deift

KUNGL. TEKNISKA HOGSKOLAN

I STOCKHOLM

HYD ROM E KA NIK

EXPERIMENTS WITH A SURFACE-PIERCING FLAT PLATE

OVE SUNDSTRÖM

THE ROYAL INSTITUTE OF TECHNOLOGY IN STOCKHOLM

DEPARTMENT OF HYDROMECHANICS

TRITA-HYD-78-02

EXPERIMENTS JITH A SURFACE-PiERCiNG FLAT PLATE

by

O. Sundström

SUMMARY

Using an oscillating mechanism, hydrodynamic coefficients of side forces and moments on a vertical surface-piercing flat plate of reflected aspect ratio 1 are determined.

Angle of attack (yaw) , depth, Froude number, and reduced

frequency are varied. Details of the design, construction and use of this mechanism are described.

The primary purpose of this research is to obtain a greater understanding of the influence of a restricted water depth and a free surface on the coefficients involved.

-1-TABLE OF CONTENTS Page SUMMARY LIST OF SYMBOLS 2 DESCRIPTION OF PLATE 3 TEST APPARATUS 3 TEST PROCEDURE 5

REDUCTION AND PRESENTATION OF DATA 7

REFERENCES 9

DRAWINGS AND PHOTOGRAPHS 10-17

LIST OF SYMBOLS Symbol Definition b Width of plate f Frequency g Acceleration of gravity 9. Length of plate

Half-distance between the pairs of plate springs

t Time

CLJ Side force coefficient

CMqJ Moment coefficient

CLIJ(x) Magnitude of nondirnensional side force

CM(x)

Magnitude of nondimensional moment

Fn Froude number = U/v

L Side force

LF Force on forward plate springs

LR Force on rear plate springs

M Moment

U Velocity

EL Phase angle of side force

EM Phase angle of moment

X Reduced frequency parameter = rrf2/U

p Density of fluid

Angle of attack

'Po Amplitude

w Angular velocity

Refers to quantity out of phase with the displacement

)S

_{Refers to quantity in phase with the}

DESCRIPTION OF PLATE

The rectangular steel plate had the dimensions

500x

440x 4 mm. It was sharpened at leading and trailing

edges and had been painted to prevent corrosion during the tests. The lower edge of the plate was immersed

250 min. Thus the aspect ratio was

0.5

and the reflected

aspect ratio 1. Although tests showed that it had no effect, a turbulence stimulation device was maintained during the entire research. It consisted of studs

cemented on both sides of the plate 7 cm from the leading edge.

TEST APPARATUS

The test facilities and towing carriage are shown in Fig. I.

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.

In order to obtain side forces and moments and to gene-rate an oscillating motion, a special balance fitted to the towing carriage was used.

The main principle of the construction was that two pairs of aluminium plate springs held the plate restrained in side-sway but was free to move forwards and backwards. This was accomplished by arranging ball bearings as shown

in Fig.

2.

Forward and backward movement was prevented by

thin steel wires.

A general impression of the layout of the oscillating mechanism is given in Fig. 3.

A 180 W electric motor drove a 1:20 David Brown worm gear, and on the shaft there was a flywheel. The movement was then transmitted by a 5-miri wide toothed belt. The uniform rotary motion was converted into periodic translatory motion by means of a scotch-yoke. The translatory motion was transmitted by a metal strip.

To achieve a smooth movement, the plate was counter-balanced by an iron bar with lead weights fitted to both of its ends. For the frequencies to be used, tests showed

that the movement was constant. The speed of the electric motor was controlled by a rheostat.

On the plate springs, material was cut away and strain gauges were cemented. They were mounted in a

temperature--compensated wheatstone bridge with two strain gauges on

-3-each side of one and the same plate spring, and the gauges were also protected from humidity. The gauges were supplied with current from a voltage regulator installed on the

carriage. The electric signals from the strain gauges were recorded on an ABEM Ultralette 5651 photorecorder standing on the towing carriage.

Later on, there was a break to set up a i 6-channel PCM

Sys-tem for transmission of the electric signals from the tow-ing carriage to a control beside the towtow-ing tank. Financial support for this system was given by the Gösta Lundeqvist Foundation for Ship Research, KTH.

Tests manifested that vibrations from the towing carriage made the signals difficult to read. Two integrators were

built which effectively filtered these disturbances. It was thus possible to record four signals for the side force. The mass of the test equipment that could work upon the

gauges was about 30 kg. Thus a difference of 1 mm in level

between the rails corresponded to a side force of about 300/3000 =0.iN. Measurements of each rail on 91 points

showed that the maximum inclination was 5.4 mm. If standard deviation was used as a measure, then the inclination was 2.62 mm. The rails were taken away and thin sheet-metal shims were inserted to reduce the difference in level. The track was reconstructed and a new measurement was made. The values were now 1.0 and 0.32 min respectively.

For the static tests, an angle of reference was measured by a telescope with a hairline cross. It was applied to both sides of the plate, and the aiming points were marked on the wall of the towing tank. The deflection of the tele-scope on the support was separately measured by applying it to both sides of a piece of metal which was also rotated 180°. The angle was determined by trigonometric calculation based on the measured values. The remaining angles were de-termined by measuring the displacement of the two large beams with a micrometer.

For the dynamic tests, the peak value of the angle was meas-ured by allowing a light source to be rotated with the plate. The light irradiated upon the end wall of the towing tank at a distance of 10 meters. When the plate made slow oscilla-tions, the turning points were marked. The distances were measured and the amplitude calculated. During the tests, the angle was continuously recorded on the phototape by a signal from a wire-wound precision potentiometer.

The towing carriage speed was determined in the following way. A circular metal disc in which radial slots had been made was mounted on the rear wheel of the towing carriage. The centre line of the polar axis of the disc coincided with the centre line of the wheel axis. On one side of the disc a light source was mounted from which a focused ray of light

emanated perpendicular to the disc.On the other side, a photoelectric cell was installed so that when the wheel ro-tated, the ray of light passed through the slots and

excited the photoelectric cell. The signals were recorded on the phototape. Calculation of the velocity was based on the diameter of the wheel and the timekeeper of the recorder. When the towing tank was empty dynamic equilibration was

made. Weights were systematically located so that the signals from the integrators drew straight lines on the phototape. So that varying bottom depths could be simulated a plate of aluminium of the dimensions 2000x1000x10 mm was used. The plate was sharpened in front and was attached to the towing carriage by two streamlined struts and also by two 2 mm thin wires of bronze. The bottom plate had a keel. The

depths of the bottom was adjustable as the struts were slot-ted, and the wires were fixed to blocks running in slotted arms. The plate was horizontally adjusted by means of a water-level.

According as it appeared to be necessary, Mr. M. Gällstedt reconstructed and corrected the balance with its electric equipment which he had previously built for use in other research. A circuit diagram showing in principle the same measuring equipment electronics will be found in Ref. 1.

3. TEST PROCEDURE

The plate springs were individually calibrated with stand-ardized weights. When the towing tank was empty and the towing carriage at rest, the test equipment was loaded with oscillating forces to check measurement facilities and the coupling between the two pairs of plate springs. The force was allowed to attack at different points. It was possible to measure the four signals of the side force. The frequen-cy was varied over all those which were to be used in the dynamic tests. The force derived from the integrated signal was not quite identical with the force derived from the direct signal. Based upon the test series, a correction for each integrator with regard to frequency was made. For the most rapid frequencies that would be used, the correction

was about 10%.

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

plate was towed slowly back to the starting position in the basin. A waiting period of 15-30 minutes between the beginning of successive runs was taken to allow the water in the basin

to become free of waves and currents. The _{sequence of tests}

Table 1.

Each angle was carefully measured. For the hydrodynamic co-efficients used the angle was of little consequence. Thus variations of ±0.10 were regarded as negligible in the summary. The Froude 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 tests, the Froude number and the reduced frequency were varied for each slot. Of all possible ways of combining these two quantities, three reduced

frequen-cies were selected and for each of these the Froude number was varied. A test program was made based on the empirical knowledge of the speed of the towing carriage and the

elec-tric motor and the controls were set to these predetermined values. The mean values from these settings are shown in

the following table. Standard deviation values are used as a measure of the precision of the settings.

Slot mm Angle of attack, degrees

5.10 3.5 1.8 54.8 5.10 3.5 1.8 34.8 5.10 3.5 1.8 15.0 5.10 3.5 1.8 1.1 ±0.2 5.10 3.5 1.8 Table 2.

Slot mm Reduced frequency

0.382 ±0.015 0.679 ±0.020 0.978 ±0.025 55 0.387 ±0.018 0.672 ±0.021 0.972 ±0.031 35 0.381 ±0.018 0.667 ±0.023 0.954 ±0.019 15 0.382 ±0.020 0.669 ±0.022 0.964 ±0.024 i 0.385 ±0.019 0.675 ±0.020 0.978 ±0.029

-6-To make the dynamic tests more complete, three Froude numbers were selected while the reduced frequency was varied for

each one. This was done for the slots and 1 mm. The mean

values and the standard deviation values of these settings are shown in the following table.

Table 3.

Besides the natural phase shift of 90° between the integra-tors and the angle of attack 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 num-ber 2rïaf 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.

4. REDUCTION AND PRESENTATION OF DATA

The total side force (L) and moment (M) acting on the plate were obtained from the two plate springs as follows:

L=LF+LR; M= (LF-LR)

where the subscripts F and R denote forces measured at for-ward and rear plate spring, and ii is the distance from

centre line of the plate to each plate spring (500 mm). They were made dimensionless in a way generally used in aerodyna-mics, leading to the coefficients

CLy=L/(1/2)pU2th sin P; CMU =W(1/2)pU22,2b sin 1f

Centre of pressure, which could also be derived from CLIJ and CMIJá have been plotted in a separate diagram. The space between the leading edge and the centre of pressure was made dimensionless by L.

For the slots and 1 mm, photographs showing the shape of

the waves have been attached.

Slot mm Froude number

1 0.239 0.244 ±0.002 ±0.002 0.388 0.390 ±0.002 ±0.004 0.531 0.512 ±0.005 ±0.002

For the dynamic tests the angle of attack varied sinusoidally,

(t) ='Yo sin wt,(o =

4970)

The side force was the sum

L = LF + LR

where LF and LR had been divided into different components

s c

LF=LF sin Wt+LF cos uit;

s c

LR=LR sin uit+LR CO5 uit

The side force could thus be divided in the saine way:

s.

c

L=L sin wt+L cos uit and further

L=V(Ls)2+ (LC)2sin(wt+CL)

where

-1 c s

EL=tan

(L /L

The moment acting on the plate was obtained from

I

M=(LF_LR)i=MSsinwt+MCcos uit=V(M5) +(Mc) sifl(wt+EM) where

EM = tan1 (Mc/Ms) and MS = (L

- L)

Mc _{= (L - L)}

The quantities were made dimensionless in the followingway

L/(1/2)pU2ib sinoCy(x)sin(wt+E)

M/(1/2)pU2i2b

Sfl

Po CM(X)Sifl(Wt+EM)

For the i min slot two independent tests were made. To

as-certain whether turbulence along the bottom plate could affect the measured values, a string was stretched across the bottom plate in front of the plate. It was found to have no effect. Thus no attempt was made to separate the two experiments by different marks in the diagrams.

Mr. M. Gällstedt took the photographs. Mr. T. Byquist made the drawings and helped with the diagrams. I am grateful for their valuable assistance.

REFERENCES

BYQUIST, T., "Wave-making Resistance of a Series of

Bodies of Revolution," Department of Hydro-dynamics, The Royal Institute of Technology, Stockholm, (1973)

SMITH, F. E. and SEVIK, M., "An Investiqation of the Forces on an Airfoil Oscillating in Pitch about the Quarter Chord," Pennsylvania State

University, (1964)

BERG, H. and van de VOOREN, A. I., "Experimental Deter-mination of Aerodynamic Coefficients of an

Oscillating Wing in Incompressible, two-dimensional Flow," National Luchtvaart-laboratorium, Report F.104, (1952)

-9-LQ .-' SIDE VIEW OF TOWING CARRIAGE WITH THE TWO- COMPONENT BALANCE AND THE MECHANISM FOR OSCILLATORY MOTION OF A FLAT PLATE

0 1 2 3 4 5 5 7 8 9 10dm O 2m

Fig. 2

Free moverient Novement restrained

by plate springs

ciement locked out by wires

1

For static loading the pairs of plate springs do not act upon each other

/

-SCHEMATIC ARRANGEMENT OF THE MECHANISM FOR OSCILLATORY MOTION OF A FLAT PLATE

Fig.3

Fig.4 Photographs showing the test facility from

various viewpoints. 12

-Fig.4

Contd.

-= 0. 1 6

= 0.34

= 0.38

Fig.5 Shape of waves in the immediate vicinity of

the plate at 5° angle of attack. Slot between bottom and plate:

-= 0.46

F

= 0.68

Suction side

Pressure side

Fig.5

Contd.

-= 0 . 1 6

= 0.34

F = O . 38

Fig.6 Shape of waves in the immediate vicinity of

the plate at 5° angle of attack. Slot between

bottom and plate: i mm

-F = 0.46

F = 0.68

Suction side Pressure side

Fig.6 Contd.

-Q Q Q o t 4 2 18

-Slot b.twsan lowly dga of flat SiRte .04 bottom;

Y.w anglo: 5 10 0 4350 o R m1 cP _00z 0OOS

0 °

-

-Slot botwoon lower .4g. of llar plato and bottom: 54 Bmw Yaw logro: 05 10 0 o3'50 O n180 3 c°

-

8 oso ç .6g. bottom: O o s =3:50 =1:Ro Slot b.tw180 lowur of flat SIRte led

34.5 mm Yaw °R° 4 s:to -0

°

0 0 01 .0 0 R s I O s s 01 0.2 0,3 0.4 0.5 os 07 OR FROUDE NUMBER F,,

Fig.7 Nondimensional side force obtained from

Static angle tests.

oj 0,2 0.3 0.4 05 06 07 05 FROUOE NUMBER F,, 0.1 0.2 03 04 05 0.6 07 08 FROUDE NUMBER F,, Q z Q e Q Q o o 4

4 2

-

19

-I .1 I lowe, edge end bottow. S 510 O =3;5o O 1:80 s Slot between of flat plate 15.0-153wro Yaw 00g10 u 11

ass

Slot between lower edge of fiat plate end bgttow,

1.0 Yaw Ongle 5:10 O =350 D 1:eo s D ems 5 C Co s O D n o s . °0 O O O e _s 0,1 02 0.3 0.4 05 0,6 0.7 0,8 FROLJDE NUMBER F0 0,1 02 03 0.4 06 0.6 0,7 0,8 FROUDE NUMBER F

Fig.7

Contd.

(J z 'J 6 o o o A

- 14 z U 1,2 o t'o o 3 0,8 0.6 0.4 0,2

-

20 -I

Slot between lower edge of flat plate and bottom:

Yaw angla 5:10 o =350 U d=1:oo e o s - !° ... 3 c ctP 0b edge bottom: O O e =3:50

-Slot betw000 lower

of llar piare and 548 nr,,, Y,w ongle Sb #='l:Bo lO 00 ,, e 9. i

-Slot between lower odgo

,,f flat pl ato and bottom 34 8 n., Yo atgle 5:10 o d=3:50 o di;Bo s o o o e . s o s .o . o Q o o o

_{° n}

01 02 0.3 0.4 0,5 0,6 0.7 0,8 FROUDE NUMBER F,, 0,1 0,2 0,3 0,4 0,6 04 0.7 0,8 FROUDE NUMBER F

Fig.8 Nondimensional moment obtained from

static angle tests.

01 0,2 0.3 0,4 05 0,6 0,7 08 FBOUDE NUMBER F,, U z 1,4 La 12 o 1,0 o 0,8 0,6 0.4 0,2 2 U - 14 z U 1,2 o U 1,0 z 2 o 2 0,8 0.6 0,4 0,2

1,2

-

21

-III L

_{lower edge}

end bottom ttto, = 5:10 o d=3:5o D =i:Bo Slot between of flot Elote 15 0-15.3 VOW ongle: o n o O O n o . sa o e o o . Q o a s o e .t't_ °° O _ao_0 O o o _Oa0

jlIj-r" Slot between Iowtr of flat plate and lOteo, Yew anglo 5:10 - -. ' -n3'5o i:8o edge bottom: 0 D e O e a '-00 O so cP s o o S o a 0,1 0,2 0,3 0,4 0,5 06 0.7 0,8 FROUDE NUMBER o 02 03 0,8 0.6 0.7 0,8 FROUDE NUMBER F0

Fig.8

Contd.

0,6 0.6 0.4 0,2 S U U 14 z U 1,2 o U 1,0 S o 0,8 0,8 0,4 0,2

0.8 o - 0.5 = Q 0,4 0.3 f-z 0.2 0.1

-

22 -lome, edge and bottom: 9 - 5t0 o 4=3:50 0 9=1:80 Slot between of flat piare Yew engle: -OSo o oa U o° o'S 00 - -.

-- --

I

Slot between lower edge of flat plate and bottom:

54.8mm

Yaw aegle: 5:10 o

9=3:50 0

4=10 e

LII

Slot between lowet edge of fiar plate and bottom.

34,8mm Yew angle: 4-5:10 0 9=3:50 o 4=1:80 S ° 01 02 03 04 0.5 0,8 0,7 0,8 FROUOE NUMBER F,, 01 0,2 0.3 0,4 0.5 0,6 0,7 0,8 FR000E NUMBER Fe 0,1 0,2 0,3 04 05 0,6 0.7 0,8 FROUDE NUMBER F

Fig.9 Centre of pressure obtained from

static angle tests.

o o o o C 5 o o C o C f-z U 0,7 0,6 0.5 04 0.3 0,2 0.1 0,7 0,6 0.5 04 0.3 0.2 0.1

-

23

-Slot botwnnn lawnr edge of fiat plato and bottom:

15.0-15.3 tttnt Yew engin,d51Ø o o350 O d=i:oo e Os o 0 SO o o

il

Slot between low., edge of fiat plat. nod bottom;

1.0 ntro Yaw engin. d 510 O 350 O di:Bo . . SO O s S a °Sb S g U 000 00- _o 0,1 02 03 0.4 05 0,6 0,7 0,8 FROUDE NUMBER F0 0,1 02 03 0,4 0,5 0,6 0.7 0,8 FROUDE NUMBER

Fig.9

Contd.

o o O z o 4 2 e O o 0,7 0.6 0.5 0,4 0.2 0,1 o o i, 0,7 z o 4 0.6 3 o 0.5 04 0.3 z 0.2 0:1

U o o 4 Q Q o 4 z Q o Q Q z o 0 9 5 4 2 01 0,2 03 04 05 0,6 07 08 09 10 11 12 13 REDUCED FREQUENCY E 24

-Slot between lower edge

o:flb;:

Ft k

UUUUUUR

u...

Ii

11111

n -J. -Slot betwear of flat plate 112mm Fraude number L lower edge and bottom -uO.24 A 0.39 a Q51 O T e a a £ A A A L e A A L a ° n z 0.1 02 03 0 4 0.5 0.6 07 0,6 0,9 1,0 1.1 1 2 1.3 REDUCED FREQUENCY D

Fig.1O Magnitude of nondimensional side

force versus reduced frequency.

o

70 60 50 40 30 20 lo. -20 01 02 0,3 04 05 0,6 07 08 09 10 It 1,2 13 REDUCED FREQUENCY A

Fig.11 Phase angle of side force

versus reduced frequency.

-Slot b.tw.on lower edge

Freude numb;,

Fn:O39 £

IiiiIIIAI

!.

lUl.

_lUi.

.

U..

Slot between lower edge

Of flot plate and bottom:

1 -1.2 errn FroodOno robe, 100.24 A F0n 0.39 £ I. o.51 A A A A n a A £ 8 L L o £ o ' a L o 0 01 02 03 04 05 0,6 07 08 09 10 11 ¶2 13 REDUCED FREQUENCY E 60 C 50 40' 30 20 10

01 0,2 0,3 04 05 0.6 07 0.8 09 10 Ii 12 13 REDUCED FREQUENCY

01 02 0,3 04 08 06 07 08 09 1,0 11 12 1,3

REDUCED FREQUENCY X

Fig.12 Magnitude of nondirnensional moment

versus reduced frequency.

26 -adg. bottom U a O

Slot beteen lower of flat plate and Proud. numb.r; o024 F0 0,39 o053 £ a

'

E A £ A A A D D Slot batwee flat alato toot Fro0d. nutnber of 1-12 n lowor edge and bottow o024 A 0.39 a o051 O a £ A 6 a a A _A o a e o o _a D 0 O 1.8 Q Q = 1,6 5 C 1,4 Q 1,2 o 10 S o S 0,8 0,6 04 0,2 S i,s o = 1,6 C 14 Q 0,8 0.6 0,4 0,2

- 70' -60' 01 0,2 0,3 04 0,5 06 07 08 09 10 11 12 13 REDUCED FREQUENCY b 27

-81,1

ff1,1between l,wnn dg.plate and bottom:

number: o 0.24 A F1u 0.39 a L 0,53 0 Fraude a o L £ --1'-o A A o A A £ t A

Slot btw000 lower edge of flat plate and bottom

1 12 mm Fr oudo nu nr ber: 0.24 A Fn/o 0,39 a 0S1 O a a a a e e e e e A A n A A 01 02 03 04 05 06 07 09 09 10 11 12 13 REDUCED FREQUENCY A

Fig.13 Phase angle of moment versus

reduced frequency. -40' - 30' - 20' -9o. 4 -BO -70' -60' -40' - 30' - 20'

o o

:8

2 C 6 o u o

-

28 -edge bottom £ O A

Slot botween Omet at tier plate and

=

Reduced tre quency

uoO,37 )oO,67 lo 0,96 A A A AA AA A b 00 AA 4 00 000 0 a £ A £4 t- - a

-- .

Slot bntween lower edge

__________u..

-

_______u.-_

Reduced lreq00000

__u...

_...

u...

_u....

u...

u...

Slot between lower edge of flat plata and bottom.

35 te te Reduced frequency' uoO,37 L co 0,67 o uoO,96 A -L A C A A C

i

a a a w o au A a I OB _A 4 A 01 02 03 04 0,5 0.6 0.7 0,8 FROUDE NUMBER F0 01 0.2 03 04 0.5 0.4 0,7 0,8 FROUDE NUMBER F0 01 02 03 0,4 0.5 0,6 0,7 0,8 FROUDE NUMBER F

Fig. '14 Magnitude of nondimensional side

force versus Froude number.

u 9 o o 2 4 z 6 o u u o a 4 9 o 4 C) 6 o u o 3 2

29

-Slot between lower dge

Roducod frequency A A. L A

:

_:A

Slot between lower edge of flot plato and bottom:

i - 1,2 mm Reduced frequency c,,037 L u067 o l0.96 A A a AA A à O a M°a o A ° a A

t

A a A o o a a

::

01 02 03 04 05 0,6 07 0.8 FR000E NUMBER 01 02 03 04 0.8 0,6 0,7 0.6 FROUDE NUMBER F,,

Fig.14

Contd.

I., 9 o D B C z o U U o o ₄ 3 2 9 8 4 3 2

-

30

-

4-'

-Slot between lower edge ei flat plate and bottom:

Reduced frequency c0,37 £ cuo,67 o l.w 0,96 A A A A AA 86 A O ° A A A e ce

:aae

e

::;a

A 4 4 J ¡

I between lower ndge

Of hat plate end bottom

55 r,, er

Reduced ire qonnu y 0n037 a u 0.67 0 n0,96 A t, A o A A -o e a e a £ a S a A A e A A i a L

Slot between ewe r edge

of flat plate and bottom

35 re w Reduced frequency: 0.37 a 000.67 0 0u 0,96 A A a e a A L -S a LA o e A A a 0.1 02 03 04 0,5 0,6 0,7 0,8 FROUDE NUMBER 01 02 03 04 0,5 0.6 0,7 0.8 FROUOE NUMBER F 01 02 03 04 0,5 0,6 0.7 0,8 FROUDE NUMBER F0

Fig.15 Magnitude of nondimensional moment

versus Froude number.

1,4 Q 1,2 o Q 0,8 0,6 0.4 0,2 o 1,6 S C 1,4 Q 1,2 o Q 1.0 S O S 0,8 0.6 0,4 0,2 ° i.e O = 1,6 S C 1,4 U 1,2 o Q 1.0 S o S 0,8 0,6 0,4 0,2

U _1,8 o 1.6 1.1 1,2 o 1.0 S o S 0,8 0.6 0,4 0.2

-

31

-Slot between lower Rdg. of flot plate end bottom: 15 tot, Reduced t ,oqueflC X,c03? A Xc 0,67 0 uoo.96 A A A A A _a 8 o a o O A A A o a 4 A 'A '

-Slot between lo,., edg. of flat plate cod bottom:

1- 1,2 mm Reducod t roUeflty: .c037 a flcO,67 O X 0.96 A A A A

t

:

a A

t

Laja

2

"8

A _'A 0' , A A 4, o a 00 01 02 03 0.4 05 0,6 0.7 0,8 FROUOE NUMBER F,, 01 02 03 0,4 ₀₅ 0,6 0.? 0,8

FROUDE NUMBER E,,

Fig.15

Contd.

1.6 S C lo S o S 0.8 0,6 0.4 0,2

-20

32

-Slot between lower edge of flot plate and bottom

Reducndtrnquoflcy 1o067 o coO,96 A a A _A AA oo B o o a A L t,1 A A £ o a a a a r

--Slot betwon,, lower edge of flat DIeto end bottom.

55to,,, Oeducnd ttequerCy: 100,37 £ 1o067 O 100.96 A A A A A L £ - a a A a o L A A o u. a_s ) _o a a

Slot between lower .dge of flot plate end bottom:

35 m'e, Reduced f ,oqutnc y n 0.37 L loOR? o 100.06 A A A A a A L a a A a _A O £ O 1 L

Li

L £ 01 02 03 04 0.5 0.6 0,7 0,8 FROUDE NUMBER F,, 01 02 03 0.4 0.5 0.6 0.7 0.6 FROUDE NUMBER F,, 01 02 03 04 0.5 0,6 0.7 0,8 FROUDE NUMBER

Fig.16 Phase angle of side force

versus Froude number.

o z 60 C 50' 40 30 20 10 0 o z 60 C 55' 40' 30' 20 10 o -Io -20 60 C 50 40' 30' 20 i 0 0' -10' -20

-20 40' 30 20 10' 0 -20'

-

33

-Slut between lower edge of flat plate and bottom

15 n'nt Reduced frequency to037 a l,c 0,67 0 c 0,96 4 A 8 6 £ a a A A 4 A A A A A A a _e a a a a

-Slot between of flat plate t - 1,2 te Reduced end t,oqueocy un037 tn0,67 'n' 096 lower .4g. bottom; a O 4 8 n no %

t

L A a

04

L A n'o ° O a a a-a 01 02 03 04 0,5 0,6 0.7 0,8 FROUDE NUMBER F,, 01 02 03 0,4 05 0.6 0,7 0,8 FROUDE NUMBER F,,

Fig.16

Contd.

60 50 40' 30' 20 10'

- 70

-9o.

C

34

-Slot bet00fl lomar edg. of flat plato and bottom:

Reduced traqueany u 0.37 & c0.67 o u 0.96 A AA A o o A _o A 00 o 0 A a A a a £ L £ B A A AA o a _{a a} a 00 a a a a -

-lot botmeon lomar Odg.

of flat plate and bottom

55 mrO Reduced f reqaonu t oO.37 L o0,67 o uoo.96 A -A A o a B A 0 A a a a A a A a o o a £ a

Slot betweeo lomer od8. flat bottom. of plate and 35mm ROdcod frequency: x0,37 a l0,67 o fu0,96 A A B A A o A a a o A oA 0 5 4

a'

A o g ° * L a o' 02 03 04 0.5 0.5 0.7 0, B FR0000 NUMBER F0 01 02 03 04 0.5 0.6 0.7 0.8 FROUDE NUMBER F0 0, 02 03 0.4 0.5 0.6 0.7 0.8 FROUDE NUMBER

Fig.17 Phase angle of moment versus

Froude number. -50 -40' - 30 -20 - 10 S -90. C 80 -70 -50' -40 - 30' - 20 -10 -70' -60 -50' -40 - 30' - 20' - 1o

-70' - 70 -60 -50' -40' 30' - 20 -10'

-

35

---

-' -Slut between lower edge

of flat plate and bottom:

15 root Reduced f roquency t.n037 £ coo,67 o unO,96 4 A A O A A L £ A o a £ £ £ _L

Stat between lower edge

of flat plate and bottom: 1-1,2mm Reduced frequency xu037 £ un0,67 u tc0,96 A A o O ° e A A A A A

i

£ a A u o A o o L a u A A A u A A u o o aa à L f ° aL 01 02 03 0.4 0.5 0,6 0.7 0.8 FROUDE NUMBER 01 02 03 04 05 0.6 07 0.8 FROUDE NUMBER F0

Fig.17

Contd.

-50 -40' - 30' - 20' - 10