Wave height at the side of two-dimensional body oscillating on the surface of a fluid

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ARCHEF

Wave Height at the Side of Two-Dimensicna1

Body Oscillating on the Surface

of a Fluid

By Fukuzö TASAI

Reprinted from Reports of Research Institute for Applied Mechanics, Kyushu University

Vol. IX, No. 35, 1961

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Reports of Research Institute for Applied Mechanics

Vol. IX, No. 35, 1961

WAVE HEIGHT AT THE SIDE OF TWO-DIMENSIONAL

BODY OSCILLATING ON THE SURFACE

OF A FLUID

By Fukuzo TASAI

Summary

The wave height at the side of two-dimensional body with Lewis-form section oscillating on the surface of a fluid was calculatej.

lt is defined that is the ratio of the wave amplitude at the side of

a cylinder produced by heaving to the amplitude of heaving and is the one in case of swaying.

is the ratio of the wave amplitude to where B is the breadth of a section and O is the amplitude of rolling about the origin in the free surface.

and Ci are given in many figures.

The phase difference between the motion and the wave at the side of

a cylinder, that is, r,, r and re,. were also obtained. Introduction

When a ship sails over in a bad weather, owing to the violent motion and the shipping water, it happens that the ship weakens the stability and suffers the damage on the upper or bridge deck, so that the speed of the ship is reduced

purposely. With regard to the phenomenon of shipping water several researches have been done, for example, J. L. Kent [1], R. N. Newton [2], and R. Tasaki

[3] etc..

R. Tasaki [3] carried out experiments on the shipping water during experiments of the propulsive performance of tanker and liner models in regular head waves. He measured the amount of shipping water per one cycle. Then he gave a formula for estimating the amount of shipping water.

Through the experiments of forced pitching Tasaki also obtained an experim-ental formula to evaluate the height of the dynamical swell-up of water surface due to relative vertical velocity of the bow and waves.

In this paper, in order to analyse the dynamical swell-up of the water surface due to ship motion, calculations of the wave height at the side of two-dimensional body oscillating (heaving, swaying and rolling) on the surface of fluid were carried out for cylinders with Lewis-form section. In using the results mentioned above for the evaluation of the shipping water of a ship, we must take into consideration a considerable amount of the effect of the three-dimensional motion of fluid. These two-dimensional results, however, will be used as a means of the analysis as regards the shipping water.

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122 F. TASAJ

L Calculating formula of the wave ampjitude

1. 1. Heaving oscillation

When a cylinder with Lewis-form section, which was dealt in [4], performs heaving oscillation two kinds of waves are created, namely, the standing wave and progressive one.

Both of them contribute to the wave height at the side of a cylinder. The

two-dimensional wave height at the side of a cylinder can be calculated using the velocity potential given in the authofs paper [5].

Suppose now that an infinitely long cylinder oscillates vertically with a small amplitude y,=hcos(ut+) (See. Fig. 1).

Zplane

Fig. 1.

The fluid motion is two-dimensional. lt is assumed that the depth of water is infinite, the fluid is invicid, incompressible and the fluid motion irrotational.

in case of the heaving the velocity potential of the fluid is expressed as

follows

=(_ (ø(s, a, O)cos

OJt±b3(eB,

a, )sin

ut

'

e2'

+cos

wtp[e2cos2mo+

cos(2m-1)O

a1

+ 2m-ie_m1

os(2m±1)0-3

+)c cos(2m±3)8}j 2m ± 3

f sin Ú)t q2m[e_mcos2mo+ _{1±a1+a31 2m-1}eB f

cos(2ml)()

aie_(lm)

3a3et23

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WAVE HEIGHT AT THE SIDE OF TWO-DIMENSIONAL BODY

OSCILLATING ON THE SURFACE OF A FLUID 123

where is the amplitude ofpgressive wave and .rB=

D, and D3 are the potentials by the two-dimensional source at origin O. Letting C denote the amplitude of the surface wave (taking positive downw-ards), we have the follwing equation in the linearised form;

gtJy=o

(2)

Then, putting a =0 the wave amplitude at the side of a cylinder can be

obtained.

Writing C0 the amplitude of wave at the side of a cylinder we get

c0=

_{700(eB4)sin r+ø80(eB,)cos}

(,)t ¿ çcos(2m-1)O

sin

_-cos2mô+ 1±a1±a31

2,n1

3ascos(2m+3)Otl 2m+1 2m+3 JJe=i2 B cos(2m-1)O

±coswtq2[cos2mO±

_l±a1+a3( _2m-1 rn=1 ±zicos(2m+1)0 3aecos(2m±3)O1 2m±1 2m+3 JJo=,,2 By virtue of

cos2m()=(_1)m

cos(2ml)=O

we obtain

co=

[

o(B,)±(_1)mp2m}Sin

at+

_{+(_l)rnq2cos cullI. (3)}

Write now

-- _{Co=ocos(cut+s'),}

where '9 and the phase difference e' become as follows:

e,=

_/{

)±(_1)mq2rn}2± {øco()±(_1)"P2rn}2

tan =

ir

Then we put

Co = 2t90 cos(cut+e') =90 cos(cut±e±e),

where o, _{is the phase difference between the motion of the heaving and} _C0. Finally we obtain

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124 F. TASAL provided that E1=ø30()±_m1

(-1)" qm, E2(P0()±

₂

_n1

(l)"Pm.

Putting E/E1 =k we get 1(kB0 A0\

5h=tan _kA0±B0) (IO)

where A0 and B0 are defined by the equation (31) in [5} and A0/B0=tan o. (11) Accordingly, the ratio of the wave amplitude at the side of the cylinder to the amplitude of the -heaving is given by the following equation;

- (Co

-

ç

Ch

h -

- 111L O

where A, is the ratio of the progressive wave amplitude produced by the heaving oscillation to the heaving amplitude.

The is given as a function of = generally.

When eB tends to zero and o, tend to zero.

Therefore,

in the case of CO, when a cylinder plunges

into its lowest

position the water surface at the side of the cylinder comes down its lowest

postion, and when the cylinder heaves up to its heighest position, the swell-up of

the water surface will become maximum.

1. 2. Swaying and rolling oscillation

For a cylinder with Lewis-form section swaying with a small displacement

x=Scos(ot±r) (Fig. 2),

its velocity potential of the fluid motion is given in

[6] or [7]. That is,

Zp'ane

Fig. 2.

= _IrE,)ìø(c, a, O)cos wt+ø(ß, a, Ô)sin oit

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WA VE HEIGHT AT THE SIDE OF TWO-DIMENSIONAL BODY

+ cos sin(2m + 1)0 + sin 2mO

a 3a e(2m+4

-± _2m+2

sin(2m+2)0

2m±4 sin(2m+4)0}j

+sincot Q2m(en)[ " 't 't 't (13)

J

where ¿P and are the potentials by the two-dimensional horizontal doublet at the origin O.

in Fig. 2, C0 denotes the amplitude of the wave at right-hand side of the cylinder, that is, _{the one at G and is positive downwards. Since the wave at J,}

the left side of the cylinder, has the same magnitude and adverse phase to CO3 it

is sufficient to take into consideration C0 only. Using the potential (13) and

putting a=O in the equation (2) we can obtain the C0.

Writing Co=7.ocos(wt±r-t-s) by the same reduction as done in case of the heaving, we get q90= Jr ) 1(kQ0Po

e=tan

_1Q0+kP0)

where P0 and Qe

are defined in [6] and i° =tan r, and

E1 =$O()+(_l)mQ2rn E2co()±,1(_1)mP2m

k=E2/E1

Then the amplitude ratio , is given by the following equation

-

Col

-Ct

where A is the ratio of the progressive wave-amplitude produced by _swaying

oscillation to the swaying amplitude.

These and eo should te the function of f= generally.

When C, tends to zero, . and o. tend to zero. That is to say, in the case of a slow swaying oscillation, when a cylinder sways to the extreme position in the

right-hand direction the surface wave at G falls in and swells _{up at J, and next,}

in the extreme negative position of the cylinder the water surface _{is swelled up}

at G and falls in at J.

in the case of rolling oscillation having _{a small angular displacement} _0=0e

cos(wt+ô) about the origin O in the clockwise direction as is shown in Fig. 3,

we can dealt with the problem according to the same manner with the swaying

oscillation.

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Fig. 3.

the Lewis-form section is given in [7].

Letting denote the amplitude of the wave at the right-hand side of a

cylinder taking positive downward, the ratio of to is given by the

following equation

-jC0

_=Ao

(17)

where is the ratio of the progressive wave-amplitude produced by the rolling

BO0

oscillation to

With regard to t90 the same equation with (14) and (15) hold. That is,

9 v'E12±E22

Ei=()+(_1)m.Q2m

L

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E2=0()+(-1)m Pm

Using the calculated results given in {7J we can evaluate E1, E2 and accord-ingly ? for the cylinder with Lewis-form section.

Writing the phase difference between the rolling motion and CO3 namely,

C090 coS(øt±±eR)

we have

i( kQo-Po\

e,=tan

_1Q+kP)'

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OSCILLATING ON THE gURFACE OF A FLUID 127

where P) and Q are defined in [7] and

_tanä=

_{and k=E2/E1.}

û B

_{cd=°T for}

Finally we can obtain and as a function of

_{_-- 2 or,}

g a cylinder with Lewis-form section.

when tends to zero, _{tends to zero and} _{to Ir. That is to say, with a}

small _{when a cylinder rotates to the extreme position in the clockwise direction,}

the water surface at the right-hand side of the cylinder G swells up to the maximum, while in the adverse position of the cylinder the water surface at G falls in the lowest position.

The phase difference R decreases with the increase of .

IL _{Numerical calculation and the results}

Using the results given in [5] and [7], the_{numerical calculations were carried}

out for several cylinders with Lewis-form section. Fig. 4 shows for the several

elliptical sections as a function of B . These curves have a similar

tendency to the curve of À10.

Fig. 5 shows the , as a function of cTh.

Suppose now that the ratio of the half breadth of a ship at the midship

B0 2 B0

section to the draught T, _{that is, Hß*=2T equals 1.25 and OEB*=}

_{-.--- at the}

natural heaving or pitching period is _{equal to about 0.9. And the draught T is}

assumed to be constant throughout the length of _{a ship.}

Then _{of an arbitrary section of a ship is given by the}

following equation:

(H0

where H) is the ratio of the half breadth of an arbitrary section to the draught T. Then, eB _{of the ship-sections at H0=0.2 and H0 =2/3}

becomes 0.144 and 0.48

respectively.

Reading the _{for the sections H)==0.2, 2/3 and 1.25 from Fig. 5, we have}

It will be seen that all e, _{are nearly the same.}

for the sections of fore part of_{a ship will suffer the effect of the three}

dimensional motion of the fluid, they will, therefore, differ considerably from the above values.

In the Figs. 6 and 7, the effet of the area coefficient _{ro are shown for the}

section with H0=2/3.

From these figures it will be found that the fuller the section is the smaller

the ? is, however, on the contrary, the _{h becomes larger as the section is fuller.}

In the next place, in Figs. 8 and 9, _and e by the swaying oscillation are

given as a function of _{for five elliptical sections.}

In the Figs. 10 and 11, ., and e are given as the function of _T.

H1 0.2 2/3 1.25

0.144 0.48 0.9

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I.8

1.6 .4

1.2 LI

-s.

Pd-V4PU

'f

Ho 0.2

02 1i

ß-

2

Heaving

Ellipse

Fig. 4. i I i I I I I I I

0.2 0.4 0.6 0.8 1.0 ¡2

1.4 1.6

18 2.0 2.2 24 2.6

12S F. TASA!

I.0

0.8

0.6

0.4

(11)

o

-180

-le

-140

o

-I 20

-loO

-8C

-20

WA VE HEIGHT AT THE SIDE OF TWO- DIMENSIONAL BODY

OSCILLATING ON THE SURFACE OF A FLUID - 129

Heaving

a.

AM

A. A. MUM

a..

;IY

I

Ellipse

-\.s

I I I i

o

0.2 0.4 0.6 0.8

1.0 1.2 1.4 1.6 1.8

2.0 2.2

2.4 2.6

Fig. 5.

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_1600 -140° -120°

-loo

-80° 60° 400 _200 Fig. 7. 1.0 0.8 0.6 0.4 0.2

Heaving

_H0 2/3 I I Fig. 6. =O.7854

°0.9463 -2 1.6 2.0 2.4 ¿ 2 I I i I 0.4 0.8 ¡.2 ¡.6 2.0 2.4 0.4 0.8 1.2

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1.8 1.6 1.4 1.2 I.0 0.8 0.6 0.5 0.4 0.3 0.2 0.1

:4

--

-._.

1 A

iivi dUU

.__

Auu..

ivirnuuu

III FL VA4

IVA

,ir

Swaying-Ellipse

01 0.2 0.4 0.6 0.8 1.0 '.5 2.0. Fig. 8.

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M

lau

I4UU

¡un

\4UU

iuriv

iiuuriu

iuurnrurn

iiiiiiuu

liii VA

iirau

M'va

FA

'w,

Swaying E

Ellipse

B 2 2.0 0.2 0.4 0.6 0.6 1.0 1.5 Fig. 9. 132 F. TASA! -9 0 s o -70 o -6 0 o -50 -30 -20

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V'

WA VE HEIGHT AT THE SIDE OF TWO- DIMENSIONAL BODY

OSCILLATING ON THE SURFACE OF A FLUID

au

:AUUU

I 4i ,AI

PAUII

AI

r

Swaying-Ellipse

133 0 _{0.! 02} _0.4 0.6 0.8 1.0 1.5 2.0 Fig. 10. ¡.8 L6 '.4 ¡.2 ¡.0 0.8- 0.6- 0,5- 0.4- 0.3-0.2 0.!

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_900 .800 700

0.4

A. IWS

VA 191

2/3 o' 0.4

o'0.2

A,,

Swaying E

Ellipse

134 F. TASAL 0.2 0.4 0.6 0.8 1.0 1.5 _2.0 Fig. 11. -50 -4 d

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1.6 '.4 1.2 l.0 GB 0.6 0.4 0.2

WAVE HEIGHT AT THE SIDE OF TWODIMENS!ONAL BODY

OSCiLLATING ON THL SURFACE OF A FLUID 135

As seen in the Figs. 8 and 10, becomes larger as H0 is smaller contrary to

the heaving oscillation.

When we discuss the shipping water in bad weather, is so large on the

section with small H0 that we should take into cosideration C by all means.

Comparing Fig. 4 with Fig. 8, it is found that is larger than generally,

however, for the section with large H, is larger than in the case of small

eB.

Figs. 12 and 13 show the effect of the area coefficient a on the _and _e.

The fuller the section is the larger the , o are and the tendency of _is

opposite to the case of the heaving.

In the Figs. 14 to 16, are shown as a function of

c.

In Fig. 14, ,, becomes larger as the section is fuller.

has the same tendency in the case of H0=2/3 and 1.0 also. However, at

H0=1.5, on the contrary, we have large R with the finer section.

r. -H0 =

2/3

0.1 0.2 0.3

0.4

0.5

0.6

0.7 0.8

0.9

S.0 Fig. 12.

(Swaying)

uui

rdN u

A

.O

0a

z-

_d"fT

(18)

I 36 F. TASAL

O 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 13.

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WAVE HEIGHT AT THE SIDE OF TWO-DIMENSIONAL BODY

4.0 30 2.0 1.0

ai

_Ai

rirai

ii,

V-inri:

u.,,

jur

VMIA

IMi

W1VI

r-j.

.,

f

H0=O.2 (Ro ¡ ¡¡ng) 0 0.25 0.5 0.75 Fig. 14. r 0. 7854 r0.6440

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138 F. TASA!

in the case of H0 =0.2 is so large and H0 at the fore part of a ship is so small that R in the neighbourhood of the stem will become large. But in the stem, as the is generally small the effect of the yawing will be larger than the one of rolling with respect to the problem of shipping water at the fore part of a ship.

Figs. 17 and 18 show 0!? which increases as the section is finer.

H0= 2/3

(Rolling)

-i

0.4 0.3

VÀU

..

a

4!

I cilS 0.3 0.6 0.9 Fig. 15. 0.7 0.6 0.5 0.2 o. i

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0.25-WA VE !-IE!GHT AT THE SiDE OF TWO-DIMENSIONAL BODY

0.I 02 0._3 0.4 0.5 0.6 0.7

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-'

16c5 O.623O ,4c O.94

Ui'.

rn

N

Fig. 17.

H0 2/3

(Rolling)

OO 78 54 - CO.623O o.94

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01 90

j

r 6! { 90 t'O 0 0 01 80 90 t'O O

(uiIjoeJ)

- 00 I Pt" 9I Ç9I

(uIg I°H)

j

; -- _pG o 0001 oot' I - 991 008 f

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142 F. TASAI III. Conclusions

Calculations of the wave height at the side of two-dimensional body oscillating on the surface of water were carried out for the three cases of heaving, swaying and rolling on the Lewis-form cylinder. The summary is those that follows:

is generally larger than s except for the case of large H with small . Except for the

of If=l.5,

s and become large as the section is

fuller. While . becomes small as the fuller the section is.

When we discuss the shipping water of a ship in bad weather we must take into consideration the swell-up of the water surface at the side of a ship due to the ship motion, namely, Co produced by the heaving, pitching, rolling, swaying

and yawing. As known from the results calculated above the effect of ., is so

large that we should not forget due to the swaying and yawing motion of a ship.

Throughout these works, the writer is very gratefull to Messers. Arakawa and Yamasaki for their help in numerical calculations as well as in presenting the manuscript.

References

[fl J. L. Kent: "The Design of Seakindly Ships" North-East Coast Institute of

Engin-eers and Shipbuilders in Soctland. 1938.

R. N. Newton: "Wetness related to Freeboard and Flare ", T. I. N. A., Vol. 102,

No. 1, 1960, pp. 49-81.

R. Tasaki: "On shipping water ", Monthly Reports of Transportation Technical Research Institute Vol. 11, No. 8, Aug., 1961.

F. M. Lewis: "The inertia of the water surrounding a vibrating ship ", S. N. A.

M. E., 1929.

15] F. Tasai: "On the Damping Force and Added Mass of Ships Heaving and

Pitch-ing ", Reports of Research Institute for Applied Mechanics, Kyushu University, in Japan, Vol. VII, No. 26, 1959.

F. Tasai: "Hydrodynamic Force and Moment Produced by Swaying Oscillation of Cylinders on the Surface of a Fluid ", J. S. N. A. in Japan, Vol. 110.

F. Tasai: "Hydrodynamic Force and Moment Produced bySwaying and Rolling Oscillation of Cylinders on the free Surface ", Reports of Research Institute for Applied Mechanics, Kyushu University in Japan, Vol. IX, No. 35, 1961.