On wave-free floating-body forms in heaving oscillation

On wave-free floating-body forms in heaving oscillation Y U S A K U KYOZUKA National Defence Academy, Hashirimizu, Yokosuka 239, Japan and KIYOTAKA YOSHIDA Department of Mechanical Engineering, Defence Agency of Japan, Japan (Received 1 May 1980; revised 30 July 1980) Some wave-free heaving forms are obtained in a manner similar to Bessho's earlier work by using a source and vertical dipole combination to generate bulbous-shaped two and three-dimensional bodies which have zero radiation damping. Then the combinations of higher singularities, such as triple-pole and quadra-pole, are considered. Using the thin-ship approximation, where the boundary condition is satisfied on the centreline, a simple procedure for generating any arbitrary shape that has the same property is established. The existence of wave-free frequencies of such bodies are verified numerically, in two dimensions by a standard singularity~istribution method based on the actual body-contour and in three-dimensions by using vortex-ring distribution for an axisymmetric body. Experiments are carried out for the models obtained theoretically and the results support the theoretical predictions fairly well. Finally, some suggestions are made to apply such a wave-free body to a moored buoy to suppress its heaving oscillation.

INTRODUCTION

FORMULATION OF THE PROBLEM

In recent years offshore firms are becoming more active in the construction of drilling-rig platform, storage and production units. Some of these are of a semi-submerged type have a wave-free frequency at which the wave exciting force vanishes. In the design stage, the form of the vessel should be chosen to have a wave-free frequency equal to the average frequency of the waves at the sea where the vessel really works, to reduce its motions in waves. To obtain the wave-free form, Bessho 1 -3 investigated theoretically, using the wave-free singularities in motion of six degrees of the freedom and Motora et al. 4 found the actual form by combining circular and rectangular sections in the 2-dimensional heaving problem, sphere and cylinder sections in the 3-dimensional case. Frank 5 verified the existence of such wave-free bodies numerically; Maeda 6 and Sao et al. 7 presented charts of Motora form to get principal dimensions for a desired wave-free frequency. Further, Bessho 3 considered the wave-free form having two or many wave-free frequencies but it is not clear whether such a form really exists or of what form it is. This paper deals with the wave-free form in two and three-dimensions, having many wave-free frequencies of heaving motion, and the experiments are carried out for the bodies obtained theoretically. Finally, some consideration is given for applying such a form to the practical use of the moored buoy.

2-Dimensional problem

0141-1187/81/040183-1252.00 ©1981 CML Publications

According to the relation of Haskind-Newman, the body which does not generate the progressive wave when it oscillates in a certain mode in still water does not experience the exciting force of that mode from incident waves. Therefore, to obtain the wave-free body form, it is

ty o

Incidentwave ~~A>x <

Figure 1. Co-ordinate system of 2-dimensional problem

Applied Ocean Research, 1981, Vol. 3, No. 4

183

I~2n,e:li'ee /hmting-hody lm'm.s. }: K vozuka and K. }oshida 0.5 0"0

i

'

i

i

10

~

i

'

i

i

x

because the boundary condition of heaving oscillation is givcn by': ¢D1

=-~

('11

QF

on C

(4)

¢ It

]'hen the velocity components of the flow field is given b): -0.46 U=

-05

~X

~q~

(5)

the stagnation point is obtained by solving equation (4), putting x = 0 ?O -10

_~-

t: = ?ii; I. = 0 = 0

* ~ Singularity

Figure 2. Single wave-free form (half-section). 2K-I: K e =1.5

sufficient to find out the form which does not radiate wave by its heaving oscillation. Now, although Bessho~'2 introduced wave-free singularities by using the complex potential, we shall rederive them more directly• The velocity potential at P(x,y) due to a source singularity of unit strength at Q(x',y') is given as follows:

, 1 (r 1~ 1 i ek°'+Y')c°s k(x-x')dk ~olx,v:x',v )= ~ l o g , , lim ' " " 2~ \r2/ ~ ,~0 k-K+il a 0

(1) where r~ = ( x - x') 2 + ( y - y,)2, r21= (x - x') 2 + (y + y,)2 K =eo2/g, a)= frequency of oscillation, g =gravitational acceleration. To obtain wave-free singularity at wave number K~, it is sufficient to do the following operation: q~(x,y)=

~-

't

=2~-

1 log

~ x,y;

v'

~t +

r2

2 /

2(K 1 - K)lim f ek°'+Y'lc°s

,.22

(6)

Starting from the neighbourhood of the stagnation point, a streamline is obtained step by step by equation (5) and the dividing-streamline is approximated by it for sufficient small steps. An example of the wave-free form is shown in Fig. 2. This is made up of the simplest wave-free singularity of source and doublet placed on the y-axis. The body form changes as the location of the singularity, as shown in Fig. 3, which also shows that there exist many wave-free forms for one wave number. Further, such a wave-free singularities can be multiplied by and added to each other. Therefore the wavefree form exists in infinite variety even if one fixes the displacement of the body or the breadth at waterline. Next, we may introduce the wave-free singularity having two different wave-free frequencies at wave number K 1 and K 2 as follows:

0'5

OC

10

(2) -0.5

k(x-x') ,. t

o

where the last term vanishes when K = K 1 and we call this the wave-free singularity of source and doublet. By making use of this wave-free potential, we may obtain the body form by tracing the dividing streamline, if we introduce: @ = y + ~o~(x,y)] ?qb ? ~h = 0 on C J

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Applied Ocean Research. 1981, Vol. 3, No. 4

-1-C

(3)

Y

Figure 3. Various .[orms of the single waveff~ree (ha(/: section). 2K-I; Ke=0.5

Wave-free floating-body forms: Y. Kyozuka and K. tbshida 1.0

0.5

0.0

i

i

i

i

4 X

0'377 -0.5

-1'0

Y

f

It is noted that this double wave-free singularity does not always represent realizable form for arbitrary wave number. Namely, there is the case that the dividing streamline goes into the singularity and the realizable form cannot be obtained as shown in Fig. 5. The reason is that the source strength and the doublet strength become so small as compared with the quadrapole strength for small wave number. Of course, there are some ways to avoid this difficulty by changing the location of the singularity or adoption of continuous distribution of wave-free singularities. In the same way, we introduce the wave-free singularity having three different wave-free frequencies at wave number K I, K 2 and K 3 are:

~o3(x,y;x',y')= K1KzK 3-(K1K 2+K2K 3+ K 3 K 1 ) ~ 7, + ~2

~3 "} 0

(8)

(K1 + K2 + K3)c?y,2 ~y,3~tPs(X,y x,) )

Figure 4. Double wave-free form (half-section). 2K-2; K d =1.5

0 02 ) , , , + K2)~hy,+ ~ q ~ 2 ( X , ) ; x , Y )

tP2(x'y;x"Y')='~f

-2--[K1K2 l o g ( ~ ) + ( K l + K 2 ) ~ x 2 _ ( y _ y,)2 +

r~

+ ( 2 K - K 1 - Kz)Y- +rZz y'

An example of the triple wave-free form is shown in Fig. 6. The wave-form singularity of multi-wave-free frequencies can be introduced easily like this but it cannot represent a definite body form in general except when all frequencies coincide with each other.

3-Dimensional problem The same method can be applied to the 3-dimensional problem as investigated in the preceding section. The velocity potential at P(x,y,z), due to the source singularity of unit strength at Q(x',y',z') in the co-ordinate system of Fig. 7, is given as:

X2 _ (y + y,)2 +

r24

=

~b=(P,Q)

1 1 ..... ~ eklz+='lJo(kR)_, znum • - - - - oK r, r 2 .~o) k-K+ip

(9)

0

- 2 ( K 1 - K ) ( K 2 - K ) I i m f ek"+'"COS k(x-x')dk] .~o j k - K +ip 0

0.5

1.0

0.0 X

The last term of equation (7) vanishes when K = K ~ or K = K 2 as in the former case. This singularity is clearly composed of source, doublet and quadrapole. However, if K 1 :p K2, this does not represent one definite form because of the third term of equation (7). Because we have this term as:

(2K-K1-K2)~(KI_ r2 for

K )Y+~Y' 2 r2

-0.5

K = K 1 but for K = k 2 it changes sign as: (2K - K t - K2)

r2

' ~ - ( K 1 - K2)Y ~Y'

Therefore we should make K 1 tend to K 2 to obtain one definite body form and we call it the double wave-free form. An example is shown in Fig. 4. It has single wavefree frequency but may be expected to have low wave exciting force in the wider band of wave frequency than the above.

-1.0 L Y

Figure 5. Stream-lines of double wave-free singularity when the realistic form is not obtained. 2K-2; K e = 1.0 Applied Ocean Research, 1981. Vol. 3, No. 4

|85

Wm,e-l)'ee floating-body.lbrms:

};

K.vozuka and K. Yoshida 1.o

0.5 00~*

--

,

,

,

,

,

,

streamline, we introduce the following velocity potential:

X

qJ-=--4 9~ ]

(11)

? , = 0 on C because in this case the boundary condition of heaving oscillation is given as: -05

on C

(12)

This point singularity represents an axisymmetric body and the same method can be applied to obtain the body form as in 2-dimensions. An example of the various wave free forms of axisymmetric body for one wave number are shown in Fig. 8. To obtain wave-free singularity having two different wave-free frequencies, the following operation is suitable: -1.0

Figure 6.

~Pz(P,Q) = KIK2 - ( K 1

Triple wat'e-]ree Jbrm (ha!]-section). 2K-3; K d

(-2 t ~(P,Q) + K 2)~-~ +(-~Z,~

(13)

=1.5 In this case also, one definite form is not obtained when Z

K 1 ¢ K z for the same reason as the 2-dimensional case. Therefore, we must make K 1 tend to K 2, and we obtain the double wave-free form of axisymmetry as shown in Fig. 9.

Y

0

),x

Method Jor the slender body Chou s presented the optimal design method for a slender buoy and a vertical float-supported ocean platform, as derived from minimizing its motion in seas. The method of calculus of variations, together with the penalty function method, is used to determine the optimal form which minimizes the buoy's motion in a random sea. Although his method is very complicated even under the slender body approximation, the following method may be easier than his method. For simplicity, we consider the 2-dimensional problem. The wave exciting 05

Figure 7. Co-ordinate system of 3-dimensional problem

10

O'

where r 2 = ( x - x ' ) 2 + ( y - y ' ) x + ( z - z ' ) 2 r22=(x_x,)2+(y - 3¢)2 + (z + z') 2, R z = (x - x') 2 + ( y - y,)2 Jo(kR); Bessel function of 1st kind. By the same procedure as in the 2dimensional problem, the wave-free singularity at wave number K1 is introduced as follows:

tP,(P,Q)=(KI -~, ,)O~(P,Q) ,

'Z

(

1 -i:

= - K 1 .;'l

12)-bz-z' .~I!

/ g,

- 2K(K 1 - K ) l i m [

z+z' .3 12

(10)

ek(Z+Z'~Jo(kR )

0

when K=K~ the last term vanishes; this is obviously wave-free. To obtain the body form by tracing the dividing

186

Applied Ocean Research, 1981, Vol. 3. No. 4

z Figure 8. Various jorms ~?J the single wave-free oJ' axisymmetric body (ha!f-section). 3K-l: Kd= 1.5

Wave:l'reefloating-body forms: Y Kyozuka and K. Yoshida 0.5

00

,

,

,

'

'

'

'

'

i

1.o g R

the source distribution is given directly for the heaving oscillation as

a(y) = 2dx = 2 f~,(y)

(19)

oy

Therefore, if we take the function re(y) under the condition of equation (16), the body form may be calculated easily by equations (15) and (19). For example, let us take re(y) f o r d = - I as:

m(y) = cy(y + 1) for - 1 ~ y ~ 0

-0.5

(20)

and putting x = 0 at y = d = - 1, we obtain:

3 I[K~ ) K~} x = o ~K~ ~-6- y + 2 ~ T +1 y2 +~Y 1 2

--0""

z

Figure 9. Double wave-free form of axisymmetric body (half-section). 3K-2; Ka= 1.5 force of heaving oscillation is represented by Kochin function under the thin-ship approximation as:

F= f_-opgaH(K)

}

m(O)=m(d)=~m(O)=~-fm(d)=O

(22)

the source distribution is given as:

where F -- wave exciting force of heaving oscillation, H(K) = Kochin function, a = source distribution, a = amplitude of incident wave. Let us give the source distribution by the following operation:

a(y)={Kl + ~}m(y )

where c should be determined by the given condition of the displacement of the body or the breadth at waterline. In Fig. 10, the wave-free forms of various wave numbers are shown for c = - 1. This method is easier and more direct than Chou's under the same linear approximation. The choice of the auxiliary function may be arbitrary under the boundary condition (16). In the same manner, we introduce the double wave-free form as follows. Let it be supposed that re(y) is continuous up to its first derivative and that it satisfies:

(14)

H(K) = ~ a(O,y)ekYdy

(21)

a(y)= ~K,K 2 +(K, + (

2)~y +

(23)

Then Kochin function becomes by equation (14) as: 0

H(K) = (K - K1)(K - K 2)f m(y)ekYdy

(15)

(24)

d

where the auxiliary function m(y) is continuous in d ~
m(d)=m(O)=O

o

o

1

4 X

(16)

Then the Kochin function becomes: 2

B

0

d

(17)

= (K I -

K)f

m(y)ekYdy

Therefore, the wave exciting force vanishes for K =K~, that is, wave-free. It is easy to obtain the body form under the same thinship approximation. Namely, expressing the off-set of the body as: x =f(y)

(18)

-1

Figure 10. Single wave-free forms ,for various wave numbers by thin-ship approximation Applied Ocean Research, 1981, Vol. 3, No. 4 187

Ware-ti'ee lloatinq-hody lorms: y K yozuka and K. )bshida 0-0

00

10

Here, we introduce the Green function of 2-dimensions Slx,y:x',y') and obtain the following expression for the diffraction potential: i? S qoa{P)= [(?qda--~padi,){P,Q} dS~,

128}

where

1 /r~ \ - l_lim [ ek"+Y"cos k(x-x'} S(,,Q)=27?Og~rz) rt,~o d . . . . k - - - K +ip- - d k

IO

0

r~z = ( x - x ' ) z + ( y -

y'l 2

(29}

r~ = {x - x'} 2 + {y + yl}e

- 1 0 , JY

Making use of equation (27) and the following relation:

Figure 11. Double wave-Jree jorms jot various wave numbers by thin ship approximation. K i d = 1.5(fixed)

C(pO -

Cpo

-

¢;

S(P,Q) dS o = 0

(30)

c

An example is shown in Fig. 11, taking auxiliary function we obtain the following equation for equation (28):

as:

re(y} = cy{ 1 - cos 2r~y)

{25)

In this case, we can obtain the wave-free body at two different wave numbers Kad and K2d , but there does not exist such a body in exact theory as was considered in the former section. Although this method is very simple and easy, it should be noted that it is applicable only to the slender body. It does not coincide with the body form obtained by tracing the streamline for the given source distribution. When the wave number is small, there appears negative breadth as shown in Fig. 11 but there may exist the realistic body form by tracing streamline for this given source distribution.

d (~Po+ q°a~nS(P,Q} dSo

~°a(P) = -

(31}

c

Further we introduce a new function T(P,Q) instead of S(P,Q) by the relation of Cauchy-Riemann:

dnoS(P,Q)=g~-oT{ P,Q) (32}

&~T{p,Q)= -

sf{P,Q)

and obtain

CALCULATION OF THE EXCITING FORCE

2-Dimensional problem ]'he wave exciting force is calculated numerically by solving the integral equation which is introduced by singularity distribution on the body surface. We investigate if the body obtained in the former chapter is really wave-free at the designed wave number. When the water depth is infinite, the incident wave potential and the diffraction potential are expressed as follows: iqa ~90,d= ~; (POd

{26}

188

on C

Applied Ocean Research, 1981, Vol. 3, No. 4

(271

(33}

c

Finally, we can formulate the following Fredholm integral equation of the 2nd kind with respect to the singularity when P approaches to the body surface.

l 2~Oo + ~oa)+ [(q~0+ q)a)dT(P,Q)= qOo(P) c

where ~o = incident wave potential, Od=diffraction potential, rpo = exp.[Ky + iKx], a = incident wave amplitude. The boundary condition on the body surface is: ? ?n{Oo + ~oa)= 0

qoa(P) = - ((~oo + ~oa)dT(P,Q)

where

T(P,Q)=-2~O , +02)+l-lira [ ekIY+""sin k{x-X')d k • rc,,~o ! k---k-+it~ ~ 0

01 =tan

1Y - 3[' X -- X"

O2=tan_ ~y+y' X -- X I

{34)

Wave-free floating-body jbrms: Y. Kyozuka and K. Yoshida We obtain the diffraction potential on the body surface by solving equation (34) and exciting force can be calculated directly by the following formula because the pressure is proportional to the potential.

Fi el = pgBa

I(~°°+ ~°d)~nndS OXi

(35)

On the other hand, we have the following expression for the outer point P, introducing the regular function (z + l/K) in the inner region of the body which satisfies the free-surface condition and the body-boundary condition:

O:ff{~n(Z+K)-(z+l~}G(P'Q)dSQ

L ¢

g

where suffix i means the direction of the exciting force

i=l i=2

heave, sway,

(39)

Oxt

8y

On

On

8x 2 _ 0 x 8n 8n

Adding equation (38) to (39), and using the body boundary condition: O~0

#z

gn

On

-

on S

(40)

we obtain the following expression:

The advantage of this method is that we can obtain the velocity potential itself directly. In actual numerical calculation, half side of the body surface was divided into twenty segments.

3-Dimensional problem

8

q~(P)= - f f #(Q~G(P,Q) dSo

(41)

where

Generally, it is difficult to calculate accurately the hydrodynamic forces for arbitrary body among waves in the 3-dimensional problem because of the computing time and storage. But for the axial symmetric body, it is easy to calculate the hydrodynamic forces as in the 2-dimensional problem. Moreover, to obtain only the wave exciting force it is sufficient to calculate the heaving motion problem because of the Haskind-Newman relation. For this purpose, we introduce a new method in which the velocity potential is represented by the doublet ring, and the unknown function of the boundary integral equation is directly the potential itself so that the solution permits us to calculate the hydrodynamical forces. Let us consider the forced heaving motion of the body in calm water and describe the motion of the body, the velocity potential, the pressure and the force as follows:

1

I~(Q) = q~(Q)+ z( Q) + ~

This is the expression of the velocity potential due to the doublet distribution. When the body has axial symmetry, we can integrate equation (41) around its latitude. The relation of equation (A3) gives the expression (see Appendix):

q)(P) = - f la(O)dGv(P,Q)

r' 2n

G~(P,Q,:ff~G(P,Q,rdOdr 0

(36)

P(x,y,z,t) = Re{p(x,y,z)e i°)'}

(43)

where

Z(t) = Re{he i°''} qo(x,y,z,t) = Re {io~hq)(x,y,z)ei'°t}

(42)

0

c denotes the longitude of the body. Further, let us introduce Stoke's stream function T~(P,Q) which satisfies the following relation:

F(t) = Re{fe i°'} 8rp- rp ~Zp ~G~, 1 ~T~ 8Zp rp Orp

where h denotes the amplitude of heaving motion. As 1s well known, the Green function of the 3-dimensional problem for the source singularity of unit strength is given

(44)

as:

ek(Z+z'))Jo(kR) k-K+'l~ 6@(37,

G(P'Q'=ln{l+l+2Kluimf

Then the stream function corresponding to equation (43) becomes:

0

= - f .@d rv(P,O)

The velocity potential is expressed generally by Green's theorem as is well known,

(45)

¢

where

q~(P)=ff(~n-q~f--~n)G(P,Q)dSe

(38)

T~(P,Q) =

-rp f ~Gv(P,Q)dz p

(46)

Applied Ocean Research, 1981, Vol. 3, No. 4 189

Ititre-;l/ee /hmtinq-I~ody /orms: Y Kyozuka and K. }bshida "lahh' 1.

7r

Principal particular~ r f the model~

2-dimen,~iomdmodel

d; depth (m) Bw; breadth [diameter) at waterline (m) Bin: maximum breadth (diameter) (m) Displacement (kg) Kd; wave free wave number

exciting force, A,,. = 4 B~,.: waterline area. a = amplitude of

4xi.s3:mmetric model

2K-I

2 K - 2 2K-3

3K-2

0.239

0.239 0.160

0.239

0.231

0.200 0.210

0.210

0.280 0.247 0.246 33.8 28.9 19.6 1.5 1.5 1.5

0.258 8.86 1.5

1 /2pOo ( K=

152}

The boundary condition for heaving oscillation of unit velocity amplitude is given as: r 2

~9(p). . . . ~

(47)

Substituting equation (47) for (45), we obtain the Fredholm integral equation of the I st kind to be solved as:

-~i=-flJ(Q)dT~,(P,Q)

(48)

c

Finally, the distribution of velocity potential is obtained from equation (42):

~P(P)= I4P)- (z(P) + IK)

(49)

The hydrodynamic force is obtained by integrating the pressure distribution over the body surface, and the added mass and the damping force are obtained as follows:

.f=-

p(x,y,z)-?ndS s

= pm2hf2rtqgdr

incident wave. And it is also given by 111c relation of H a s k i n d Newman as follows:

(5O)

This relation is used to check the numerical accuracy of equation (51). Finally, although we solved the integral equation (48), it is to be noted that the expressions (41) and (42) may be considered as the boundary integral equation with respect to the unknown function IL

M O D E L S A N D THE EXPERIMENTAL RESULTS Experiments were carried out at the sea-keeping tank of the National Defence Academy (L x B x d = 9 " x 1.2" x 1m) as shown in Fig. 13. The principal particulars of the models are shown in Table 1, and its notation is indicated in Fig. 12. We select the model size which might be wavefree at the wave length 1 m (the non-dimensional wave number Kd = 1.5, where d denotes the depth of the model) and also might be large enough to enable us to measure the wave exciting force accurately. If the length of the tank is long enough to use long wave, one may choose the models that have lower wave-free number which is more interesting from a practical point of view. As shown in Fig. 13, two waterways were set-up for the 2-dimensional models in the middle part of the tank to create the 2-dimensional condition. The wave exciting force was measured by the 3-component load cell which was installed in the model, and the incident wave was measured by the wave gauge in another side of the waterway to the model. All measured records were analysed numerically by Fourier analysis. Although the super-harmonic components appeared in measured records of the forces especially in the experiments of short wave length, we did not consider them but we dealt with the fundamental frequency component only. The comparison of the calculation and the experimental results are shown in Figs. 14, 15, 16 and 17, where the wave exciting force is normalized as follows:

= pu)2hlj~ + iJ~) where PJc denotes the added mass, - p t o J s denotes the damping term. The wave exciting force is calculated from the solution of radiation problem by the Kochin function as:

e=

F w

pg A,,a

L

= -H(K) (51)

s

where ~%=exp.[KZ+iK(x.cos 0 + y . s i n 0)], F w = w a v e

190

Applied Ocean Resemvh, 1981, Vol. 3. No. 4

Fi,qure 12. Notation q/the principal particulars

Wave-free floating-body Jbrms: Y. Kyozuka and K. Yoshida Beach

a body is really wave-free experimentally as was theoretical predicted in the preceding sections. These forms may be directly applicable to practical use of, such as, tension leg platforms which does not oscillate in heave among waves. But in such a type of stable platform, the buoyancy exceeds the weight and the necessary vertical equilibrating forces are supplied by vertical tensional anchor lines 9. On the other hand, it is more important to consider the resonant response of heaving motion in case of a floating platform. On that point, there are several reports

Wave maker

ot +

I

4000

I 9000 05

2K-2

e

Figure 13. Experimental set-up

2K-1

e

05

~.~, 0.0

05

1.0

k...~/.--~ ,---rT--~*~ 15

20

25

Kd

Figure 15. Comparison of wave exciting Jorce of double wave-free model between experiments ( • ) and calculation 00

)

05

10

15

20

25

Kd

Figure 14. Comparison of wave exciting force of single wave-free model between experiments ( • ) and calculation

(

0-5

e

2K-3

)

F w

e=pgLB~ Fw

e=pgAw ~

\ for 2-dimensional model ]

for axi-symmetric model

J

(53)

j

00

Generally, the experimental results show good agreement with the calculation. To compare the results of the model 2K - 2 with those of 2K - 1, we find that it is effective to reduce the wave exciting force for wide wave number especially near the wave-free wave number. Contrary to the expectation, the triple wave-free form 2 K - 3 is not effective in both calculation and experiment, but the reason is not clear. In Fig. 17, we see that axial symmetric model 3 K - 2 is excellent for reducing the wave exciting force for wide range of wave numbers as in 2 K - 2 . Consequently, the double wave-free form is superior to the single one but the triple one is not so. We could not conclude whether the still higher wave-free forms (than triple) is effective, but it seems that multi-wave-free may not be advisable since its singularity becomes stronger and it is more difficult to obtain a realistic body form. SOME CONSIDERATION TO THE HEAVING MOTION OF MOORED BUOY It is confirmed that we may obtain the wave-free body form at arbitrary wave number theoretically and that such

05

10

15

~

20

25

Kd

Figure 16. Comparison of wave exciting .[brce of triple wave-free model between experiments • ) and calculation (---)

10

e

3K-2

05

05

10

15

20

25

Kd

Figure 17. Comparison of wave exctting Jorce of double wave-free model of axisymmetric body between experiments ( • ) and calculation ( )

Applied Ocean Research, 1981, Vol. 3, No. 4

19|

W~u:erl)x'efloatmg-hody lbrms: Y K vozuka and K. Yoshida motion, and for this purpose equilibrating weight must be reduced from the buoy to keep its statical draft. Then, the equation of the heaving molion of such a condition is given as follows:

30 ¸

2K-1

(Free}

W (1 + m o - c)5 + pN- + (p#A,, + k):. -

f~.l"

e ~I....

2o

Zst =

cH /

(

(561

kq

where k = c o n s t a n t of the restraining spring, Zst = initial deflection, c = subtracted mass coefficient. In this case, the wave number of the natural period is given as:

1.O

Kod= (pgA,,+k)d W(l +mo - c ) •

Q

0.0 0.5

1.0

15

2-0

(57)

In our experiment for the model of 2K-2, the constants and the wave number of the natural period were as follows:

Kd

Fi,qure 18. Heaving response of single wave-fi'ee model in waves. , Calculated: e, experimental

c

=0.264

k=125.0 concerning with the optimization scheme for motions of an ocean platform in seas 1° ~2. For example, as seen in Fig. 18, we show the amplitude response of heave for the model 2K-l, free-floating in waves, it does not oscillate at the wave-free period, but it oscillates at very large amplitude at the natural period. This means that such a body has little radiation damping over a wide range of wave number, so that it may be a disadvantage in practice. It might be desirable to make the natural period of the heaving motion coincide with the wave-free period, but it has already been proved impossible to do so for the body free-floating on the water'*. Therefore, by making use of the mooring system so as to make the natural period coincide with the wave-free period, it may be possible to reduce the heaving motion. For simplification we consider the 2-dimensional problem, but of course, the same idea may be applied to 3dimensions. The equation of the heaving motion for the free body among waves is given as follows: W (1 + m o ) 5 + p N Z + p g A . , Z = F w e~''' H

(54)

kg/m

Kod = 1.60 The comparison of the heaving motion between the calculation and the experimental results for both cases of the free and moored buoy is shown in Fig. 20. It might be effective for reducing the heaving motion to make the natural period coincide with the wave-free period by making use of the mooring system. In our experiment we had to choose the wave-free number of the body comparatively higher, due to the restriction of the experimental installation, so that the natural period differed by far from the wave-free one, and a strong spring was necessary to make both periods coincide with each other. But, in practical use, the wavefree wave number should be chosen smaller generally, so that it may be closer to the wave number of the natural period, because both wave numbers are very near when the wave-free wave number is small. Therefore it seems that the more weak spring might be sufficient enough to make both wave numbers coincide with each other.

where W= displacement of the body, m o = a d d e d mass coefficient, N = damping force coefficient, A,, = area of the waterline of the body, F w= wave exciting force in heave. Then, the natural period is given in the non-dimensional wave number as:

Ko" = (@t = pyAAt ~/ W(I +m0)

Applied Ocean Research, 1981, Vol. 3, No. 4

Wave

(55)

the wave number Kod of the natural period for the wavefree model 2K-2, is 0.78 and its wave-free wave number is Kd = 1.5. We make use of the mooring chain for the body and regard it as an equivalent spring as shown in Fig. 19. If we choose an adequate spring constant, it is possible to make the natural period of the heaving motion coincide with the wave-free period. It is necessary to have statical tension in the spring for the entire extent of the heaving

192

(58)

/

/ N

// // /

////// Figure 19.

/

/ /

/

/

/

/ /

/

/

/

Conceptional figure o['moored buoy

/

Wave-free floating-body forms: Y K y o z u k a and K. Yoshida

II

3.0

I I

k 2K-1 3K-2 G(P,Q) Gv(P,Q) T~(P,Q) re(y)

2K-2

I I I I

2.0

V/a

/

t I

n

S(P,Q)

"1

T(P,Q)

./ 10

W

J

•

(x,y,z)

l

(r,O,z)

l

0.0

(PO

05

1'0 Kd

15

2.0

Figure 20. Heaving responses when the buoy is free and moored in waves. Free: e, experimental; - - - , calculated. Moored: ,t, experimental, • - - , calculated

CONCLUSIONS

q~d q~s

q,s 0 O"

P O9

W e investigated the various wave-free b o d y forms in 2- and 3 - d i m e n s i o n a l p r o b l e m s in heaving oscillation a n d o b t a i n e d the following conclusions. (1) W e i n t r o d u c e d the wave-free singularities having m a n y wave-free frequencies. But, they did n o t c o r r e s p o n d to a single form in general a n d we should equalize such wave-free frequencies to one value to o b t a i n physically realizable form. W e investigated the d o u b l e a n d triple wave-free bodies in 2- a n d 3 - d i m e n s i o n a l cases, which m e a n s that its K o c h i n function of h e a v i n g m o t i o n has d o u b l e or triple zero p o i n t at t h i s wave n u m b e r . (2) W e also o b t a i n e d wave-free forms u n d e r the a p p r o x i m a t i o n of the thin-ship by i n t r o d u c i n g an auxiliary function. It is simpler a n d easier to c o n s t r u c t wave-free forms t h a n C h o u ' s m e t h o d . (3) A m e t h o d was p r o p o s e d a n d verified e x p e r i m e n t a l l y to reduce the heaving m o t i o n of the m o o r e d b u o y a m o n g waves by m a k i n g use of the m o o r i n g system to m a k e the n a t u r a l p e r i o d coincide with the wave-free period, which is an i n t e r m e d i a t e type between a floating a n d a tension leg type.

(D o

a

Aw B d e

F F~, .f(Y) g h

H(K) K

= a m p l i t u d e of incident wave = waterline a r e a = b r e a d t h of the body, b = B/2 = d r a f t of the b o d y = n o n - d i m e n s i o n a l wave exciting force = force acted b y fluid in heaving oscillation = wave exciting force = sectional b r e a d t h of the b o d y = g r a v i t a t i o n a l acceleration = a m p l i t u d e of force heaving oscillation = K o c h i n function = ( D 2 / g = wave n u m b e r

Q = weight of the b o d y = C a r t e s i a n c o o r d i n a t e system = c y l i n d r i c a l c o o r d i n a t e system = velocity p o t e n t i a l of the wave-free singularity in 2-dimensions = velocity p o t e n t i a l of the wave-free singularity in 3-dimensions = n o r m a l i z e d incident wave p o t e n t i a l = n o r m a l i z e d diffraction p o t e n t i a l = source p o t e n t i a l of 2-dimensions = source p o t e n t i a l of 3-dimensions = s t r e a m function of a x i - s y m m e t r y = source d i s t r i b u t i o n = artificial frictional coefficient or vortex-ring distribution = f l u i d density = 2 n i t = circular frequency = c i r c u l a r frequency of the n a t u r a l p e r i o d = 3.141592654

ACKNOWLEDGEMENT T h e a u t h o r s wish to express their i n d e b t e d n e s s to Prof. M a s a t o s h i Bessho who m o t i v a t e d t h e m on this thesis and for his g u i d a n c e a n d e n c o u r a g e m e n t t h r o u g h o u t this work.

REFERENCES 1 2 3 4 5

NOMENCLATURE

= c o n s t a n t of the restraining spring = single wave-free b o d y in 2-dimensions = d o u b l e wave-free b o d y in 3-dimensions = G r e e n function of 3-dimensions = G r e e n function of v o r t e x - r i n g ---stream function of v o r t e x - r i n g = auxiliary function to d e t e r m i n e source distribution = unit n o r m a l into fluid = G r e e n function of 2-dimensions = c o n j u g a t e function of S(P,Q) with respect to

6 7 8 9 10 11 12

Bessho, M. On the wave-free distribution in the oscillation problem of the ship, Zosen Kiokai, 1965, 117, 127 Bessho,M., On the theory of wave-free ship forms, Memoirs Defence Acad. 1967, 7, (1), 263 Bessho,M. Could any floating body be wave-free at two periods?, Prive Note, 1965 (unpublished) Motora, S. and Koyama, T. On wave-excitation free ship forms, Zosen Kiokai 1965, 117, 115 Frank, W. The heaving damping coefficients of bulbous cylinders, partially immersed in deep water, J. Ship Res. 1967, 11, (3). 5 Maeda, H. Wave excitation forces on two dimensional ships of arbitrary sections, J. Soc. Nat'. Arch. Japan 1969, 126, 55 Sao, K., Maeda, H. and Hwang, J. H. On the heaving oscillation of a circular dock, J. Soc. Nat'. Arch. Japan 1971, 131), 121 Chou, F. S• A minimization scheme for the motions and forces of an ocean platform in random seas, SNAME Trans. 1977, 85, 32 Horton, E. E., MacCammon, L. B., Murtha, J. P. and Paulling, J. R. Optimization of stable platform characteristics" Offshore Technol. Conj., Houston, 1972 Hooft, J. P. Design platforms for minimum motion, Ocean Ind. Dec. 1970 Hooft, J. P. A mathematical method of determining hydrodynamically induced forces on a semisubmersible, Trans. SNAME 1971, 79, 28 Minkenberg, H. L. and Van Sluijs, M. F. Motion optimization of semi-submersibles, Offshore Technol. Con£, Houston, 1972

Applied Ocean Research, 1981, Vol. 3, No. 4

193

Wiwe-li'eelloatinq-hody /orms. Y Kyozuka and K. Yoshida APPENDIX

Using equations (A3), equation (43) is obtained from equation (41)as:

The Voltex-Rino Potential ]'he velocity potential of vortex-ring of radius r' is introduced for axial symmetric body by using the Green function as in: r' 2n

G,~(P,Q):ff ~(ZQG(P,Q)ro,drqdO 0

Then

~41)

~°~P'=- f ~ 2~

(A1)

0

dO dS~ c 0

G(P,Q) and G~,(P,Q)satisfy the following relation: 2tr

~G~ =-f#(Q)~-~
?re )o r°~G(P'Q)dO 2r~

(A2)

OzQ- f rQ (P,O_)dO 0 ~G,,

c

0

Therefore we obtain the following expression. 2~

2rr

frq?~QG(P,Q)dO f ['?r°- '~G ?Ze'CG~do = re)one'?re -~ (,~Q CZQJ 0

0

~G v

~?SQ

194

Applied Ocean Research, 1981, Vol. 3. No. 4

= - I#(Q) dG~(P,Q)

(A3)

(43)