Behaviour of vertical bodies of revolution in waves

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Ocean Engng, Vol. 13, No. 6, pp. 505-538, 1986. 0029-8018/86 $3.00 + ,00

Printed in Great Britain. Pergamon Journals Ltd.

B E H A V I O U R OF VERTICAL BODIES OF R E V O L U T I O N

IN W A V E S

KONSTANTIN KOKKINOWRACHOS

Technical University Hamburg-Harburg & Technical University Aachen, Federal Republic of Germany

SPYRIDON MAVRAKOS

National Technical University of Athens, Greece

and

SAMPSON ASORAKOS

National Technical University of Athens, Greece

Abstract--This paper presents the so-called macroelement method by means of which the complete linear hydromechanic analysis of arbitrarily shaped bodies of revolution with vertical axis can be carried out. The development of a special method for this wide class of structures which are common in offshore designs is of great advantage for the engineering work.

The method described here is based on the discretization of the flow field around the structure by means of ring-shaped macroelements, the velocity potential in each element being approximated with Fourier series. For the matching of the solution between neighbouring elements Galerkin's method is applied. Both the diffraction and the radiation problems are solved. a al all, a33,a55 bp

bll, b22, b33

d dt

F,o, F3o, F~,,

g

hp

H Hm In I,,, k kt K,,, n N O M E N C L A T U R E

maximum radius of the axisymmetric body (see Fig. 1)

small radius of the /-th ring element of type II, l = 1, 2, . . . , L (see Fig. 1)

added masses for surge, heave and pitch, respectively

small radius of the p-th ring element of type III, p = 1, 2 . . . P (see Fig. 1)

potential damping for surge, heave and pitch, respectively water depth

distance between the bottom of the l-th ring element of type II and the sea bed, l = 1, 2 . . . L (see Fig. 1)

amplitude of the exciting wave forces and moment acceleration of gravity

gap between the top of the p-th ring element of type III and the sea bed, p = 1,2 . . . L (see Fig. 1)

wave height

m-th order Hankel function of first kind

m-th order modified Bessel function of first kind m-th order Bessel function of first kind

wave number defined by Equation (5)

imaginary root of Equation (48) for the /-th ring element of type II m-th order modified Bessel function of second kind

unit normal vector of the body's wetted surface in average position pointing outwards

(2)

~(1(~ ~ONL-,LkNIIN ~<)KKIN<)\t+RA(H()N e[ (/], r r, 0, z t xl

ZkCz), Z,,(Z)

Z~,(z), Z~,(z) O{, Oq E p (?Po ~7 ~bl, ]' = 1 , 3 . 5 CO position vector cylindrical coordinates time variable motion components (] :- 1. 3. 5)

amplitudes of motion components (] = 1,3, 5)

functions defined by the Equations (33) and (34), respectively functions defined by the Equations (44) and (45), respectively real roots of the Equations (37) and (48), respectively phase shift angle

wave length (X :

2v/k)

fluid density

velocity potential of the incident wave diffraction potential

surge (j=l), heave (j=3) and pitch (/=5) radiation potentials wave frequency.

1. INTRODUCTION

FOR THE hydrodynamic analysis of a body in a regular wave analytical and semi- analytical approaches, the method of multipoles, the finite-element- and the boundary- element-techniques and the method of integral equations have been widely used. The analytical and semi-analytical methods are restricted to simple geometrical shapes. The method of multipoles has been used by Havelock (1955) for the determination of the added mass and damping coefficients for heave of a floating sphere in infinite water depth. The same method was used by Wang (1966) for a sphere in finite water depth. A frequently employed approach is the method of integral equations based on a suitably selected Green's function (John, 1950). Kim (1965) presented added mass and damping coefficients for a semi-ellipsoidal body with its origin on the free surface in water of infinite depth.

Garrison (1971, 1972, 1974), Lebreton and Cormault (1969), Milgram and Halkyard (1961), van Oortmerssen (1972), Faltinsen and Michelsen (1974) and Chakrabarti and Naftzger (1976) developed several numerical schemes based on the three-dimensional sink-source techniques for bodies of arbitrary shape. Black (1975) and Fenton (1978) exploited the axisymmetry of vertical bodies of revolution to reduce the surface integral equation to a one-dimensional equation. Bai and Yeung (1974), Chen and Mei (1974), Yue, Chen and Mei (1976), Zienkiewicz, Bettes and Kelly (1977) and Mei (1978) presented numerical procedures based on the finite-element variational method.

The three-dimensional sink-source technique and the finite-element variational approach can soon become complex and, moreover, extensive in the numerical treat- ment. For this reason semi-analytical methods are justified for certain body types. As the effect of the geometrical configuration on the flow is partly taken into account in the basic formulations of these methods, the amount of calculation necessary is usually reasonable.

The main object of this paper is to present a semi-analytical approach for the hydrodynamic analysis of large bodies of revolution with vertical axis in water of finite depth. The method is based on the discretization of the flow field around the structure using coaxial ring elements, which are generated from the approximation of the body's meridian line by a stepped curve, (Fig. 1). An outline of the method and some numerical results were firstly reported by Kokkinowrachos (1978), whereas the detailed theoretical

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B e h a v i o u r o f v e r t i c a l b o d i e s of r e v o l u t i o n in w a v e s 507

procedure was presented by Kokkinowrachos, Asorakos and Mavrakos (1980). The method used could be considered as an extension of the one employed by Miles and Gilbert (1968) and Garrett (1971) for the diffraction problem of a circular dock as well as by Kokkinowrachos (1974, 1976) for the case of floating or submerged, simple or composite cylindrical bodies. On the other hand, Sabuncu and Calisal (1981, 1984) have in extension to Garrett's (1971) method presented results for the hydrodynamic coefficients of simple or composite vertical cylinders at finite water depth. A solution for the radiation problem, as far as the simple vertical cylinder is concerned, has also been given by Yeung (1981).

In the discretization scheme used in the following a distinction is made between "finite" elements, which cover bounded regions of the fluid, and the "infinite" one, whicla extends horizontally to infinity, (Fig. 1). There are two types of "finite" elements in this idealization depending on whether they extend to the free surface, (type II), or whether they are bounded by the sea bed, (type III). For the "infinite" ring element, (type I), the selected functions for its velocity potential approximation have to satisfy the radiation condition.

. . . 7 r - l r - - - l r T F - I r . . . .

7r-]

I I I I I I | I I

i!

If II '1

~1 /

I ~" II _14, _~ _I.~1 II _I I ] _I ' , '", t 1 • 11~ I I

_,

_,,r

!

'I

""

L r o I - -

.--©d k

I

i

'

Ip

I~ l,

I

n II Ii I t U I I i I I I I I i i' I )I T-- t I I i ~ I I ÷ I o . ii I II 11 • CLi I o. I Ill i i I I /

I

'~

i

il

/

i: , , ,{ m .-- - ~ _ J L _ _ J L _ _ _ il _ _ _ J L _ _ l i F . . . . 7 F ~ l , r 7 F - - - T - l r . . . I , I "1 II II I I I II I I I I I I I f~'~ I U .I I ~ I I i 'I I I L S _ J i ' II I II ~ II _r"

L Y

i

I I ~ r I I I I I I h , I I II I i I1 ~ II I I _{I I} II ~ I I i I ' Z I ! I I I ' I I I II ! ( ( I i I I I II I I I I I I i I ~ I # i I I

,i _ ^ I _

L--J

ii X d I

(4)

5 0 8 K{)NSIANIIN K{JKKtN(}WRA{'II{}S ('1 ~11

For each type of element the Laplace equation is formulated in cylindrical coordinate~ and an eigenfunction expansion for the velocity potential is made. The requirements for continuity of the potential function and its first derivative (hydrodynamic pressure and velocity) at the boundaries of neighbouring ring-shaped macroelements are satisfied using the Galerkin Method.

2. FORMULATION OF THE DIFFRACTION AND RADIATION PROBLEMS We consider a freely floating rigid body of revolution with vertical axis in a regular wave, (Fig. 1). For this type of structure the introduction of a cylindrical coordinate system (r, O, z) is suitable which in this case is chosen with its origin at the sea bed. Viscous effects are neglected and the assumption is made that the fluid is incompressible. The finite water depth d is constant; the free surface is infinite in all directions. The motions of the body and the fluid are assumed to be small, so that the linearized boundary value problems for the diffraction and the radiation are considered.

Under the assumption of a symmetrical mass distribution a vertical body of revolution performs under the action of a regular wave a three-degree of freedom motion in the

wave propagation plane, i.e. two translations (surge xj, heave

x3)

and one rotation

(pitch xs). The first order total velocity potential for steady-state condition can be expressed as follows:

q~(r,O,z,t) =

~,(r,O,z)e- j''' + ~7 (r,O,z)e i<o, + ~

xs,,~j(r,O,z)e"

-,~t

(1)

i 1 3 , 5

where ~oe -i'°' is the velocity potential of the incident harmonic w a v e , q)7 e - i ' t is the diffraction potential for the body fixed in the wave, {pie -j~'', (j= 1,3,5), is the radiation potential resulting from the forced body oscillation in the j-th mode of motion with unit velocity amplitude and ks, > is the complex velocity amplitude of body motion in the j-th direction.

The diffraction problem is unequivocally described by the velocity potential

• o(r,O,z,t)

= tpD e -*'°' = (~,, + q~7)e-"°'. (2)

The velocity potential of the undisturbed incident wave can be expressed using Jacobis's expansion as

• H c o s h ( k z ) l ~

emimj,~(kr) cos(mO)le_i~,

(3)

q~o(r,O,z)e - ~ ' = - t o ~

k s i n ~ ( k d ) L .... {, J

where

Jm

denotes the m-th order Bessel function of first kind and ~-m the Neumann's

symbol:

~o = 1, ~.,=2 (m->l). (4)

Frequency (o and wave number k are related by the dispersion equation

o~ 2 = gktanh

(kd).

(5)

In accordance to equation (3) the total velocity potential of the flow field around the restrained structure can be written in the form

(5)

Behaviour of vertical bodies of revolution in waves 5(19 The fluid flow caused by the forced oscillation of the body in otherwise still water is symmetric with reference to the 0 = 0°-plane and antisymmetric to the 0 = ~r/2-plane for surge, ( j = l ) , and pitch, (/=5), whereas it is symmetric with respect to both these planes for heave, (/=3). Thus, the corresponding velocity potentials for these modes of motion can be expressed as

~1 e-i'~t = tllll(r,z ) cosO e -itot (7)

~p3 e-i°~t = ~!/3o(r,z ) e -i°~t (8)

q)5 e-i~'' = 051 (r,z) cosO e -i~' . (9)

In the functions Ojm of the Equations (6)-(9) the first subscript j=D,1,3,5 denotes the

respective boundary value problem whereas the second one indicates the values of m which must be taken into account within the solution of the corresponding problem. Thus, the functions 0Din, (m=0,1,2 . . . . ), 011, 03o and 051 remain the principal unknowns of the problem.

The complex velocity potentials ~i (/=0,1,3,5,7) have to satisfy

02q)j 1 1 O2q)j O2q:)j

Aq~j = -ff~ + r OtpjOr + r 2 00 2 + ~ = 0 in the entire fluid domain (10)

Oq) j

--O')2q)j +

g~-z = 0 on z = d (11)

0% _ 0 o n z = 0. (12)

0z

Furthermore, the potentials % with j = 1,3,5,7 have to fulfil the radiation condition

lim V~r (Oq)j )

r - - ~ \ Or - iktpj = 0. (13)

Finally, the kinematic conditions on the wetted surface So in the average position of the body have to be satisfied, i.e.

0q)7 SO- _On 0q)O SO _On or 0q) D St) _{O - r} = 0 (14)

and

O~j So On = nj, ( / = 1,3,5). (15)

In the Equations (14) and (15) O/On denotes the derivative in the direction of the

outward unit normal vector n to the surface So of the body and nj is defined as

(nl, n2, n3) = n , (n4, ns, n6) = r × n (16)

where r is the position vector with respect to the origin of the coordinate system. Considering now the particular geometry of the idealized axisymmetric structure, (Fig. 1), the condition for impermeability on the wetted surface of the body as expressed in the Equations (14) and (15) can be separately formulated for the diffraction ( j = D ) and radiation (j=1,3,5) problems as follows:

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51(I

horizontal b o u n d a r i e s

w h e r e

KONSIANflN KOKKINOWRA(tlOS t'l d[,

for l = 1 2 . . . L and P = 0.1 . . . . P ( b o = 0, aL+l

O~jtn

_{Oz = Vj} _{on z = d / f o r}_{a~ <~ r ~ a/ + i}

on z = hn for b n <~ r ~ b p ~ l

V D = V ~ = 0 , V ~ = l a n d V s = - r vertical b o u n d a r i e s

for l = 1,2 . . . L and p = 1,2 . . . P (do = d): o n r = a f o r h p ~ < z ~< d~

O+j,,~ =

Or Uj o n r = at f o r m i n { d/, dl l } <- z <~ m a x { dt, dl_ l }

on r = b e for min { h n, h p - i } <~ z <~ max {hp, h p _ l }

w h e r e = a): (17) (18) (19) UD = U3 = O, U , = I, U s = ( z - e ) . (20)

T h e f o r c e d pitch m o t i o n , ( j = 5 ) , of the structure is c o n s i d e r e d to be p e r f o r m e d a b o u t a h o r i z o n t a l axis lying at an arbitrary distance z = e a b o v e the sea bed.

M o r e o v e r , b o t h the velocity potential and its derivative & p / O r , ( j = D , 1 , 3 , 5 ) , must be c o n t i n u o u s at the vertical b o u n d a r i e s of n e i g h b o u r i n g m a c r o e l e m e n t s , (Fig. 1). This results in .f,,, (a.z) = *},2 ( . , z ) o,~(,, , o,', ~L~ , - Or ~ Or ~ ,, Ol~//m r - , u ~3'h(P) r Or - - O r ... *}2 (.,.~) = *~,;,; '~ (", , .z) OJff(I) "41171 r z ~'°dlqt - , , q j h ( I - 1) , Or ~ ,q Or = ,, *~/; (b,,, z ) = * ~ ; , - ' ) ( b , , , z ) Or - - - O r = % f o r dL ~< z <~ d for 0 <- z <~ h r for m a x (dl,d1-1 } ~ Z <<- d l = 2 , 3 . . . L for m a x { d / , d t _ l } < ~ z < ~ d / = 2 , 3 . . . L for 0 ~ z ~< min { h p . h p _ ~ } p = l , 2 . . . P . (21) (22) (23) (24) (25) (26) (27) (28)

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Behaviour of vertical bodies of revolution in waves 511 In the notation used above the superscripts l and p imply quantities corresponding to the l-th and p-th macroelements of types II and III, respectively, whereas the superscript I is associated with those of the infinite ring element.

Starting with the method of separation of variables for the Laplace differential equation appropriate expressions for the velocity potentials, i.e. for the functions +jm, in each macroelement can be established. These expressions in the form of Fourier series are selected in such a way, that the kinematic boundary condition at the horizontal walls of the idealized body, the linearized condition at the sea surface, the kinematic one on the sea bed and the radiation condition at infinity are fulfilled.

3, DIFFRACTION AND RADIATION VELOCITY POTENTIALS FOR DIFFERENT TYPES OF RING-ELEMENTS

For each type of macroelement the following expressions for the functions +D,,, 0~1, 030 and 051 defined in Equations (6)-(9) are derived.

(a)

Infinite ring-element

Type I (r t> a, 0 ~< z ~< d)

F

Kin(err)

8jqilj (r,z) = g~m (r,z) + ~,~ j , m , , K ~

Z,(z)

(29) where

Jm(ka),

I Zk(z)

(30)

glDm (r,z)

= Jm

(kr)

H~(m(~a~l'-lm (kr) d Z~k(Z)

g~l (t,z) = g~o (r,z) = g~, (r,z)

: 0 (31) 8 o = 81= 8 3 = d, 8 5 = d e (32)

and

Hm

and Km are the m-th order Hankel function of first kind and the modified

Bessel function of second kind, respectively.

Furthermore,

Z~(z)

are orthonormal functions in [0,d] defined as follows:

where

Zk(Z ) = g k

1/2 c o s h

(kz)

(33)

Z~(z)

= Nffl/2 c o s ( a z ) , a : r e a l (34) sinh(2kd)]

N k = ½ 1 +

2kd J

(35)

sin(2~d)] N ~ = ½ 1 + 2ad J ' a = r e a l (36)

and ot are the roots of the transcendental equation

(O 2

- - + a tan (ad) = 0 . (37)

g

Equation (37) has two imaginary and an infinite number of real roots. Here, the

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512 KON>;IANIIN KOKKiNOVvRA('ItOS {'l O[.

o~ = - i k

in Equations (36) and (34), Equations (35) and (33) can be directly obtained. The velocity potential approximation for +~,,, as formulated in Equations (29) and

(39),

coincides with Garrett's (1971) representation.

(b)

l-th ring element o f c p e 11 (al ~ r ~ al,

i, dt ~< z ~ d, l = 1, 2 . . . L)

~i +~)'' (r,z) = g}~ (r,z) + ~ [R,,,,,~ (r) F~

I

+ R,,,~,~ (r) F},,j Z~, t (z)

(38) where 8j is defined by Equation (32).

g~),,, (r,z) = g]'?

(r,z) = 0 (39)

,y

g~_{t (r,z)

= d - 1 + co- g~ (40)

g~Z¢(r'z)= - d r { ( z - d ) + g

(41)

1,,,(alr)K,,,(o#a,) - l,,,(ala,) K,,,(e~,r)

R,,,,,,(r) = l,,,(~xta,+,) K,,,(e~,a,) - L,,(c~,a/)K',(a,a,~,

(42)

l,,,(ala,, ,) K,,,(a,r) - K,,,(ata, ~, ) l,,,(atr)

(43)

R,,,,4r) =*

' f

,i

/,,, is the m-th order modified Bessel function of first kind.

Z,,~(z)

are orthonormal functions in

[z=d-dl, d]

defined by:

Z,, (z) = N g ''2 cosh

[k,(z -

d,)] (44)

Z~, (z) : N,,, ''-~ cos [o~/(z-d,)], a / : real (45)

where

s!nh[2k~(d_d/!] ]

N , , = ~ 1 + 2k,(d-d,) J (46)

{ s'n[2°~'(d7 d')] ]

N~, = ½ 1 + 2~t(d-d/) ,cx/" real (47)

and ~ / a r e the roots of the equation

(.02

+ cxl tan [~,(d-d,)] = 0 (48)

g

with the imaginary one ~1 =

-ikl

considered as the first.

In Equation (38), ~,q) represents particular solutions for the different modes of OJl?l

motion, which satisfy the respective kinematic conditions on the horizontal boundaries of the elements of type II,

[z=d~,

d], i.e. Equations (11) and (17).

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Behaviour of vertical bodies of revolution in waves 513

For this type of ring element the potential function t ~ ) (r,z) is approximated as follows:

oo

(np,rrzl

10(p ) (r,z) = gj(Pm

) (r,z) + ~ e,p [Rm% (r) Fj,m% + R,p (r) Fl.mnp ]

COS (49)

np=O

where ~j is defined with equation (32).

g ~ (r,z) = g~) (r,z) = 0

z 2 _ (1/2)r 2

g~) (r,z) =- 2hpd

-r[z2-(1/4)rq

g~) (r,z) =

2hpd 2

Rm°(r) = {bp+llm ( bp )m

\~p ]

\bp+l,

(,; (r)m

_b_{5 _} _{_}

- - - - , R , * . o ( r ) = ( b p + , ]*

m

~ , f o r n p = 0

\ b~-p /

\bp+ l/

(np~rbp] (np~rr)

(npTrbp) (np'rrr]

Km\ he ]Im \-~-p ] -Im \~h~-p ] Km \ ~-p ]

Rm. p (r)=

[.p'ITbp+ 1

)

(np~rbp]

(l'lpqTbp) ( npTfbp+I] , I'Ip :7 & 0

I., t- ~p

Kink hp ]--Im\ hp ]Kink

hp ]

(50)

(51) (52)

(53)

(54)

(npTrbp+l]

(nprrr]

(np~rbp+ (npTrr]

Ira\ hp ] K m k ~ ] - K m \

hp l)Im\h~-p }

R,p (r) =*

, np 4: O .

(npw_bp+,)

(np~rbp)

(nprrbp] (npTrbp+,

Ira\

hp ]gm\hp-p / - I m \

hp ] g m \

hp )

Especially for p=O the velocity potential is given as follows:

(55)

[no3,,rr\

)

1

/ \

"vjm (F,Z) g~O, (r,z) + Z e.o Fjm~l Im~ h(?

c o s

(56)

' ( n , , l r b , ) \ h o o ]

W

no=o

Im

where g~)) is defined in Equations (50)-(54). The modified Bessel function Km does not appear in the expression for this type of element with p = 0 in which no source or sink is existent. The function K,, has namely a contribution proportional to

(bl/r)"

for m:/:0 and log (r) for m = 0 at r---,0, (Abramowitz and Stegun, 1970). Nevertheless, considering the limiting forms of Bessel functions for

bp----~O

w e note that the velocity

potential described by Equation (56) can be directly obtained from Equations (54) and (55). In this case the following limiting values are obtained:

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5 1 " I K O N S I A N I I N K O K K I N O W R A f H O S ~'f (11. lim R,*,,,,,,

(r) ~ (}

]

b , ~ 0 I I [ tlpTfr~ lim

Rmn p

(r) = i / n p'rr b t, , i\ ~

b,- o

) i

( 5 7

The potential functions t~D,~, expressed through Equations (29), (38), (49) and (56).

have the advantage that they can be described for all the vertical boundaries r = a .

(/=1,2 . . . L), r=bp, ( p = l , 2 . . . P) and r=a by simple Fourier series in the following forms:

and

d Ohm (a,z) = ~, F~ .... Z,~ (z), for 0 ~< z ~< d (58)

c~

~t O~,, (at,z) = ~ F*o .... , Z~, (z), for d, ~< z ~< d (59)

c~ /

1 l

3 O~)m (a/+l ,z) = ~

FD,m~ Z~, t (z),

for dl ~< z <~ d (60)

c~ /

1 -~ (np~rZ 1

O~)m (by,z) = ~ en, t~O,m,p COS , ~ - - - - , , 0 <~ Z <- hp (61)

3

_np=O \ t~p /

(npTrZ]

l O~)m (bp+ l,z) = E ~np Fo,mnp

COS \ h T J ' O~ Z ~ hp . (62) np=O

Moreover, the solutions for the functions

~Jjrn,

(j=D, 1, 3, 5) are selected in such a way

that the conditions at all the horizontal boundaries of the elements are a priori satisfied.

The kinematic conditions at the body's vertical walls, Equation (19), as well as the requirement for continuity of the potential and its first derivative, Equations (21)-(28), at the vertical boundaries of neighbouring elements remain to be fulfilled. These conditions deliver the system of equations for the unknown Fourier coefficients. The numerical procedure is outlined in the following chapter. Once these Fourier coefficients

have been found, the functions qJjm(r,z) and hence the velocity potentials for all fluid

regions can be obtained.

4. N U M E R I C A L P R O C E D U R E

Considering two neighbouring ring macroelements of type II, e.g. the elements I and 1-1, it is useful for the sake of brevity to note with Ix the element which is characterised by the greater distance between the horizontal boundary of the body and the free surface, whilst h denotes the other one. The same notation is introduced for neighbouring ring elements of type III, e.g. p and p - 1 . Thus, the notations da = max {dt,dt_l}, d~, = min {dr, d~_l} and h~ = max {hp,

hp_ 1},

ha = min {hp, hp-I} are used for the ring elements of type II and III, respectively.

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Behaviour of vertical bodies of revolution in waves 515 The condition for continuity of the potential at the boundaries of neighbouring elements is expressed with the Equations (25) and (27) for the elements of type II and III, respectively. Considering the elements (l) and ( l - l ) of type II, both sides of Equation (25) are in a first step multiplied with Z ~ ( z ) / ( d - d x ) , where 13x is an arbitrary root of Equation (48) for the h-th ring element. Then, after integration over the region of validity of the equation, i.e. dx ~< z ~< d, the following set of equations relating the unknown Fourier coefficients of two neighbouring ring elements of type II can be obtained: Fj,,,,~,x = Qj.,~x + ~ L=x.% Fi,m~ ~, at r=at, dx <~ z <~ d where (i) for dt < dr-1: (ii) for d l > dr-l: Furthermore (63) k ~ Z - 1 , Ix-- l, Fj,,,,~x = Fj,tnal_l ,

F].nlctp. =-F~j,mc~ I

h =- l, Ix =-l-1, Fj..,~ - b-~j.,.~,, Fj,m% =-- El,real_ 1 • (64) (65) 1 fd~ g~g) (at,z) - °~')(al,z)) Z~,x (z) dz Qj,m,~ h -- d Z d x ~ ~'J" (66) ( 6 9 ) (70) (71) (72) and 1 I h~ [n~ovz~ n~a'rz 1 f l ~ (n~,rrz~ Q/.m,,a = ~ (g}~m) (bp,z) - g~'Q (bp,z)) cos \ hx-J dz where (i) for hp < h p _ 1: (ii) for hp > h p _ l : Furthermore k =P, IX = P - I, Fj ,,,,,~ ==- I,m,,p, Fj ,,,,,~ -- Fi , _ h -- p - 1, Ix =- p, F/,,,,,~ =- Fj ,,,,,p ~, Fj . . . . mn~ ~ F* J ,mttp • and L~.~,% - d--d~, Z % (z) Z,~ (z) dz . (67) h

The expressions of Qj,,,,.~ and L~x,% obtained from the above integrals are given in the Appendix.

The same procedure can be applied to Equation (27) which expresses the continuity of the velocity potential between two neighbouring ring elements of type III at r=bp, ( p = l , 2 . . . P). Using the weight function 1/h~ • cos (q~rrz/hx) we obtain

Fj,,nnx= Qj.,,,.~+ ~ % Lnx,. Fj.mn~,atr=bp, O<-z<<-hx (68)

(12)

516 KONSIAN fiN KOKKINO'&RA('HOS ~'I all.

L,.,x.~ ~

and

Qi.m~a

are given in the Appendix.

Especially at

r=a,

i.e. at the outer vertical boundary of the structure, Equations (2l} and (23) result, after multiplication with

Z~,(z)/(d-dt.)

and

l / h r .

cos

(nr~rz,"hrk

respectively, and integration over their regions of validity, in

Fj . . . . L = Q J . . . . l-{-

~__~L,.,,~I~

... a t r = a , d c < ~ z < ~ d and (73) Fj ... l , = QI . . . . p +

~ L,,,,~ F~

... at r = a , O <~ z <~ h r (74) where ~, Z~ (z) Z~, (z) dz Lo, o : 3 = < ,,.

L.,,,.- 14p. m Z,~

(z) c o s \

hp )dz

QJ..,,~,c :

d-dt~l f",c (gfm (a,z) - o, mO{L' (a,z)) Z~,. (z) dz

1 ( h'"

(nt~rrz]

QJ .... I" = heoo (gf.,

(a,z) - om,°~e)

(a,z))

cos \ he / dz.

(75)

(76)

(77)

(78)

Now, the conditions for the continuity of the first derivative of the potential as expressed by the Equations (26) and (28) for the macroelements of type II and III, respectively, as well as the kinematic condition on the vertical boundaries of the idealized structure as described by Equation (19) must be fulfilled, too.

Considering the ring element of type II with a boundary at

r=at,

(/=2,3 . . . L), both sides of the Equations (26) and (19) are multiplied with the function

Z~(z)/(d-d.),

in which 13. is a root of Equation (48) for the I*-th ring element. After integration of each equation over the region of its validity, i.e. for dt ~< z ~< d and d . ~< z ~< d, respectively, and after adding of the resulting expressions, the following set of equations is obtained.

d-d~,

O~ h at

r=at,

d~ < z < d (79) where (i) for d l < dl- 1" ORm(~l r=al *

Or r=a~

(8o)

(13)

Ogmoq_ 1 r=al ORm,~t_* 1 r=al

= - - =-- Amal-I Or

~n~, x ~ Amp, t_ 1 at Or , ~*m~,x * = a t - - (81)

The expressions for R,,,~ and R*,, t are given by Equations (42) and (43); (ii) for dt > d l _ 1:

~lmap. ~ Areal_ 1 , "Y'm%. =- A'at_ 1, ~ma x ~ Omat, ~na x =- O*m~,t (82)

Furthermore

a~ (a Og}Xm) " Z ~ ( z ) dz + at fa t Og}Z )

ej

_{" ~ -} _{d - d ~ Jax Or}

I r=al ~ __p~ O r r - r=al Zap(z ) dz

a l ( dx

8j(d~d~) Jd~ Vj Z% (z) dz (83)

where ~j and Uj are defined by Equations (32) and (20), respectively. The values of P;,,,,~ are given in the Appendix.

"'Especially at the vertical boundary r=al, where, according to the body's geometry, only the kinematic condition, i.e. Equation (19), must be fulfilled, the Equation (79) is reduced to

D,,,,~ Fj,m,~ + D*m,~ l Fj.,,,,~i + Pj,m~q = 0, at r = al, dl ~ z ~ d (84)

where Dm~4 and D*m,~ are defined by Equation (80) for l= 1 and _ dZ-dlJdl (d

ag}5)

r=al Z o t l ( Z ) d z al (d

Pj,mc~l 8j(d__d,) jd UJZ~l (z) d z . (85)

For the ring elements of type III the requirements at r=bp, (p= 1,2,...,P), as expressed by Equations (28) and (19), are treated by the same procedure as before, using 1/h~

c o s ( n ~ r z / h , ) as weight functions. An analogous set of equations can be obtained:

h h • F* + Pj,mn~ = K E E'nX (~mnxFj.mnx ~lmnp. Fj,mnp. "-]- "Ymnl x l,mnp. p. nh:O + ~*mnxF~,,,,,,~)L,,~ n~ at r=bp for0 ~< z ~< h . (86) where (i) forhp < hp_l: r:bp * r=bp OR,,,,,._~ • OR m,,._~ (87)

"Ymno =-Amnp_ 1 = bp Or ,'Ymn =-- A~nnp_l = bp Or

OR,,,.. OR*mnp ~=bp

~""x =-- D,,,.p = bp ~ ~=bp' ~*'~ =- D * . p = bp ~ (88)

(14)

5] g KONSIANTIN KOKKINOWRA(HllS (,[ aZ

"'mno#(r)

= --

and R,*,,,,. (r) - (I

(Sg)

(n,,~,b, t

(see also Equations (56) and (57)).

Thus, using Equation (86) for

r=b~,

the value

Am.o*

= 0 has to be taken into account,

(ii) for hp >

hp l:

"~mnp.

=

D,,,.p, .y.~, = D.p,

~mnh =

Am.e_,, ("x = A.p_,.

(90)

Furthermore

htx

Pjmno. = - b P

(hhO-g(l~m)

COS (n~-z) dz

_ bp

l Ujcos ( ~ - )

d z n ~ r z

' h~jo Or ~=.p ~jh~

Jh~

bp Ih. O g} ~m_

)_

COS ( ~ Z ) d z

(91,

+ h~ Jo

Or Ir=ap

For the outer vertical boundary of the structure, i.e. for r = a , Equations (22), (24) and

(19) are multiplied by the function

Z~(z)/d,

integrated over the appropriate limits and

the resulting equations are added to each other. This procedure leads to the following

set of equations:

DmotFjmoAf_ n j m c Jr_ e j m a _' _' _" _:

d-a

_d

_{.... ~ ( A m ~ L F J ' m ~ L + A * m ~ L F ~ i ' m ~ L )} _{L ~ L "~} o~ L h

e t~.~ ° e.,, L,,,,x,

(Amnt, + d n

Fj'mnp + Am.)*

at

r=a, 0 <- z <<- d

(92)

where

Dm,,

= aa K~ (oLa)

Km(aa)

(93)

a

l'/ OgJm r=. Z~(z) dz

(94)

&m =d

or

t'j,m --

a l"

ag}~)

r=.

Z~(z) dz - a 1% Og}Pm

'

r=a

d Sc Or

d Jo

O r

Z=(z) dz

d~j

Uj Z~ (z) dz.

(95)

P

The functions Bj,.,~ and

Pj,m~

are given in the Appendix. Am.L, A*.L and

Am.p,

mmn P a r e

defined by Equations (81) and (87) for I = L + I and p = P + l , respectively,

with

aL+j=bp+l=a.

The total number of the unknown Fourier coefficients is 2(L+P+ 1). This is also the

number of the linear sets given by Equations (63) and (79) for 1=2,3 ... L, (68) and

(15)

(86) for p=1,2 ... P, (73), (74) and (92) for r=a and (84) for r=al. Thus, the Fourier coefficients can be determined. For that the infinite series will be truncated after sz, Sp and So terms, respectively. The numerical aspects of this procedure are discussed in Section 6.

Relations between the Fourier coefficients of the p-th ring element of type III can be obtained in the following matrix form by introducing Equation (68) in Equation (86) and using successively the resulting expressions for r=bl, bz . . . bp:

{F~np}] --= {Mnp}j q- [Snp,np] ] {F.p}jforj = D, 1,3,5 (96) where {F~n },, {Fn }j are both complex vectors, the sp elements of which are the unknown

• P ~ • P

Fourier coefficients of the considered ring element, {Mnp}j is a complex vector and [S ... ]i a complex matrix as given in the Appendix.

15oflowing the same procedure for the elements of type II by introducing Equation (63) into (79) and starting with Equation (84) the following equation is obtained:

j = D,1,3,5 (97)

{F~,,}i = {M~,}] + [S~,,~,]i {F~,}j for l = 1,2, . . . , L

The vector {M,~}j and the matrix [S,~,~]j are given in the Appendix.

It should be mentioned that Equations (96) and (97) are valid for each value of m. For the diffraction problem, (j=D), these equations must be formed for m=0,1,2 . . . Contrary to that, in the radiation problem, Equations (96) and (97) will be formed for m = 0 for the heave motion, (j=3), and for m = l for the surge, ( j = l ) , as well as the pitch motion, (j=5).

Finally, substituting Equations (73) and (74) into Equation (92) and considering the Equations (96) and (97) for p = P and l = L , respectively, we obtain the following system of linear equations

[E~.]j {F~}j = (B}j, j = D,I,3,5 (98)

with [E~.]i and {B}i as given in the Appendix.

After determination of the Fourier coefficients Fj,,.~ of the infinite ring element the Fourier coefficients in the other elements can be obtained using the relationship between the coefficients as given before•

and

5. EXCITING AND HYDRODYNAMIC REACTION FORCES The exciting forces F, and F3 and the moment F5 can be expressed as follows:

G(t) = -- f s j Pnk dS = - it°pe-'°" fs f ~°nk

fsf

= --tO2p ~ e -'t°t ~ em im t~Dm(r,z) cos(mO)nk dS for k = 1,3

- - m = O 0

(99)

(16)

52O with K O N S I , X N r l N K ( ) K K I N f ) W R A ( ' H ( ) S ~l a].

L J

i,f

Mk(t)

= - p(r x nk)dS = -itope ' ~ ' tpD (r × n~) dS I1 I)

= - oj2p

He "°' ~ em i" [ ~ D.~(r,z)cos(mO)(rXnk)as.*

(101)

• ~ t = O J,~O J

Furthermore, the hydrodynamic reaction forces Pll(t)and

FS(t)

and the components

M~(t)

and

M~(t)

of the hydrodynamic reaction moment ~ ( t ) are given by

~(t)=-o~29e -i'~' ~ xh, l (~im(r,z)cos(mO)nkdSfork=l,3

] = 1 , 3 , 5 JSIIJ

~(t) = M~(t) + M~(t)

(102) and with (103) M~(t) = - o ;

pe -i't ~ xj. fs f +j"~ (r,z)cos(m~)(rxn,)dS.

(104) j-- 1 , 3 , 5 0

For the calculation of forces and moments the values of the following integrals are needed: and

fk/= (

f t~jm(r,z)cos(ma~)nkdS

(105) 25;o2

Mkj=fs~I~jm(r,z)cos(mO)(r×nk)dS

(106) ) with ]=D,1,3,5.

Introducing the respective formulations for the potential functions in the above integrals, one obtains from Equations (105) and (106) after the integration over the idealized wetted surface of the body following expressions:

Flj = ~rSj a - N21/2 Fi la (sin(ahL) - s i n ( o ~ h p ) ) Ot

+ ~ [f~t~ + (tx-h)a,~ 1 N_,/2F/.l. sin(a~(dx_d.) )

(107) l= 1 ~ Ol.~ ~

+

j + (Ix-h)bpFj.w~(h~-h~) -

2bp(p.-h) ~ - Fj ~,, sin! . . . . / ! M ~ j = "a% a . . . . Ot '

(17)

Behaviour of vertical bodies of revolution in waves

521 ,=,

'~ c%

.

~-~ cos

(a~(d~,-d~))+dx

sin(oq(d~-d~)) - ~

1 2 2

+ ~ mq,j+ ~(h~-hObp(lX-X)Fj.,o+2bp(~-X) ~, h~E

p = l nfx= 1 rIg'T~ j ' l n t x

[ 2 ~ (

[n~'rrh~,~ h~,sin(n~'nhx]]ll eFu.

x

( - 1)% - c o s , - - '

-

\ h~ )

\ h~ )]JJ

(ao8)

gj{l~l{)~!}

--1N-'/erF

(A

+ a * , ~-* ,r~

* )]}

F3j = 2~

+ Y" 04 ~' t j.,,~,~ ,,~, "~,,.,J-. j.,,.~,~.-',,~., + D,,~,

a I

+1

- - ~ {f~.] ~D(x)e[Fj.ooe{b2p+l(ln(bp+,/bp)-l/2)+~b2p}

p = l

+ F;,,~,, { ~ b~ +, +

+22 (-1~"~ h,:

_{, , ~}

_{[Fj.o.,,(Ao,,p+Aoe)-F)o.}

_{p (Do..}

+D~.,,)]

}

np + 1 Flp "~T-

1 h 2

}

-fl})j - ~bZ Fj.,x~,- 2 ~ (-1)"On~'~-~Ao,,,,Fj.,,.

o

nO= 1

(109)

and

1

+ t~j,,~,, (at+, A~{,, + at-atD~{,~,)]

+

m~] + ~bpbp+~(bp+~-bp) 1

1 + 2 ~ (-1)% h~,

_{[ F ] , l n o ( b p +} _{l A l n . - b p +} _{l - h p D l n p )} np ~ l Fl~g t ~ - r

+ ~l~p (bp+xA;,,, + G - G

Di"~p)]

lb3

_ _ _

}

• n(,=l n27r2 b a ( a l n ° - l ) F j " " .

In these equations the following definitions have been used

l~s} = fs f g}Z (r,z) nk dS

o

(110)

(18)

522 KONSTANTIN KOKKINOWRACIIOS £I ¢/[, and j = 1,3,5 for S=(I,1,2 ... P o r S = 1 , 2 . . . . L

mV, k= f~ f g,~(r,z)(r x nk)dS

(112) 0

the functions g}~ being defined in Section 3.

6. N U M E R I C A L R E S U L T S

Since the macroelement method has been established in its formulation for arbitrary bodies of revolution with vertical axis, (1976), a great number of computations has been carried out the results of which have been compared with other theoretical and experimental data. The cases discussed in the following are mainly selected from the point of view of the verification of the macroelement method.

The ten-year experience with this method developed at the Ocean Engineering Division (LMT) of the Technical University Aachen shows that the macroelement approach provides a very efficient numerical tool for the hydrodynamic analysis of this wide class of structures. Concerning the influence of the number of the Fourier coefficients used in the numerical procedure on accuracy and computing time, Garrett's experiences have been principally confirmed.

In the series expansions of the velocity potential 20 terms were used for the ring elements of type I and II, i.e. s = s/ = 20, (! = 1,2 . . . L), whilst for the element of

type III

sp

= 50, (p = 0,1,2 . . . P), was chosen. An error less than 1% could be

proved by use of these numbers of coefficients. All the computations were carried out on the CYBER 175 of the Technical University Aachen.

In Fig. 2 numerical results from the macroelement analysis of a hemispherical tank are compared with the theoretical and experimental data reported by Garrison, Rao

] ~

_{'If I}

(~ (dlo)=2

d ( ~ (d/o) : 3

I

@

1.0 0.1 0.1 f x =

0.1 g

I I1

to 1o.o o.1

FX9

fz = go2(H/2)

U

1111111 I I[111 IIIII ! IIIill ~1~111t111 1.0 k"~'lO.O

g02(1"1/2 )

FIG. 2. Amplitudes of the horizontal and vertical exciting waves forces on a hemispherical tank ( - - present method; 27 O, + : Experiments by Garrison, Rao, Snider (1970)).

(19)

Behaviour of vertical bodies of revolution in waves 523 and Snider (1970). T h e body has been idealized using five ring elements of type II and the infinite one. T h e numerical results coincide with those given by Garrison, R a o and Snider and agree satisfactorily with the experimental data.

Further comparative investigations have been carried out for the Khazzan-Tank, (Fig. 3), for which the exciting forces and the overturning m o m e n t have been calculated by Chakrabarti and Naftzger (1976), using a sink-source technique. Also in this case the accuracy of the macroelement approach is confirmed.

2.0

1.0

0.5

0.2

0.1

0.05

0.02

0.01

r - - "

: .TsC-

I klJ ~ ~, , , , 0.1 0.2 0.5 1.0 2.0 5.010.0

FIG. 3. Amplitudes of the exciting forces and moment on the Khazzan type storage tank (

method; O, : Chakrabarti (1976)). : present C - F3° 1.0 3- l~ggQ2 H C3 t / / / ~ - / / / s - / / / - / / / / /

~ ~ ( , ~

C a s e dla 1

1.5

2

2.0

0.5(3, 3 3.0

4 1 0.0

0.2! - 0.5 1.0

FiG. 4. Vertical forces on a floating hemisphere (

k o

1.5 20

(20)

3 _ 4 K O ~ , S I , ~ N I ' I ',, K O K K I N O W R A ( t H ~ S ~'I a/.

Figures 4 and 5 show the transfer functions of the vertical and horizontal wave forces

acting on a floating hemisphere for four different ratios water depth to radius (d/a).

The agreement with Garrison's (1974) results for the case d/a=10 is satisfactory. For

this floating hemisphere the added mass and damping coefficients for heave and surge

are plotted in the Figs. 6 and 7 for the cases d/a=l.5 and d/a=lfl against the

dimensionless frequency parameter to2d/g. The results are compared with those given

by Kim (1965) for infinite water depth and Garrison (1974) for d/a=l.5.

The fluid region around this half-submerged sphere has been discretized using twelve ring elements of type Ill and the infinite one. The effect of a smaller n u m b e r of ring elements has also been studied. Thus, by way of example, in the case of seven ring elements all the hydrodynamic parameters with exception of the added mass for heave show discrepancies in the range of t to 1.5%. The added mass for heave shows approximately 5% higher values than those computed with twelve ring elements. The CPU time increases linearly with the number of ring elements.

In Fig. 8 the computed transfer functions for the surge, heave and pitch motions of a disc buoy are plotted and compared with experimental data given by Mercier (1971). The numerical results are identical to those reported by Garrison (1974).

In the Figs. 9, 10, and 11 the transfer functions of the vertical forces on three rotationally symmetrical floats are plotted. These numerical results are in satisfactory agreement with the experimental data given by Mercier (1970).

0.~ 0.4 0.2; k . a _ _ J _ _ i i I 0.5 1.0 15. 2.0

(21)

B e h a v i o u r of vertical bodies of revolution in waves 525 011 m b~ 0 7 5 05 0.25 -- : 011 - : b l l

/ ~ ~---

- _ ._.. , ~ / I / / / o / P / / u 2 a / g & " - " o's ;o ,5 20

Fro. 6. Dimension_less added m a s s and d a m p i n g coefficients for surge for a floating h e m i s p h e r e (t~, = a l / ( p V ) , b . = b u / ( t o p ~ ) ; - - : present m e t h o d ; • : Kim (1964); © : Garrison (1974)).

o10

b33 : 033 0.75 - - - : b33 0.50

"'"\~,~

T----.

0.25 /

~'-.,"-~).

,,'

/>

--~:--...

W2OI~ i I L L ..- 0.5 1.0 1.5 2.0

FIG. 7. D i m e n s i o n l e s s a d d e d m a s s and d a m p i n g coefficients for heave for a floating h e m i s p h e r e (fi33 =

(22)

526 KONSIANIIN KOKKINOWRACHOS eta/. 2.0' 1.5 1.0 0.5 0.5 1.0 (~ (rad.sec -~)

FIG. 8. Responses in heave (x3o), surge (xHj) and pitch (x~.) of a disk buoy in deep water ( method; O, • : experiments, Mercier (1971)).

: present 1.0 0.8 0.6 O.t+ 0.2 F~a , 11 -oo',",=

j=,.t o,.

0.2 O~ 0fi 0.8

FIG. 9. Amplitude of the vertical exciting force on an axisymmetrical float in deep water ( method; (3 : experiments, Mercier (1970)).

: present

Finally, in the Figs. 12 to 18, the results o f the h y d r o m e c h a n i c analysis o f two vertical floating c o m p o s i t e cylinders are given. Besides the transfer functions and the p h a s e shift angles o f the horizontal and vertical wave forces and o f the pitch m o m e n t , the a d d e d mass a n d d a m p i n g coefficients for surge, h e a v e a n d pitch are p l o t t e d as well as the transfer functions f o r the h e a v e m o t i o n .

(23)

Behaviour of vertical bodies of revolution in waves 527 0.75 0.5 0.25

~ 2 ~

0:25

o15

w2a/g 0.75

FIG. 10. Amplitude of the vertical exciting force on an axisymmetrical float in deep water ( method; O : experiments, Mercier (1970)).

: present 0.8 0.6 O.Z, 0.2

T ~ - - + + t-

I I \ i / I 'I''

l/

0.2 0./,

,

O.G 0.8

FIG. 11. Amplitude of the vertical exciting force on an axisymmetrical float in deep water ( : present method; O : experiments, Mercier (1970)).

(24)

~ KOINslANIIN K.OKKIN(~'~R,\CItOS C( d/. ' C I - cl I 4°1 ~ ~ h 1.50 d/a~--6; afla2= 2 C a s e !h/a 1 hl/h 2 ~oo

f < C f \ \

~

0.50 0 k.Q 1 1.0 2.0 30

Fro. 12. Floating composite cylinder. Amplitude and phase shift angle of the horizontal exciting wave force

•3o

c3: ~:gga~(H/2} c3: - - £F3 ... /.[

L¢

-

075 r ![[i

i

air

;

/iJI ~

[

o.,oV,,,/

o

,i//-X \,

1.0 2 0 30

(25)

3oT

2.0 1.0 0.75 ~ o cs = ~gga~(H/2) cs: EFs: k a 1 I I 1.0 2.0 3.0 _.g

FIG. 14, Floating composite cylinder. Amplitude and phase shift angle of the exciting pitch m o m e n t about a horizontal axis lying at a distance z = h from sea bed (see Fig. 12).

Qll Qll :

bll b11: . . .

0.50

0.25

J I I

1.0 2.0

k Q I

I =

3.0

FIG. 15. Floating composite cylinder. Dimensionless added mass and damping coefficients for surge (a,, = a,,/(p~7), b,, = b,,/(copV)).

(26)

530 KONS'IANT1N KOKKINOWRACHOS 6l tll.

I °3%

o3%:

b a 3 x l 0 b33: . . . 15 I I x \ / \x I \ ! I \ I \ 1.0 ! \\ I \ I "\

l

/ \\ I \\

[

',

O.St I

/"'"<(~

",

o . , , 0 10 2.0 k.o~ I 3.0

FI6. 16. Floating composite cylinder. Dimensionless added mass and damping coefficients for heave

2.0 1.0 055 055: bss bs5: ... i I "'xx l I 1.0 2.0 k.o 1 3.0

FIG. 17. Floating composite cylinder. Dimensionless added mass and damping coefficients for pitch. The pitch forced motion is carried out about a horizontal axis lying at a distance z = h from sea bed

(27)

Behaviour of vertical bodies of revolution in waves 531 X3o

(HI21

1,~

1.0

0.5

1.0 2.0 3.0

FIG. 18. Floating composite cylinder. Transfer function of the body's response in heave.

R E F E R E N C E S

ABRAMOWlTZ, M. and STEGUN, I.A. 1970. Handbook of Mathematical Functions, 9th edn. Dover Publications,

New York.

ASORAKOS, S. 1981. Ein potentialtheoretisches Verfahren zur Erfassung der Wechselwirkung zwischen Elementarwelle, starrem K6rper und elastischem por6sem Boden. Dr.-Ing. Thesis, Technical University Aachen.

BAI, K.J. and YEUNG, R. 1974. Numerical solutions of free surface flow problems, O.N.R. Symposium MIT. BLACK, J.L. 1975. Wave forces on vertical axisymmetric bodies. J. Fluid Mech. 67, Part II, 369-376.

CALISAL, S.M. and SABUNCU, T. 1984. Ocean Engng I I , 525-542.

CHAKRABART1, S.K. and NAFrZGER, R.A. 1976. Wave interaction with a submerged open-bottom structure, OTC 2534, Houston.

CHEN, H.S. and MEI, C.C. 1974. Oscillations and wave forces in a man made harbor in the open sea. Tenth Symp. of Naval Hydrodyn., Boston.

FALTINSEN, O.M. and MICHELSEN, F. 1974. Motions of large structures in waves at zero Froude number.

Proc. Int. Symposium of the Dynamics of Marine Vehicles and Structure in Waves, University College,

London.

FENTON, J.D. 1978. Wave forces on vertical bodies of revolution. J. Fluid Mech., 85, Part II, 241-255.

GARRETT, C.J.R. 1971. Wave forces on a circular dock. J. Fluid Mech. 46, Part I, 129-139.

GARRISON, C.J. 1974. Hydrodynamics of Large Objects in the Sea. Part I, No. 1; Part lI, No. 2. J.

Hydrodynamics 8, 5-12.

GARRISON, C.J. and CHow, P.Y. 1972. Wave forces on submerged bodies. J. Watways Harbors Coastal Engng Div. Proc. Am. Soc. cir. Engrs 98, No. WW3, 375-392.

GARRISON, C.J. and SEETHARAMA RAO V. 1971. Interaction of waves with submerged objects. J. Watways Harbors Coastal Engng Div. Proc. Am. Soc. civ. Engrs 97, No. WW2, 259-277.

GARRISON, C.J., RAO, V.S. and SNIDER, R.H. 1970. Wave interaction with large submerged objects, OTC 1278, Houston.

HAVELOCK, T. 1955. Waves due to a floating sphere making periodic heaving oscillations. Proc. R. Soc. 231,

(28)

532 KONSTANTIN KOKKINOWRACHOS et al.

JOHN, F. 1950. On the motion of floating bodies. Part I and II. Communications of pure and applied mathematics. Part I, Vol. 2, Part II, Vol. 3

KIM, W.D. 1965. On the harmonic oscillations of a rigid body on a free surface. J. Fluid Mech. 21, Part 3,

427-451.

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A P P E N D I X

Values o f the coefficients." L ~ . % , L,~., , L,~..~ and L ~ ~ (definition e q u a t i o n s are (67). (72).

(76) a n d (75), r e s p e c t i v e l y ) " N~,I/2 N~, 1/2 L % . % = (d_dx)(c(2 ~_~2) [c(a s i n [ ~ a ( d - d a ) ] c o s [ c % ( d - d ~ ) ] - e% c o s [ ~ x ( d - d a ) ] s i n [ e ( ~ ( d - d ~ ) ] + ~ s i n [ e % ( d x - d ~ ) ] ] for ~ : ~ { x ~ N-J/~N 1/2 1 L,~,.% - 2(3-~d~i- [ ( d - d , ) c o s [ ~ , ( d , - d , ) ] + s i n [ ~ a ( 2 d - d x

~d~]

1 - 2c~ s i n [ o ~ ( d ~ - d ~ ) ] ] for c ~ = ~

(A.I)

( A . 2 )

(29)

Behaviour of vertical bodies of revolution in waves Ln~..~ = 1/2 for n.~r/h~ = n~r/hx 4:0

L . s , % = 1 f o r n . = n x = 0

• 1 "~ n~'rrh~h~ . [n~rrh~ for all other cases

L . ~ . = ( - ) n2~r2~ ~ - ~-rr2h2 sin t h T )

L.p.,~ = NgV2/2 for a = neTr/hp 4:0

L"e~' = (__l)np N~_I/2 _• _{a 2 h ; T nZ,rr:~ sln(othe) for a 4:}s h e . new~he

N • l / 2

N-l~2 ~L [asin(ad)cos[aL(d-dL)] - otsin(etdL) L~L,,, = ( d _ d t ) ( a 2 _ a 2 ) - C~L c o s ( a d ) sin[aL(d--dt)]] for c~ 4: a L N~- 1/2 N - 1/2 1 ~L [(d-dz.) cos(adD + sin[a(2d-dD] L~'L'C'-- 2 ( d - d L ) 1 sin(adL)] for a = a L .

Values o f Qj,m~, Qj....~, Qj,,.np, and Qj,,.% (see Equations (66), (71), (77) and (78),

respectively)

F o r the diffraction p r o b l e m it holds:

OD,ma~. = QD.mn~. = QD,mne = QDm~,t. = 0 while for the radiation p r o b l e m s

QI.,,~. = Q l , l n x = Q l , l n p = Q l , l a L = o Q3,o% = 0 1 Ng 1/2 Q3'°~L -- dLd a2L Q3,0nx : ( _ 1 ) . 1 h 2 ( 1

1(;

3 for n× 4:0 for nx = 0 Q3,0np : h p

-(-1)°,,~

h p a 2 - 6~ + 4 ~ e for np 4 : 0 for np = 0 533 (A.3) ( A . 4 ) ( A . 5 ) ( A . 6 ) (A.7) (A.8) (A.9) (A.IO) (A.11) (A.12) (A.13) (A.14) (A.15)

(30)

534 K()NS'I'AN'I'IN KOKKINOWRACHOS el al. Qs.l~. = 0 _ a N,~],': ( A. 16) (A. 17 Q5,1nh dZn2~r- ~ ~-~ - for n~ ¢ 0 - ~ b P ( h2\g - ~ ) ( l ~ - l x ) f o r n x = 0 (A.18) QS,lnp ( - 1 ) " e a he n2p,tr2d 2 for n p --/= 0 ah e a 3

6d 2 8hed 2fornP = 0 (A.19)

Values of Pj,m%, Pj,.,~,, Pj.,~% and Pj.m~ (see definition equations (83), (85), (91) and (95),

respectively)

For the diffraction problem it holds:

e D , m % = PD.m~,, = eD,mn~. = PD . . . . = 0 (m.20)

while for the radiation problems:

N ~ I /2 P , j % - a ( a ~ d . ) sin[%(d~-dx)] (A.21) N~-I 1/2 Pm~l - d ( d - d l ) sin[cq(d~-d)] (A.22) P l , l n ~ = & s i n ( ~ h~) for n . : # 0 - d ~ for n~ = 0 (A.23)

P1 la _' = - N ~ 1/2 a 1 _{d daa [sin(adL) -- sin(cthe)]}

P 3 , o % = P3,o, h = 0 b e 1 . n~.'trh~ 2h~d n ~ sm ( ~ - ) P3.on v- = 0 for n . ~ 0 for n . = 0 (A.24) (A.25) (A.26) N £ 1/2 a 2

e3.o~ -- 2hea d 2 sin(a hp)

(A.27)

_ 1 %, /

e s . , . (d--d.) d2a~ [ 2cos[%(d~-d.)] + % + 2 d h - d - e sin[%(d~-d.)]-2

(31)

Behaviour of vertical bodies of revolution in waves 535 P S , l . , = -

(d_dl)

d 2 ~ 2 2 c o s [ ~ - l ( d - d l ) ] "~ t211 --t- d - e

sin[~l(d-dl)]-2 (A.29)

- - / l - - / 2 b p h "

[

/n~Irhx\

.

{3hxn.~r

1 h.

~2n2d2[C°S\ h. ] - ( - 1 ) ~ + \4 h.

2hx~rn.

3 b~n~r

enter I . {n.~rh~]

Ps,x.~ = 16

hxh~ - 2h-~]

s m ~ - ~ - } ] for n~ 4:0 (A.30)

6ff~P~(ha-h.)(4hx+4h~-6e)

for n~ = 0

Ps.I, = - ~" (a d) 2 2 cos(~dL) - 2 cos(~he) +

2adL-eLd

+ ~ - ae sin(~dL)

+

~ h--p + ae - ~ ~he + ~

sin(ahe) . (A.31)

Values of

Bj.,,, (see Equation (94)) 2i

Bo.,~. = -

gk.~

(A.32)

~rdH,.(ka)Z'k(d)

where gk,~ is the Kronecker symbol. For the radiation problems according to Equation (94) and

the expressions for g~,.

(r,z),

according to Equation (31), one can obtain

Bl.la = B3,o~ = ns,la = 0. (A.33)

Matrices

[S~...p]j

and

[S.~.~]j (see Equations (96) and (97)) For the/-th ring element of type II it holds:

(i) for

dt<dl-1, l>~2

[S~,,~,]j = [G~,.j -~ ['D,..r] (A.34) where

d-dH[L""~'-'][ ['Am~H]

. . . . ] - " " (A.35) (a) [S.,,.,l

d-d,

+ ['A*., ,][S., , . , ,lj [L., , . , ] - [ ' O ~ . , l is a (st x st) square matrix.

(b) [L.,.~_~] is a (stxsl-~) real matrix with elements given by Equations (A.1) and (A.2). (c) [L.,_~,j is its transpose matrix

• " * " " * r

(d) [

Drear], [ Drear], [ Amat_r], [ Ama,_r]

a e diagonal matrices, the first two being (st x st) whereas the last two (st-~ × st_~) matrices. Their elements are given by the Equations (80) and (81).

Especially for

l=l,

from Equation (84) the following is obtained:

[ S a l , a , ] j = - [

"Omal,1-1 [ "Dmal, ]*

(A.36)

(ii) for dt > dr-l, 1 I> 2

d-dr

(32)

536 where

KONSTANTIN KOKKINOWRACHOS

el al.

[G~, ,.~,_,] = ['Am~, ,.] + ['A*~, ,] [S~, ,.~, ,]j

d-dr

"D*

(A.38)

d-dr-1

[L~, . . . . ,][ ,,,,~,.] [L,,.,~, ,]

is a (st-l x &-l) square matrix. All other matrices, which appear in Equation (A.37), are defined above. F o r / = 1 the Equation (A.36) remains valid.

Furthermore, for the p-th ring element of type III, it holds: (i) f o r h p < h

e _

1 , P = 1 , 2 . . . P

hp

[S%,%]j

= ~ [t..,%_,] ['e% ,] [G% ,,.. ,]-1

[Lnp .... p] ['~-np] ['Dmnp.]

(A.39) where

",4*

(a)

[G%_,,%

,] = ['Am.. ,.] + [

mn 1.] [Snp_ i.np_l]j

he

"D * (A.40)

hp-l [L% ,,%] ['~%,] [

m%.][L%.%_,]['~%_,] is a

(sp_j x Sp-l)

square matrix.

(b) [S.,,..,,] = [0], ['A*.o.] = ['0 ]. (A.4I)

(c) The elements of the

(sp × ~p-l)

real matrix

[L.~,.%

,] are defined by the Equations (A.3), (A.4) and (A.5).

(d) [L% ,.%] is the transpose matrix of

[L%,%_,].

"O*

(e) The elements of the

(sp × Sp)

and

(Sp-i x Sp-O

diagonal matrices

['D,,%,.], [ ,.%,]

and

['Am%_, ], ['A*%_,], respectively, are defined by the Equations (87) and (88).

(f) ['%,,land ['~.. ,] are

(sP>_2SP)

and (sp-i × sp_~) diagonal matrices, respectively, with

e,1=1 and

~kk

=2 2 for (ii) for

hp>hp_l,p

= 1,2 . . . P where G -1

[s%...b

= [ %.%]

['Din.,]

is a (Sp × defined above.

Matrices

(M%}j

and

{M~I} j (see Equations (96) and (97)) For the/-th ring element of type II it holds:

(i) for dt < dl-i, 1~>2

{M,~t} j

= - [G~,,,~,,]-' {N~,}i

(A.42)

(A.44)

[a.~.%] = ~ - [L°,,,°~ ,l[ "~_, ] ['Am.,, ,] + ['Am.~_,] IS% .... p ,b

[L.. ,..~1['~. ] - ['O*.. ] (A.43)

(33)

Behaviour of vertical bodies of revolution in waves 537 where

(a) [G~t,~t] is given by Equation (A.35).

d-dr_ 1

_{L~ ~}

_{[[['a,,,,~,_,]}

I

(b) {N~,}j = ~

[ ,, ,-1] •

[ "A *,,,.,_ ,.] [S,~, v':', ,]./] {Q':', ,}/

+ [ ram_v] {M,,_,}j - {P~,}j. (A.45)

(c) [S~ . . . . ,_,]j is defined by the Equations (A.34) and (A.36).

(d) The elements Qj m, of the vector

{Q,,-1}J

with st-~ elements are given by the Equations

(A.10) to (A.19i

d~p~ending on the problem considered (diffraction or radiation). (e) {P~,}~ is a vector with st elements given by Equations (A.20) to (A.31).

Especially for l = 1

- "D* ] - 1

{M~1}j = [ m~t {P~,I}j. (A.46)

(ii) for d l > dr-l, l~>2

{M~,}j = {Q~,}j + [L~,,~,~I] [G~, . . . . ,_,]-i

{N,~,

,}j (A.47)

where

(N,~,_,}j

=

-

{P~,_,}./- ['A*m,~,~,] {M,:,,_,} + ~

[L,,,,_ , .,:,,]

['Dm~,.]*

{Q~,}j (A.48)

is a vector with st-1 elements. The definition equation for {M,,}j, Equation (A.46), remains valid. The matrix [G~ 1~ ] is given by Equation (A 38).

• I - - ~ I - - I . , " ,

For the p-th ring element of type III we also &stmgulsh two cases: (i) for

hp<hp-l,p

= 1,2 . . . P

{M,,,,}j

=

{Qnp}j

+ [Lnp%_l][~np_,.]

[G,,e_~.%_1] -1 {U,,p_l}./

(A.49)

where

(a) [G%_1,%_1] is given by Equation (A.40).

(b) {Q..}j is a vector, the sp elements Qj,.,% of which are given by the Equations (A.10) to (A.19).

Behaviour of vertical bodies of revolution in waves