An approach to computing the second order steady forces on semi-submerged structures

TECHN1SCHE UNIVER

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IEMI FINLAND

ScheepshydromeChaflloa

Report No 16

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AN APPROACH TO COMPUTING THE SECOND ORDER STEADY FORCES ON SENISUSMERGED STRUCTURES

Tuomo Karppinen

HELSINKI UNIVERSITY OF TECHNOLOGY SHIP HYDRODYNAMICS LABORATORY

OTANIEMI FIÑLAND Report No. 16

AN APPROACH TO COMPUTING THE SECOND ORDER STEADY FORCES ON SEMISUBMERGED STRUCTURES

Tuomo Karppinen

Thesis for the degree of Doctor of Technology approved after public examination and criticism in the Auditorium Ko 216 at the Helsinki University of echno1og on the 16th of March, 1979, at 12 o'clock noon.

Pagé 8 expresSion of R2 X' instead of x'

Page. 66 next to the last röw

generalized instead of genralized Page 127 eq. (A57) after ff and f! is missing

SF SR

ISSN 0356-1313 ISBN 951-751-568-5 TKK OFFSET 1979

During the last decade naval architects have begun to get seriously interested in a new field, closely related to the expansion of the oil industry- into the ocean-: offshore engineering. Above all the methods developed for handling the problems of seakeeping of ships have found wide applica-tion in offshore engineering and evolved further to face new challenges. .1 was guided to this novel, rapidly expanding field in the beginning of the seventies by General Director J-E.. Jansson, at that time professor of naval architecture at Hélsinki University of Technology. Since my Master's thesis had concerned wave-induced motions of ships, a logi-cal continuation was to tackle similar problems in floating offshore structures.

lam much -indebted to General -Director Jansson for his warm support and encouragement during every phase of my studies.. I am also very grateful to Professor V. Kostilainen, head- of the.Ship Hydrodynamics- Laboratory, for his support and advi-ce in the course of this work.

I likewise wish to express my gratitude to Professors H. Rik-konen. and J. Sukselainen who critically reviewed the manu-script and suggested important improvements.

-ManS, thanks. go to the staff of the Ship and Water Laboratories in Otaniem.i for their help and interesting discussions.

- Particularly helpful in discussing a broad range of topics -has been Mr. P. Hervala. His advice considerably shortened

the time needed to bring the present study 1t0 a close. My thanks are also due to MIss E. Heap who painstakingly correted the English text of tlie manuscript and to Mrs. I. Halenius for typing the manuscript with her customary skill. The financIal support -of the Academy of Finland has made the present study possible.

An approach is made to determining theoretically the second order steady forces in sinusoidal waves on freely-floating, rigid multi-body structures, such as semisubmersibles,

composed of deeply-submerged, slender cylindrical and small concentrated members and of vertical, surface-piercing

cylindrical members. The fluid is assumed incompressible, inviscid and the flow irrotational and acyclic. The hydro-dynamic interaction between the member bodies is neglected.

General expressions of the secönd order steady forces are derived in finite-depth fluid ïn terms of the Kotchin func-tion. An approximative method is presented for evaluating the }(otchin function for bodies of the aforementioned type.

In determining the wave-induced oscillatory motion of thè, structure the exciting forces have been computed from the given unit Kotchin functions by the Haskind's relation. Computed results concern a vertical surface-pièrcing circular cylinder, a

and a catamaran-type firmation is shown. been made with other computational scheme

the appliabilit must be properly evaluated with model experiments and further theoretical correlations.

submerged horizontal circular cylinder semisubjuersible. No experimental con--However, in some cases comparisons have theoretical predictions. The present seems promising, b-ut the accuracy and

Expressions of the Steady Forces and Moments in Terms of the Kotchin Function in Fluid of

REFRENCES 106

3.

.

Uniform Depth .

Comparison with the Deep-Water Case

APPDXIMATE METHOD FOR EVALUATING THE KOTCHIN FUNCTION OF A SPACE-FRArIE-TYPE SEMISUBMERGED STRUCTURE

UNIT KOTCHIN FUNCTIONS FOR MEMBER BODIES OF THE STRUCTURE

26

-37

41

49

Slender Cylindrical Body 49

Concentrated Body

Comparison with Kotchin Funct ion Derivéd from

58

Singularity Distribution 62

5. WAVE-INDUCED OSCILLATORY MOTIONS 66

Equations of Ñtion 66

Damping and Added Ñass Coefficients - 67

First Order Exciting Forces and Moments 70 6. APPLICATIONS WITH NUMERICAL RESULTS 73

Vertical Surface-Piercing Cylinder 73

Horizontal Submerged Cylinder 82

Catamaran-type Semisubmersible . 99

7. ISCUSSION, AND CONCLUSIONS O2

CONTENTS

SUMMARY - -.3

NOMENCLATURE 7

1. INTRODUCTION 13

Review o Earlier Study 14

Present Paper 20

2. SECOND ORDER STEADY FORCES AND MOMENTS 22

TERMS OF THE VELOCITY POTENTIAL 115

Appendix III SECOND ORDER FORCES AND MOMENTS IN

TERMS OF THE KOTCHIN FUNCTION 119

NOMENCLATURE

A Incident wave amplitude

Ajk r Added mass coefficients :(j,k r 1,2...6)

B r Typical transverse dimension of body

3jk Damping coefficients

C r Closed contour -

-Cjk r Hydrostatic restoring coefficients

D r Distance

F. r Hydrodynamic foróe and mo,ment components -(j 1,2...6) -:

<F.> rSecond order steady .fòrce and moment components Incident wave depeident part of

r Part of. <F.> entirely dependent on body potential

r First order exciting force and moment ainp1itud

G r Green1s function

GMT, GML Trànsverse and longitudinal metacentric height = Kotòhin function, defined by euation (2.25)

d(k,)

r Deep-water Kotchin function, défined by

equation (2.2L)

-J

H!' J

r Unit Kotchin functions of ith member body

(j = 1,2...6), defined by equation (3.10)

r Diffraction Kotchin function

= Unit Kotchin functions

(j

= -L,5,6),

defined

by equation (3.12)

-r Mass moments and p-roducts of ine-rtia

('j,k -= 1,2,3)

r Integrals defined by equatlöns (6.1) r Function defined by equation (6.2)

-L Body length

L(a)

11cos a + l2.sin a (j = 1,2,3)

M Body mass

M. r Components of linear andangular momentum

j

(j 1,2...6)

Mik. r Generalized mass matrix (j,k 1,2.. .6)

r Outward two-dimensional unit, normal (N1 ,N2)

r Position vector

-Pòsition vector of center of 'mass (O,O,ZG)

r Cylindrical polar coordinates with

Xr

Rcos O and Y r R sin O, see Fig. i

r

Cylindrical polar coordinates with

X-X'

R'cos O'

and y-ye

r R'sin 8'

R2 r /(X - x')2 + (y - Y')2 + (Z + 2h + Z')2

S Surface

SB r Average wetted surface of body

SF Plane of meai free surface Z O SR Fluid bottoth Z r

-h

r

Vertical circular cylinder of large radius

r

Draft

r

Body velocity on SB

Tran'lational velocity of body t Norrnal velocity bf control surface

r Fluid velocity vector (V1,V2,V3)

r Volume

r

Normal velocity of fluid

X,Y,Z

r

(X1,X2,X3) r Coordinaté system fixed to mean f'ee surface', see Fig. i

X',Y',Z' Coordinates of point on body surface

i i i

..

iii

X ,Y ,Z0 Coordinates of origin of ( ,n ,r )- and O O

(x,y,z)-systems

X1,Y1 Coordinates of center of waterplane of ith member body

Z r Vertical coordinate of body center of mass

a Cylinder radius; also, major axis of oblate spheroid

a r Waterplane area of ith member body

a.k

r

Added massesof member body (j,k

r 1,2..

.6) b

r

Non-dimensional heave damping coefficient

c Minor axis of oblate spheroid

r

Cross-section contour

d

r

Depth of submergence e0 = Eccentricity

r

Functions defined by equations (6.10)

g = Gravitational acceleration

k ,k ,k

_{xy z}

1.

jk

= Fluid depth

also, as superscript, denotes ith member body

= Subscripts

r

Integration variables Radius of gyration

r

Direction cosines (j,k r 1,2,3), defined by equation ('4.1); for j,k = '4,5,6 defined by equation (14.5)

r

Unit normal vector out of fluid (n1,n2,n3) Components of generalized normal

(j

i,2...6)

(n,n,n)

r Components of unit normal in (x,y,z)-frame

h

i

p Pressure

r,,x

Cylindrical polar coordinates with y r s,nß - and z r rcos, see Fig. i

r,y,O

Spherical polar coordinates with x r r0sinycosU,

y z r0sinsino and z

rcosy

s Cross-section area

t

rTime

u r Aipli-t-ude of normal velocity on control surfàce x,y,z

r

Coordinate system fixed to member body, see

Fig. 1

y(x,y) r Moment distribution of vertical dipoles

= 2(e. -

À

-

e2 arcsin

Complex amplitudes of body motions (j rl,2...6 refer to surge, sway, heave, roll, pitch and yaw, respectively)

A r Wave length

u = Diredtion of propagation of incident waves (i = O for propagation along positive X - axis)

(2/)

_r

< tanh(Kh)

= Coordinate system fixed to ith member body, see Fig. I

p r Mass density of fluid

r Total velocity potential

r

Complex amplitude of total velocity potential = Còmplex amplitude of. body potential

r Complex amplitude of incident wave potential Speed independent part of potential for unit motion in jth direction (j i,2...6)

= Unit potentials of ith member body (j rL1,5,6)

r Complex amplitude of diffraction potential = Unit potentials of member body :(j l,2...6)

Circular frequency r Integration variable

i - INTRODUC.TON

A body ;floating on the free surface of water or hovering just undernéath ihe free surface -in the presence of waves

experiences both unsteady and steady hydrod'namic forces and moments. The force and moment osiÏlating with the encounter frequency of waves. excite the comidon oscillatory motions of .a fre-e body ïn six degree,s of freedom. As s

weI1-knorrn these first -order exciting forces are linearly proportional to the incident wave height. The hîdrodynamic forces with non-zero mean on the contrary are proportional to the square of wave height and -their second order magnitude is much smaller than the magnimagnitude of the first order

-forces. .

-In spite of their smallness the econd -order forces may in certain cases have practical importance-.- Por a freely floating surface vessel the steady forces can cause large excursions in the horizontal, plane -- the drift mot-ions

-due to t-he lack of horizontal static -restoring force and moment. Bodies with nil or -small waterplane area, such as

submerged object-s or semisubmezged latforms, -may -äometime experience in the vertical plane laige translational or rotational displacements in response to the. act3-on of the

second, order vertical force or monent.

In irregular seas the second order force and moment are composed of two parts: one steady and one sloly varying. -The- lätter is not present in regular waves-. The period of the slowly varying part may- easily coincide with a long natural period of a moored object -in the horizontal plane or -with a vertical pl-ana natural period of a vessel with négligible waterplané area. In such cas-es the small -second order force or moment induces large anplitude synchronous - oscillations.

force on a stationary vessel is closely related to the added-resistance in waves of an advancing ship. The longitudinal drift force is the added-resistance at zero speed.

In recent years the second order hydrodynamic force and moment have gained rapidly growing attention as a result of offshore oil operations and other actiiities related to the exploration and exploitation of the résourcesof the sea. In such activities it is usual for vessels to have to maintain a precise location on the ocean for long periods and the efficiendy of the operation is dictated by the requirement of small wave-induced motions o the platform.

Review of Earlier Study

Among the first to study the steady action of waves was Havelock. In a paper from the year 1940 Havelock1 examined the diffraction of regular waves by a restrained vertical-sided obstacle and determined the non-zero mean pressure force on the obstacle due to the wave motion. He cóncluded that the diffraction effects can account for only -a small fraction of the observed added-resistance of a ship in head waves. Havelock stressed that the wave-induced oscillations of the vessel are essential factors in determining the --added-resistance, and suggested, that interactions between purely periodic first order effects through phase differ-ences may give ri-se to steady forces. Two years later Havelock (1942) derived an approximate formula for the longitudinal drift force on a heaving and pitching ship in regular head waves. HavelockTs formula expresses the drift force in terms of the heave exciting force, pitch exciting moment and the heave and pitch amplitudes and phase lags.

The formula was derived by integrating the longitudinal

i

References-are listed in alphabetióal order at the end of the paper.

pressure ôomponent of the undisturbed incidént waves over

the oscillatin.g ship's hull, i.e. the well-known

Froude-Krylov hypothesis was used.

A few years earlier Watànabe

(1938) had deduced by the same method an expression for the

lateral drift force acting on a ship while rolling among

beam waves.

Watanabe's analysis was motivated by Suyehiro's

(192L) experiments, where he had measured the steady

hydro-dynamic force experienced by a ship model in beam waves.

Iotchin (1952)2 considered the radiation problem and derived

the steady forces, when a body was undergoing forced

oscillations under the otherwise calm free surface of an

ideal incompressible fluid.

Eis final results were both

geneal and simple due to the use of certain integrals over

the body surface, which since then have been called Kotchin

functions.

The relation between the Kotchin fu±ictïon and

the potential due to the disturbance associated with the

body resembles the relation bètween a function and its

Fourier transformation.

Maruo's (1960) análysis by momentum and energy considerations

concerning the drift forces in regular waves included both

the diffraction and radiation effects.

Marua provided

general expressions for the drift forces on an arbitrary

body in the two-dimensional case in terms of the reflected

wave amplitude and in the three-dimensional case in terms

of the Kotchin function.

If the incident wave amplitude is

set at zero Meruo's final results are in agreement with

Kotchin's.

Maruo also showed that one part of his

longitii-dinal drift force formula coincides- wit-h ±he approxinatè

formula given by Havelock (19L42).

Ogawa (1966) and Kim &

Chou (1970) applied Maruo's two-dimensional theory fbr a

ship in beam and oblique waves by -the

trip method.

Their

comparisons of the theoretical predictions with experimental

data shock good agreements.

Ogawa calculated the a.mplitudès.

of the reflected waves in the strip domain by the method of multipole expansions and Kim & Chou solved the two-dimensional radiation and diffraction problems by the source technique.

Newman (1967) re-derived Maruo's results by a different method and included in his analysis the drift moment about. the vertical axis. Newman presnted specific formulas and computed results for a slender ship and compared th theo-retical data with model experiments. Hu and Eng had already in 1966 considered a thin ship with small draft in long waves and given, without detailed derivation, a formula for the drift yaw moment, which for a thin or slender ship

represents the leading part. /

Faltinsen and Michelsen (1974) generalized Newman's results for finite fluid depth. They described the body potential by a distribution of three-dimensional source singularities

on the body surface. Faltmnsen & Michelsen presented computed results for a floating box and compared them with model experiments and theoretical asymptotic values for small wave lengths. The Latter were determined by a formula given by Maruo (1960). The computed results agree well with the asymptotic values and experimental data. moue (1977) applied the procedure suggested by Faltinsen & Michelsen and also Maruo's two-dimensional drift force formula, but did not show any computed results.

Ando (i976) presented both measured and computed drift forces experienced by a floating box and a semisubmerged platform with two longitudinal lower hulls. His computation method seems to be the same as Ogawa's.

Goda et al. (1976) considered a vertical circular cylinder in fluid of uniform depth arid compared by three differènt methods computed drift forces with experiments. Recently

betwen drift forces computed by sevGral methods and experimental data. Their comparison concerns ship-shaped -'and other floating bodies in regu]ar waves.

Kaplan and Sargent (197E) determined the drift force on a semisthmerged pipe-laying barge in regular waves by the added-resistance. formula of Gérritsm& & Beukelman (1972.). Their simple- f 'ormula is. based on Havelock's analysis and physical reasoning. The formula, involves the sectional heave damping coefficients and. vertical velocities, of the

ship. with respect to. water at each section.. In comparison' with experiments the förmula' has given good corrè'lation..

In addition to Gerr,itsma & Beukelinan addedre-sistance.

theories have been published in recen, years., för instance by

Salvesen. (197', 19-78). and Lin. &. Reed (.1976). Lin' & Reed utilize: in their derivation Kotchïn fünetion' and present an approximate method based on the strip-theory for deter-mination of the Kotchi'n function for a slender ship. They do not show. any cputed results.. Sa.lvesen's papers include computed results and. comparisons with experimental values.. His final added. resistance fórmula involves: only'terms that can easily be computed: by' existing strip-theory ship-motion computer programs. '

An aspect not considered .by other investigators of the drift

force: has been: examined, by Ohkus.u. (19:76).. He, derived by Màruo.s' two-dimensional theory, the drift: force experïenced by a ship section in the vicinity of a two-dimensional' structure änd. took approximately into account the- hydro--dynamic interaction, between the bodies..

AIÏ the aforementioned analytical. studies are. based: on the Po-

-tential theory. However, in real fluid viscous effects also exert drift.. The influence of viscosity on the' drift force has: been

studied by Mizuno (1976). He applied Maruo's two-dimensional theory and obtained a viscous correction term to the drift force formula. According to his analysis it seems to be possible at least approximately to separate in the drift force a poteiitial part and a viscous part, which are linearly additive.

1-luse (1977) explained by viscous effects the drift force opposite to the wave propagation observed sometimes in model experiments with sêmisubmerged vessels. His analysis

provides an approximate calculation method for the viscous contribution to the drift force on a semisubmersible. Pijfers and Brink (1977-) computed from the viscous drag the steady viscous drift force experienced by a tubular semi-submersible structure in regular waves and considered also the sloly varying drift in irregular waves.

Studies concerning the slowly varying drift of freely floating or moored bodies in irregular seas have for in-stance been conducted by Verhageri & Sluijs (1970), Remery & Hermans (1971) and Kim & Breslin (197E). Newman (1974) has shown that the slowly varying second order forces acting on a marine vehicle in irregular waves can be aporoximted from

a kñowledge of the steady second order forces in regular waves.

The literature dialing with the steady vertical second order force and roll and pitch moments is fairly meager compared with the bulk of papers concerned with the drift forces and

moment. In addition to the paper by Kotchin, already referred to, Goodman (1965) donsidered the second order vertical force on a slender body of revolution hovering under regular and irregular waves in head and beam seas. He assumed thè wavelength to be of the same oder of magni-tude as the transverse dimension of the body and derived an expression for the second order heave force by direct pressure integration. Lee & Newman (1971) examined a

neu-t-rally buoyant submerged body in the presence of regular waves and derived general analytical expressions for the

second order heave force and pitch moment in terms of the Kotchin fùncton. The part of the second order heave force, which entirely depends on the body disturbance potential, coincides with Kotchin's result. Lee & Newman give specific formulas for a deeply submerged slender body in long waves, but do not present any computed results.

In their papers discussing the stability requirements of semisubrnersible units Numata et al. (1976) and Kuo et al.

(1977) draw attention to the dramatic stady heel of semi-submersibl òbserved sometimes in model experiments in regular beam waves. Numata et al. also provide a theoreti-cal explanation for the phenomenon. The steady tilt is associated with the second order vertical force, which is greater on the upper pontoon or footing than on the lower one of a heeled semisubmersible. Hence, a steady heeling moment is exerted, which increases the tilt angle until an euilïbrïun with the hydrostatic restoring moment is reached.

The second order forces in regular wavès have also been derived for some specific two-dimensional bodies. Ogilvie

(1963) computed the first and second. order forces on a sub-merged circular cylinder by a procedure suggested by Ursell (1950), who had set on a rigorous basis the solution of the problem derived by Dean (1948). Dean had discoveréd the remarkable fact, that there is no reflection from a restrained submerged circular cylinder,the incident wave oiüyunder. going a phase shift when passing the cylinder. This implies that the horizontal secònd order force on the cylinder is zero. However, on a restrained submerged vertical plate the drift force is non-zero, as shown by Evans (1970).

20

Present Paper

The current practice in determining the drift force and moment on a hip or semisubmerged vessel in an ideal fluid and in the presence of regular bean or oblique waves seems to be to use Maruo's (1g60) two-dimensional theory and the strip-method. Concerning a ship in head waves one must rely on the slender body approach of Newman (1967) or on some added resistance theory. The computation method of Faltinsen & Michelsen (197) allows an arbitrary heading and is in principle applicable to any hull form, but the method is not very practical for complicated body shapes. A few attempts have been made to dètermine analytically the viscous contribution to the drift force on a semisubmerged vessel, but there seems to be no method especially intended for computation of the potential part of the second order

force and moment on a semisubmersible.

The objective of the present study has been to develop an approximative computation method for the potential part of the steady second order force and moment on a semisubmerged vessel in regular waves-. In this paper general analytical expressions for the six components of the second order force

in fluid of uniform depth are derived.. Details of the derivation are shown in the appendices. The second order forces are expressed by momentum considerations as 'integrals over the plane of the 'mean free fluid surface and fluid bottom, following in principle the method used by Lee & Newman (1971) in the deep-water case. The final equations of the second order forces and moments are given in terms

of the Kotchin function and an approximative method for determining the Kotchin function of a semisubmerged platform

is presented. The approximation is based on similar assump-tions to those frequently used in computing the wave-induced oscillatory motions of semisubmersibles, i.e. the effects of free surface and hydrodynamic interaction are omitted. The unit Kotchin functions or the velocity independent parts of

the Kotchin. functioñ are then simply obtained as a suoi of the, unit Kotchin' functions of the structural parts forming the. underwater body of the. platorm.'

Unit Kotchïn functions, are déri'ved for two basic types of structural parts of semisubmersibles-: for a. slender cylin-drical body and for a. small. côncehtrated body., The results can be regarded as an extension of the unit Kotchin functions given by Lee & Newman. The uni.t Kotch,in functions of the pIatfbrm known',, it is an easy task to, compute also the first order ecïting forces inducing the oscillatory: itotïon. of the platform from the. ui.t Kotchiri functions, by' the well-' knownjaskiñd relation. This seems to, be the first tine that. Kotchin- functions have: been utilized in, determining the forces, exerted. by waves on' a semisubmerged structure..

The' proposed computation scheme h'as been applied to a vertical, surface-piercing circular cylinder, to a submergd horizontal circular cylinder and to, a catamaran-type semi-submersible....Nthnerical results are presented. and discussed,. IJhfortunately it h'as not. ben poCsible' to' compare cämputed results- w1th. experiments due to the, lack of experimental data.,, but, in some, cases the results. are compared, with other theoretica-i data.

2. SECOND ORDER STEADY FORCES AND MOMENTS

Definitions arid -Governing Equations

The general expressions of the six components of the second order steady force and moment are derived for an arbitrary shaped body hovering under plane progressive waves. It is supposed that the body has no forward speed, but is free to respond to the incident wave system. The fluid and the resulting body motion is assumed harmonic in tiñie with circular frequency I and so small that the

boundarycondi-tions may be linearized and imposed on the average position of the bounding surfaces. The. fluid depth is assumed uni-form.

Let (X,Y.,Z) be a right-handed cartesian coordinate ystem fixed on the plane of the mean free surface of the fluid

(Fig. 1). The Z O plane coincides with the undisturbed

/ CE N TR E OF GRAVITY MEMBER BOOT GUR FAC E

Fig. i Coordinate systems and definition of the six components of rigid body motion

fluid surface and the positive Z-axis points uward. Thé

fluid is assumed iniscid, incompressible and is motion irrotational and acycÏic. Consequently the fluid notion can uniquely bè described by a velocity potential,

(X,,Y,Z;t)., which is a scalar function of the space coòrdi-nates and the time t. The harmonic fluid veIócitt is

given by the positive gradient of the potentil in the form:

(2.1) r

_{= Re[V(X,Y,Z)et]}

where. Re denotes that only the real part is to be qonsidered and i

T fluid presure p can be determined by Bernoulli's equation

(2.2) p = - -p JVJ pgZ

where p is the mass density of fluid and g the gravi-tational acceleration. The time dependent integration constant in Bernoulli's equàtion ha been merged into the time derivative term and the atmospheric pressure has been

set at zero.

It follows fron the equation of continuity of an incompres-sible fluid and (2.1) that the velocity potential thust sa.tisy Laplace' equation

(2.3)

=0

x2 y2 z2

in the fluid domain, which is bonded by the. body surface,

the free fluid surface and. tlie rigid horizoñtal bottom. Certain boundary conditibns must be imposed on these physical suraces..

The kinematic boundary condition on any physical surface, defined by equation S(X,Y,Z;t) 0, is (c.f. Wehausen &

214

Laitone 1960):

(2.'4) O on s = 0

D

-where + V V is the substantial derivative. Physically the condition (2.14) means that a fluid particle on the surface stays on he sürface. Ori the average posi-tion of the submerged surface of the body SB the boundary condition (2.14) can be written in the form

(2.5) u

n n

where demotes differentiation in the direction of the normal to the body surface and u is the amplitude of the normal velocity component of the body surface. The posi-tive direction of the unit normal vector (n1,n2,n3) is defined out of fluid or into the body. The kinematic boundary condition on the fluid bottom SR is simply stated as

(2.6) on Z - h

On the free fluid surface, in addition to the kinematic condition (2.14), a dynamic boundary condition must be

imposed. The last mentioned condition takes into account that the pressure from BeinoullitS equation must be constant

on the free surface. To obtain for the velocity potential a single boundary condition which does not explicitly i-volve the wave elevation, the two conditions must be combined. A convehient way to obtain a single boundary condition di-rectly is to utilize the fact, that the substantial

deriva-tive of the pressure (2.2) must vanish on the free surface (Lamb 1975, p. 3614; Newman 1970). In this way and neglecting terms of the order 2 and higher, the following linearized free surface condition is obtained:

(2.11)

aq)1

an an n

On physiaJ. grounds and due to lirearity of Làpläce'S equa-tien and the bàundary conditions (2.5), (2.6) and (2.7) the total potential. cari be decomposed into tuò parts

(.2.. 8.) q) _{q)1 +}

where

(2 9) q)1 g cosh K(Z+h) eÌl«X cosjj + y sini)

coshth

is. the veloci.t potential of the incident plane progressive waves satisfying (2.6.) and (2.7) and represents the

disturbance: associated with the body. The potentiál Includes both the effects of wave radiation due to motion of thé.body and scattering due to diffraction of the incident waves upon the body.

In (2.9) A is the wave amplitude, K 2irIÀ he wave nwnber;. A is the wave length and i.i the wave direction

angle relative to the positive X-axis. The wave nuthber is related to the frequency

of

waves by the dispersion rela-tionship

(2.10)

v.-rKtanhKh.

The body potential satisfies Laplace's equation, boundary conditions (2.. 6) and (2..7) arid on the.surface of the body

In addition must satisfy a radiatIon condition that the waves radiated by the body are outgoing at infinity. The radiation condition ensures the uniqueness

of

the solution and is defined ii Appendix I.

-are (2.13) d dt M. p

¡f f

M3

V

Applying Green's theorem in the fluid domain and consider-ing the appropriate conditions on the boundconsider-ing surfaces including a closing control surface at infinity the body potential can be expressed as

i

(2.12) 46(X,Y,Z)

= -

_¡f

[G(X,Y,Z;X',Y',Z')

SB

B G]dS

where the integration extends over the wetted body surface and X',Y',Z' are the integration variables. The Greèn's function G may be written in several different forms as can be séen for instance from John (1950) and Wehausen & Laitone (1960). The specific forms used in the present analisis are for convenience reproduced in Appendix I.

Expressions of the Steady Forces and Mòments in Terms of the Kotchin Function in Fluid of Uniform Depth

The second order steady forces and- moments are first expressed in terms of the wave and body potentials añd a integrals over the mean free surface Z O and fluid bottom Z -h by considering the rate of change of linear and angular momentum in the fluid domain. In addition to thé body surface, mean free surface and fluid bottom the fluid domain is considered to be bounded by a fixed control surface at infinity Sc., which is taken as a vertical circu-lar cylinder about the Z-axis. The components of the réte of change of linear and angular momentum in the fluid domain

at

_j

_dV

V.

+ p

¡f

UdS

for

j

1,2,3

whère t'he first integral extends over the fluid volume V

-iwt . .

-and r Re ue is the normal velocity of the control surface S _{SB + Sy + SR + Se,.} Here V for j r i ,2 and

3 denote the components of the fluid velocity (2.1) and is a position vector with respect to origin of the (X,Y,Z)-system. Replacing in (2.13) the fluid acceleration by an expression derived from Euler's equation combined with the continuity equation and transforming the volume integral into a surfàce integral by Gauss' theorem it is obtained' (Newman 1967): -M. n. (2.i) -p

ff[(+

gZ) [

()

] V. J. + (V -U ) ]dS for i 1,2,3 n n J

where _Vn is the fluid velocity in the direction of the normal to the control surface S.

The six components Of the hydrodynamic force and moment exerted by the fluid thotion oñ the body are

(.2.15) F. r

p ff(2. +

gZ)n.dS for j r 1,2...6

SB

where F1 is the force in the X-direction, F2 the force

in the Y-direction, F the moment with respect to X-axis añd so on. Here

ni

for j r 1,2...6 are components of the generalized normal defined by

r

fl r Yfl3 _Z12

n5 r Zn1 - Xn3

From (2.14) the following expressiàns for the hydrodynamic forces can be ôbtained:

(2.16) F. r _p . (( + gZ) { n. -

F3

_SF+SR+S .

+ (V-U) [

}]dS_ { 5 J for j = 1',2,3,

since according to the boundary condition (2.5) on the body

sui'fàce V

r U.

The pressure given by Bernoulli's

equa-tion (2.2) is substituted for p in (2.16) and the time

average is taken in order to obtain, the mean force and moment. Since the motion is periodic and there can be no net increase of momentum in the control volume from one cycle to another, the time rate of change of linear and angular momentudi integrated over one period of oscillation is zero. Dueto periodicity the term in Bernoulli's equation also disappears. When one takes into account in addition the boundary conditions: on the plane of the mean free surface r O ' and V r , on the horizontal fluid

bottom V r r U = O and ori the fixed control surface at infinity U r o and Vn r , it follows as the steady forces and moments .

(2.17') <F.>

J.

<F. > J+3 r p<ffEIV$l2 SF - . " ]dS + _.

¡f

Iv4I2

dS

(xV).

SR

(x).

:1 .. J

+.f2Rd8f°(+IVI2

.

[J

]

]dZ> for j = 1,2,3

xV).

where (R,8,Z) are cy-lindrical polar coordinates with

r R

cosO and

y r R

sinO and < > denotes the tïme average.

Since in the far-=field the body potential represents out-going waves, it can b assumed

(2.18)

r

coshK(Z+h) x,Y,o;t as + ..

Substituting the foregoing far-field form of the velocit3 potential for » in the last termof the equation (2.17)

the vertical integral from -h to O can be evaluated. The remaining integral with respect to O may be considered as a line integral around a circle closing the plane Z r O

and càn be transformed by Gauss' theorem into a surface integral over the same plane, as has been done in Appendix II.. When further in (2.17) the first terth, which is also an integral over the mean free surface Z O, is combined with the last term, the steady forces and moment acting in the horizontal plane, as shown in Appendix II,. can be written in the form:

(2.19)

<E.> r-

(tanii ii + seóh21th)<fJ(KZ J

SF

+ V)

dXdY> for j r 1,2 and 6,

ZO

where

_4-

is defined by X - Y

+ is the 'two-dimensional Laplacian. Y2

and y2

r

2

Similarly it can be obtaïned for the vertical plane force and moments:

X-Y -plane.

When in (.2.19) the iddicate time averaging is performed, the decomposed forn (2.8) is substituted for the velocity potential and it is taken into account that K1 + 0,

the horizontal plane forces become

(2.21) <F.> = - (tanh Kh + Kh sech2Kh)Re ff(K2ttB

J SF

+ VB) ¡-

) dXdY for j 1,2 and.6,

2 _Z=0

where the overbar denotes the conjugate complex. As is evident from (2.21) it is possible to divide the forces into two parts, one of which is linearly dependent on the wave potential

_h

and the other is independent. The former represents the forces due to interaction of the body gen-erated waves with the incident waves, while the latter is

entirely related to the body disturbance and can be computed

solely from B

Designating the interaction part of the

total steady force

<F1>

and the part related to the body disturbance <FBj>

then <F>

<F1> + <Bj>

The same

<F3> 1 (2.20) - <ff[sechKhlV I2 Y <PS> 2 -x 1

+ tanh21V2)

Y -

._L

(tanh2lKh 2K2 2 -x O + K2h2sech2Kh)(K2+V2) z Y. ] dXdY z=0

1l

-

JfIv

l2 Y.l dXdY> SR 2

_{-x J}

I Z -h

partitioning of the total force applies also in the vertical plane.

In the same way as (2.21) was obtained from (2.19) and using also the dispersion relationship (2.10) it follows from

(2.20) for the interaction parts of the steady forces in the vertical plane -

Il

L B I _.!.) _y (Z.22) <F11> = _{Reff[2sech21th(j_} ---+ I <F15> SF

1l

-

-i-- (tanh2Kh - 1(v28+tanh2}th V2q) I 2K2 I-X I O

+ K2h2sech2})(K25+V2$5)

-w

- _{Ref dXdY} 2 S y

and for the body disturbance parts: f<Fß3>)

a

_{B aB}

_{B aB}

1

(2.23) _{l<F5>I =} RefJ[sech21th(-

_-+y-

yI '

I<FBS>J SF _[-X I B I 1 R -X Z=-h

-+ K2h2sech2Kh)(K245-+ VB)

dXdY z=0

L.

(tanh2Kh dXdY ZrO 5 5+taith

V5)

1 '' -X 2K2 O

- _{Reff( dXdY} Z r-h

The terms solely depending on the incident wave potential have been dropped out from the foregoing expressions of the steady second order forces, since, there cannot be any force influence related to the undisturbed incident wave system. ThIs is easy to check by substituting the wave potential (2.9) for in (2.20) and performing the calcu-lations indicated.

Next the body potential will be expressed in terms of the Kotchin function and substituted into the equations of the second order forces. The deep-water Kotchin function is defined in the usual way as

(2.2L) _Hd(k,a) =

fI(!_B

)ekZ'+1XtCO5Y'5ifldS

SB

and the Kotchin function for fluid of uniform depth h as

(2.25) H(k,ci) r

ffc!!_'

._.)cosh k(Z'+hieik(X'cos+Y'sina). B 3n coshkh

B

By use of Green's theorem (2.12) and the expression (All) for Green's function, which is valid on the bottom of the

deflected around the pole at k K from belòw. An expres-sion for

B valid on the plane of the mean free surface

fluid and given in Appendix I, the body potential can be written ori the plane Z r -h in the form

(2.26) i

Jdk e_khHd_k,aeik

cosa + Y sina)

o. =

Jda

o 2i i da dk (k+v)e H(

k,a)eCO5),

J

ktanhkh

y O O

can be óbtained by substitutIng (Alo) for G in (2.12).:

C2..27) - j-dajdk

k nh kh - H(-k,o)e

cosa+Y a)

on Z O

introduing the asymptotic expression of Green's function (A18) into Green's .theorem the fàr-field form of the bod potential- becomes

(-2.28)

a-/

-1 cosh K(Z+h)

i3 2rR tarih .1li+}(b sech'1<b

HK,8)e

where 0ÇPT3"2) indicates that the remainder is of the òrder -3/2 in R.

substituting (2.27) and (2.9) in (2.21) the following equa--tions can be bbtained -after some calcilation, the details of -wh±ch are given in Append±x III, for the second order steady

horizontal forces or the drïf..t forces

C2..:29) -<E1> COS.P Re i E(-K,j) w 2ir 2 COSO 8 -- J do IH(-K,a)12

tanh -Kh + Kh sech2lth _{sino j}

--and -for the drift -moment

(2-.3O) <E > - --6 2w -8R&-K,a) e Orli p1< 1 Re 87T tanh Kh + Kh sech2Kb

-As in the dee.p-water'case (Maruo t960-, Newman 1967) the first -part in the expressions of the drift -forces -can be

iJ(_k)

H(K,o)

put in another form by energy consideration, if there is no net work being done on the body by external forces. In Appendix III it has been shown that from conservation of energy it follows then

2i (2.31) Re iH(-K,p)

:__S._

1 -

f

JH(-K,a)da. irg tanh Kh+Kh sech2Kh0

The foregoing relation enables the drift forces (2.29) to be written in the final desi±'ed form

(2.32.) <F1> pK2

<F2> tanh Kh + ith sech2Kh sinij-sin

Here the eqiations (2.21) relate the drift forces to the value of the velocity potential on the mean free surface. It would also have been possible to express the drift forces in terms of the far-field potential by choosing as the control surface instead of the meán free surface Z O the actual linearized

-at'(X,Y,O;t)

free surface Z r - Re as was done by Newman (1967). In that case, applying methods similar to Newman, the following expressions corresponding to (2.21) and

equations (27) through (29) in Newman (1967) can be obtained:

(2.33) <F1> r _(tanh + }th

sech2)J[(

li

<F2> 6K o R2 +

(.± .i + .±

3d8 - R O 8 R (2 . 3 ) <F6> -. (tanh + I sech2 2

-+

.4)Rae

+ O() ,

+ ZrO

-where now the :farfield form of the velocity -potential may be used. Substituting in (2.. 3.3) and (2.. 3') for the

sum-of (2.-9) and- (2.28) some -of the resulting .integ'ais can be evaluated by the method of stationary phase (for a short descriptióri of the method see Appendix III). After lengthy manipulations exac.tly the sane results, i.e. expressions

(2.29.) and (2.30), can be obtained. This indicates that -the final -results (2.29), (2.30) and (2.32) are also valid

for a -surface-piercing body. The validity of the derived -drift force formülas in the case of a .surf-acepiercing body

could have been directly concluded from the fact, that the drift -forces and moment can be related solely to the value of the velocity potential at large distances fÌ'on the body -and there remains no integral over the free surface.

In order to derive expressions for the steady second order vertical force and moments one needs the value of the body

inp, pgAbCK-v) - .-ReiH(--K,-u) 2w -cose-pgA(K-v) Re i, 2w .si-n:

.!

sin

kK

s

potential both on the. mean free surface (2.27) and on the fluid bottom (2.26). Substituting the aforementioned potentials together- with the incidnt wave potential (2.9)

in:(2.:22) it can be shown (Appendix III.): pgA(K-)

ReHd(-K,U)

H(-k,)

_kK

(2.35) ReHC-K,p) and - <.F1>.

r

Rei sinm -(2 .:-361) _2w

sinli

pgAh(K-v)

Re lHd(_K,u) cosp

The remaining part of the second ordér steady vertical forcé depending entirely on the body potential, as derived in Appendix III, is

(2.37) <F.,.>

Refdsfdk

k(k+v) k-v-(k+v)e lH(k,$)I2

l61T2

o

. (k tanh kh-v)2

- ReJ dJdk k e_2khlH (_k,)I2

l6i2

o o

2,r

-

_2_ReJ djdk

_{Ñ( k,)Ha(-k,)}

JL

where denotes a principal value integral.

As can be seen the body part (2.37) is proportional to the square of the Kotchin function while the interaction part

(2.35) is linearly dependent on the Kotchiñ function. Considering a slender body this means that (2.37) is of higher Order with ±'espect to the (small.) beam/lengthLratio of the body than (2.35). Since in the present analysis only the leading part of the second order steady forces will be considered the contributior of (2.37) will be neglected.

Expressions corresponding to (2.37) can be derived for the body parts of the steady vertical moments from (2.23) by similar calculations as was used in Appendix III in deter-mIning (2.37). The vertical body moments are also second order with respect tO the Kotchin function, since they are proportional to the square of the body potential, which

are of higher order in H than (2.36). The quite long expressions of thé vertical, body moments are not given here.

in the horizontal plane both parts of the 'drift forces are second. order in H, as was shown by applying the principle of energy conservation, but the drift moment has a contri-bution linearly dependent on the Kotchin function.

Contrary to the drift forces (2.32) and the drift moment (2.30) the-expressions' of the second order steady vertical force and moments., (2.35) through (2.37), may not be valid fora surface-piercing body, since they were derived fro equations involving integrals over the plane of the mean free surface.

Here' the second order steady moments have, been derived with respect to the origin of 'the inert±a. (X,Y,Z)-frame. If one. wants to calculate the moments about the center of gravity of the body, which may be supposed to be located at (O,O,Zg)' the body at' rest, one only has td replace z

in the' components n and n5 of the generalized normal in (2.15) by _{Z_ZG} and perform the same operations as in evaluating (2.36). This, ofcoirse, implies that the wave-induced motion of the body is assumed small and quantities of second order im the motion amplitudés are 'dropped out. The result is essentially the same final expressions of the moments as in calculating the moments about the origin of the (X,Y,Z)-system, as has been noted by Lee & Newman

(1971).

-Comparison with the Deep-Water Case

It may now be interesting to compare the general expressions of the second order' steady forces in fluid of uniform depth with the corresponding expressions in fluid of inifinite depth given by Newman (1967) arid Lee & Nean (1971).

and (2.30) involve, in addition to the deep-water case, the factor 1/(tanh Kh + Kb sech2Kh) and in the Kotchin func-tion, the exponential dependence on the vertical coordinate has changed into a hyperbolic dependence. However-, for a

slender1 surface-ship the fluid depth has no first order-effect on the Kotchmn function. This may easily be verified by writing the first order' equations of motion for heave and pitch, which involve just the hydrostatic restoring and the Froude-lKrylov excitation, even if the fluid depth is assumed to be of the same order of magnitude as the ship beam (Beck & Tuck 1972). From the heave and pitch equations the Kotchin function to leading order can then be formed by flux consideration in the same way as in Newman (1967).

Thus, in the case of a slender ship the fluid depth does not have any first order contribution to the drift moment and the ratio of the drift forces in- fluid of uniform depth to the same forces in fluid of infinite depth is to leading order 1/(tarih Kb + Kb sech2Kh), which has been presented graphically in Fig. 2. The ratio approaches infinity as Kb approaches zero and of course one as Kb increases. The ratio has a minimum of about 0,83 when e2Kb

(ICh + 1)1(1Kb - 1) or Kb 1,2.

From Newman's computations it can be seen that the drift forces have maximum in quite short waves, the longitudinal drift force when the wave length is of the order of half the ship length and the lateral drift force when the wave length is comparable to the ship beam. Consequently, even if the fluid depth were one sixth of the ship length., the maximum of the longitudinal drift force t-akes place when the parameter Kb is approximately two. As is ev-ident

from Fig-. 2 the effect of fluid bottom is in this case a

- few percent. Hence, it may be concluded that in most cases

The "slenderness" of a body is defined in the beginning

-Fig. 2 Functions appearing as factors in the expressions of the steady second order

forces in fluid of uniform depth h.

in practice the fluid depth does not have a remarkable effect on the drift forces and moment. In very shallow water only, the situation can be different.

The expressions (2.35) through (2.37) for the second order steady vertical forci and moments coincide with the expres-sions given by Lee & Newman (1971), when the fluid depth h approaches infinity. Compared with the deep-water case the present formulas have in addition all the terms involving .the Kotchin function Hd. Also additional is the second

term in (2.36), which in fact on the basis bf the formula (2.31) is proportional to the square of the Kotchin function, i.e. of higher order than the other terns in (2.36). When the fluid depth increases, the additional terms approach zerò for fixed K like 1 - tanh XI-i or Xh(1 - tanh 1(h).

As can be seen from Fig. 2 the factors i - tanh Kh and Kh(1 -tanh 1(h) deòreáse very fast with increasing Kh.

Thus, it seems that the fluid depth is in practice of minor importance when considering the second order steady verti-cal force and moments.

3. APPROXIMATE: METHOD FOR EVALUATING THE KOTHIN FUNCTION OF A SPACE-FRAME-TYPE SEMISUBMERGED STRUCTURE

The second order steady forces and moments acting on an arbitrary body without forward speed in regular waves can be computed by the equations given in the preceeding section, if the Kotchin function of the body is known. In order to derive an approximate method for evaluating the Kotchin- function of a semisubmerged platform, consider a group of distinct, rigid bodies moving in an ideal in-compressible fluid as though they were structural members of a rigid platform oscillating in waves.

The wave-induced motion of a floating body may be specified by three translatory displacements and three angular dis-placements. In order to define the displacements of the body from eqùilibrium a cartesian coordinate system

is introduced with origin fixed in the fluid at

-the mean position of -the center of gravity of -the body at (O,O,ZG). The axes ' and are parallel to the axes X, Y andZ, respectively (see Fig. 1). Let the complex amplitudes of the translatory displacements along the axes

and with respect to the origin be

2 and respectively, so that when the motion is assumed harmonic im time the real part of Cie_itt is the surge,

2e1Wt

is the sway and C3e_1Ct is the heave displacement.

The rotational motion of the body may be defined by a set of modified Eulerian angles (Blagoveschensky 1962). If the angular displacements are assumed mall as usual, the

tHgonoinetric sine of -an angular displacement may be replaced -by the angle itéelf and the cosine set to one. When in

addit-ion products of the small motiòn amplitudes are omitted, the Eulerian angles coincide to first order with rotational displacements about the axis of the space fixed

system. Thus, the small, harmonic angular motion of the body may be specified by rotational displacements

-iwt -iwt - -

-5e and 6e about the axes X, Y and Z, respec-tively, being the complex amplitude of roll, the complex pitch amplitude and the complex yaw amplitude.

The velocity due to wave-induced motion of the body on the surface of the body is given by

(3.1) U +

-where the translatory velocity vector is

the angular velocity vector is

= _i(C,5,6)ejWt and

(O,O,Z0). Here is a radius vector pointing from origin of the (X,Y,Z)-system to the surface of the body.

In the foregoing section the- velodity potential was expressed as a sum of twò parts: the incident wave potential and the body potential. It is now adapted to divide the body potentiell. further, into several parts. Using the common decomposition of Has}cind (1953) the body potential can be written in the form

(3.2) r ₊

potential follows by considering the boundary condition (2.11) satisfied by _B on the wetted surface of the body at the mean position.

Considering the group of distinct, rigid bodies the unit potentials may be represented in accordance with Moneim's (1976) analysis as s ums of two parts: the one part being the sum of potentials due to the unit motion of where is the speed independent pa±'t of the potential

for unit motion of the body in the jth direction ( r1,2...6

refer to surge, sway yaw, respectively). Here is

the complex amplitude of the diffraction potential repre-senting the waves diffracted upon the restrained body and scattered in all directions. The form (3.2) for the body

the' ith body in the jth direction alone in absence of the others and the other part being the sum of unit potentials of interaction. However, if the distance between th bodies is relative3y large compared with the lateral dimensions of the Î odies the interaction effects are rather small. For instance, according to classical hydrodynamics, some of its results -being reproduced by' oneim, -the unit potentials of interaction are for three-dimensional bodies in infinite mass of fluid generally of the order 'O((a/D)3),, where a

is a characteristic radius of the body -and D is the distance between two neighbouring bodies. Similar conclu-sions regarding the significance of the interaction effects can be drawn from Ohkusu's (1-97) computations, where the influence 'of free surface has also been tá.ken into account.

'Considering a semisubmerged platform, whose underwater body 'typically consists of submerged, longitudinal, hulls or small

pontoons, surface-piercing vertical columns and inter- -connecting slender braces, the assumption on large dis-tances -'between the member bodies is of course badly violated at the trusses connecting the structural members. However, it has 'been a common practice to ignore all the hydrodynamic interaction 'effects in computing the first order hydrodynamic forces and wave-induced motions of seinisubmersibles by the 'so-called hydrodynamic synthesis procedure (Paulling 1970;

for a thorough reference list see van Sluijs & Minkenberg i'97i5,)., in 'spite 'of this and other simplifying assumptions the computation procedure 'has 'given quite a reasonable oerrelation with model eperithents (Hooft 1911). Thus, due

to the evi'déntly rather small influence of intraction effects and the considerable analytical difficultis in-' volving their contribution, all the hydrodynamic 'interaction 'effects between member bodies of the platform are disregarded.

Then the body potential can simply 'be 'w±itten in the form

(3.5)

where has already been defined and is the diffrac-t.ion potential due to the ith member body in absence of the other members. The summation with respect to i extends

to all the member bodies of the structure.

The diffraction potential j -must satisfy on the surface of the ith body

i

(3»4)

- on S

Considering the boundary conditions (2.11), (3»4) and taking int6 account the velocity on the surface of the body (3.1) it can be seen, that the unit potential must satisfy the condition

i

(3.5) u- = n._n On S1 at mean position, j

where n are components of the generalized normal on the surface of the ith member body.

Introducing a right handed cartesian coordinate system (1,n',c1) the origin fixed at the ith member with direc-tions of axis coinciding with direcdirec-tions of the X, Y and

Z axes, respectively, the six components of n can be expressed as - i i i i i i i n4 (y + )n3 - (Z _{- ZG +} (Z _{- ZG +} - (X' )n i

ii

i i i n6 (X + )n2 (y + i )n1

Here the origin of the (',ri1,')-system is ssumed to be located at (X',Y',Z') in the (X,Y,Z)-systei, when the body is at equilibrium position.

(3.7-) - -- -(Z

- Z)

+ - - X14 + -i i_ -ì i i + i -

ol

6

-where

i1

and are unit potentials satisfying,

- - -4 -ii

ii

-(3.8) - ' n3 -/

ii

.ii

-- _{nl -- -n3} 6 _-

ii

i.i i n - fl n. on S

The-potentials

4',

and are unit potentials describing the flow due -to angular motion of the ith body about the axes n1 -and

e'-,

respect-ive-ly. --Substitutirigihe expl'es.sion (3.3) for the body -potentiaTi

into the -Xotchin function (2.25) the following is obtained

.C3-.9)

H(k,)

---ìw - i cosh k(Z'-h) ik(XTcosYtsins)ds j n cosh k-h e +

j.

(7

1 cosh-k(Z'+h) c(X!cos+Ysi) -. n 73n -coshkh e dS

1S1

- 6 ,:ik 14) :Cj(-5

q)i(.) cohk(Z'+h)

e'C0+Y'1'13)cis

Bc

i tr v( ik3)Sh-k(Z'+h) ik(X'cos+Y'sim)

-- 3n 3ri -coi

where in the last two terms- the superscripts1 k i and

ik . . . k

4). is the value of the unit potential 4). on the surface

of the ith member body. Respectively 4)7 is the value of the kth ember diffraction potential on S1. The lest two terms also represent hydrodynamic interaction effects between the medthe± bodies and are omitted.

If the distances between the mènber bodies are quite large and the bodies are deeply submerged the two terms neglected represent the highest order interaction effects, since the effect of a body moving in unbounded fluid is -far from the body equivalent to a dipole and thus of the order O(D2), where D is distance from the body. This becomes evident, for instance, from the equation (4.-26). Hence, to leading order, the influence of hydrodynamic interaction on the Kotchin function can be calculated without solving for the unit potentials of interaction.

Subst-ituting (Xt,Y',Z')

(X,Y,Z) +

into (3.9) nd defining the following unit Kotchin functions of the ith member body

(3.10) H(k,m)

_ff

c--J Si

cosh k(+Z+h)

_e0551fldS

j

cosi kh for- j :-1,2...7

i

i

__.i . .

where now n and ' are the integration variables and

the integration extends ovér the surface of the ith member, the Kotchin function (3.9) can be written without interation terms in the form

-1

6 .- ikCX1cos+Y'sins) (3.11) Bck,a) -1w _Z

.H(ka)e

i ik(X'cos+Y's.ino) +

H(k,)e

°

When the following unit KotchirL functions ere introduced

(3.12) H!1(k,)

_ff

(--.--si

cosh k(ç'+Z'+h) . j i

ä) o ik(

cos+n Sin)dS

coshkh

e

the Kotchin functions H+ for

j

, 5 and 6 corresponding to (3.7) may be expressed as

(3.13)

= YH - (Z

- ZG)H +

(Z

- ZG)Hl - XH +

= X'H1 - Y1H1 + Htl

6

o2

al

6

The determination of the Kotchin function for a semisubmerged platform, whose underwater body consists of several, widely spaced structural members, has now been reduced to determi-nation of the wave-induced motions of the platform in six degrees of freedom and to determination of the unit Kotchin functions of the structural member bodies (3.10) and (3.12). Unfortunately it is in general an almost impossible hydro-dynic problem to calculate the Kotchin functions exactly. Usually one must rely on simplifying assumptions. For instance, Kotchin himself sometimes substituted into the Kotchin fmction for the body potential the velocity poten-tial describing the flow around the body in infinite fluid and obtained so an approximation to the Kotchin function.

Considering the special characteristics of underwater bodies of sisubmersibIes it is also

in

this case possible to find reasonable approximations fbr the Kotchiri functions of the

deeply submerged arid vertical surface-piercing slender cylindrical members and/or deeply submerged, small con-centrated volumes. The same assumptions regarding the type and submergence of thé member bodiés are also made in the hydrodynamic synthesis procedure.

Lee & Newman (1971) calculated the unit and diffractioñ Kotchin functions for a horizontal, deeply subiiierged slender. body. In the present study their results are generalized for an arbitrarily oriented, deeply submerged slender body. The results apply also to a vertical, surface-piercing slender body. Using the same method, unit and diffraction Kotchin functions are determined in addition for an arbi-trarily orieñted, deeply submerged concentrated body in terms of its volume and added masses in infinite fluid.

Slender Cylindrical Body

Expressions of unit Kotchin functions when k r K are derived for a deeply submerged or vertical surface-piercing slender

body.. Tha slenderness of the body implies not only that a typical transverse dimension B of the body is small

compared with the length L,- but also that changes in curva-ture of the body surface are gradual in the longitudinal direction.- Tim 'ansverse dimension B is also supposed small in comparison with the wave length so that KB « 1. The body may be arbitrarily positioned, when only the dis-tance from ail points of the axis of the body to the free surface arid fluid bottom is of the same order as the length of the body or 0(1).

From the assumptions concerning the orientation of the body it follows that the inner region does not contain to leading order the free surface nor the fluid bottom, when the velocity potential describing the flow due to body motion is represented as an asymptotic expansion in -the inner and outer regions in accordance with the method of matched asymptotic expansions

(Newman 1970). For a. vertical, surface-piercing slender body Newman (.1963) has shown by a rigorous slender body analysis, that to riri _{order the free surface has no contribution to}

tha motion associated potentials.. Thus, considering the motioii sociAted potentials near' the body the fluid may be regarded as a first approximation unbounded. In addition,

i-c'rn-ding to the slender body theory the flow cari be examined i:rt every cross-section. plana of the. body as two-dimensional.

Lii order to write down the motion associated potentials a new body-fixed, right-handed cartesian coordinate system (x,y,z) is definad. The x-axis coincides with the axis of the slender body. The new local system, when. fixed to the ith member body, is related to the (1,n1,ç1)-aystem by

where 1._jk is thé matrix of direction cosines. Here the superscripts i referring to the ith member have been, dropped as unneceésary.

Since the unit potentials of a slender, deeply submerged body are known in the. (x,y,z)-systen, the potentials

for transiatiönal motion into the , n and directions and the potentials . for angular motions about the same

axis are expressed in terms of the unit potentials in the (x,y,z)-system. Let

(n,n,n)

be components of the unit surface normal on the axes x, y and z. Using the transfor-mation formula ('4.1)

('4.2) _n

= 11n4 + l2n

+ l3n

for

j

= 1,2,3

From the boúndary conditions (3.5)'it can then be seen that the unit potentials may be expressed as

('4.3)

= + l..2i2 + l3P3 for, 1,2,3

where are unit potentials due to translational motions in the xi_directions satisfying on the surface of the body

for j 1, 2 and 3

Respectively, applying the formula ('4.1) on the right-hand sides of the boundary conditionS (3.8) it can be established that thé potentials may be decomposed into the form

- 1j)44))4 + ]55 + 1j64)6 for 4,5,6

where ' 4)5 and

6 satisfy on the surface of the

(i. 1') _{n =} 111 21 131 112 122 132 113 123 133 X y z

body

- zn

zn4 - xn

xn - yn. and

It is easy to verify by direct substitution, that the poten-tials (14.3) and (4.5). satisfy the boündary conditions (3.5) and (3.8), respectively, if llj for

j

-= 1,2...6 satisf' (4.14) and (4.6).

In the expression of the Kotchin function (2.25) the integ-ration extending over the surface- of -the body ay be replaced by an integration over anî closed surface containing the body, if the body is submergéd. The proof -is based on Green's theorem (Wêhàusen & Laitone 1960). Considering .a surface-piercing body -the integration in (2.25) when k =K can be evaluated over any surface.surrounding the body and extending from the fluid bottom to the free surface. This fo11ows,.aîp-from Green's theorem, since - -

-cosh K(Z'+h) iK(X'cosa + Y'sin)

-- - - e

cosh Kh

in (2.25) is a harmonic function of X', Y' and Z' in the fluid domain and -satifies on the free surface and fluid

1144 = 122133_123132 154 = 132113_133112 1145 _{123131_121133} 155 = 133111_131113 146 132_122131 156 = 131112_132111 164 = h12123l13122 165 11 31 123 166 11i122112l21

bottom the same boundary conditions as but does not satisfy the radiation condition.

The integration over the body surface in the expressions of the unit Kotchin functions is changed into integration over a circular cylinder with x as axis. When-in addition it is taken into account that duè to slenderness of the body the integration can be carried out stripwise, the unit Kotchin functions (3.10) for j 1,2,3 and 7 and (3.12) for i , 5 and 6 when k r K can be written in cylindrical polar coordinates (r,B,x) with y r sin B and

z r r cos B into the form

21T 31 o cosh K(l x+Z h)

_eL1f

dBr(-(.7)

H(K,)

:_Jdx cosh Kl-ì L o

-

Kr(l32siñß+l33cosB) + tanh K(131x+Z +h)sjnh Kr(l32sinB o iKrtL2()sinB+L3()COsB] + l33cosB)]e for j

r 1,2...7

where for convenience

.

and respectively H

for 5

t

,5,6.

Here the abreviations ICa) rL15cosct+125SÌfl have been uséd.

If the radius r of the cylinder about the x-axis satisfies

1(1

_{» r » /, where}

s is the cross-section area of the

body, the unit potentials in the cross-section planes may be

taken in accordance with Lee

&. Newman (1971) as approximations

valid far from the body. They also give expressions of the far-field potentials that can be used here. In two-dimensions the far-field approximations of the unit potentials due to motion of the cros-section in infinite fluid into the y

& Ñetman 1971) + s)s±n. + cosß ('4.8) 'P2 p 2iTr a a32 (.-__ + s)cosß + .- sinß ('4.9), TP3 . - _2rr

where ak

is the sectional added. mass in they xk-d'±rection

due: to motion in the x.-direct±oii.. The added. mass' a.3 a32 js zero. 1f the directions y and z. co.ncde. with.

directions of permanent. translation of- the crossSection..

The potential due. to axial motion is the familiar- slender-body a-pprocimation:;

The poténtials describing the flow due to angular motion, of. the body aboüt the y- and z-axes can be talc....a the stïp-theory. approximations

('4.11) 'P'5: = XTP3

(4.12) 'P's _'

The. remaining- potential due to angular motion of the slender body about its Iongi.tud.±naÏ à.xis is of: higher order with-. respect tb- the small transverse dimension B than the other

potentials given above and can be omitted

By means of (.4.8), (4.9) and (4.lO)the- sectional potentials -to be used in. ('4.7) fôr j ;. 1:,. 2. -and 3 may be expressed in

accordance with ('4.3) as:

+ s)l,2sinB

(.. + S)l3O5ß

(4.13) i

d s()

- 2ir dx log Kr ï - - " 1. .d s. (x) dx log Kr for j, 1, 2. and. 3

(4.14)

where it has been assumed that a23 = a32 O. Respectively,

by (11.11), ('4.12) and ('4.5) the sectional potentials for

j 4, 5 and 6 may be w'itten in the form:

a22

( +

s )l. x sin$ p c 2,r a + s )l. X cos5 p C -iS for .2 iTr '4, 5 and 6

Substituting the unit potentials ('4.13) and (4.111) in (11.7) for 4,, expanding the cosh-, smb- and exp-functionS into Taylor-series for small values of Kr and carrying out the integration yith respect to ß yields to the leading, order:

coshK(l31x+Z0+h)

iKL1()x

ds0 ('4.15) H(Kc2)

J

coh Ith - a + K[l32tanh K(l31x+Z0+h) +

iL2(-- + s)l.2

+ K[l33tanh -K(131x+Z +h) + iL3(a)](--

+ s)l3}

o

+ O(K2r2log Kr) for

j

1, 2 and 3 and (4.16) H(K,a) cosh K(l x+Z +h) iKL.()X - - _cosh e .{[l2t&j1-i K(131x+Z0+h) -KJdXX 31 o. i L + + 5,:l)l6 - [l33tanh K(13x+Z0+h)

+ iL3()](+ s0)l5) +

O(K2r2) for j 4, 5 and 6

The unit Ktchin-functiOfl5 fòr k -K can be obtained fxm (4.15) and ('4.16) by setting r The deep-water unit Kotchin functions corresponding to the Kotchin-funCtiOn

(2.24) can bé derived fdi k -K from the equations

-(4.15) and ('*.16 as limiting values h approaching infinity, wheñ it is set h = -h

and

T+OE.

In the cae onsidered 'by Lee & 'Newman the unit Kotchin

functions obtained from the foregoing, equations agree with

their results.

Ín order to derive an expression corresponding to (4.'1) and (4 16) for the diffraction Kotchin function the incident wave potential (2.9) is written in (x,y,z)-coordinates

(4.17)

- 0çOsp+Ys.ini.i) cosh K(131x+132y+133z+Z0+h). e[LX(Zl

'(p ' coshEh-

-and substituted into the boundary condition (.3.4)yièlding

(4.18)

-j-iK(X0cos+Y0sin) cosh'

K(131x+132y+133z+Z0+h)

{n4[13-1tanhK(TL31x

cash Eh

+ 132y+ 133z+Z0+h) +iL1 (ii)] nI132tanh K(131x+132y+133z+Z0+h) +:iL(i.i)1

-. L1(i)x4L(p)y+L3(i)z]

+

n[l33tarth K(i31x+ly+l33z+Z+h)

+

on the surface of the member body.

Since the sÏendernes f the:body iplies

n «

the foegoing condition becomes to leading ordet with respect to KB:

(4.19)

cash K(131x+Z0+h)