Wave Drift Force and Moment on a VLFS supported by a Great Number of Floating Columns

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Wave Drift Force and Moment on a VLFS Supported by

a Great Number of Floating Columns

Masas/ii K4shïwagi

Research Institute for Applied Mechanics, Kyushu University Filkuoka, Japan

ABSTRACT

A caicülation m thod is presented of the wave-induced steady drift forcé and yaw moment on a very large floating structure (VLFS) .omprismg a multitude of floating coliimn The theory

is based on the momentum-conservation principle, and all

neces-sary integrstions are analytically implemented. Thus the resul-tant formulae include only the coefficients of the incident-wave

and disturbance potentials at a large distance from the

struc-ture. A hierarchical interaction theory developed by thé author is applied to determine the disturbance potential due to

hydrody-namic interactions among a great number of floating coliinrns and

elastic motions of a thin upper deck. Experiments in head waves are alsoconductéd using 64 truncated vertical cylinders arranged

periodically in 4 rows and 16 cohirni,s. Goód agreement is found

between computed and measured results. Furthermore, through numerical computations in oblique waves, discussions are made on variation characteristics of the steady force and yaw moment particularly near trapped-mode frequenciés.

KEY WORDS: drift force and moment, hydrodynamicinterac-tions, momentum conservation principle elastic motion, trapped

mode

INTRODUCTION

Very large floating structures (VLFSs) are categorized with the configuration undér the sea level into: (1) a pontoon type

which looks like a simple plate with very shallow draft, and (2) a

column supported type in which a thin upper deck is supported by a lárge number of floating colin-nns it is said that the pntoón type is advantageous in low costs for construction and

mante-nance, but the wave-induced motiOns may be relatively large. On

the other hand, the column_supported type has reverse features; that is, the motions in waves may be small relative to the pon-toon type, because incident waves will transmit through a gap

between columns.

HòweveE, the above recognition may not be the case. Recent studies' 2) on hydrodynamic interactiòns among many cylinders

reveal that near-resonant modes occur at some critical frequen cies and cause large wave forces on each element of the array. Accordiig to Maniar & Newmàn' , these critical frequencies are eigen-frequencies in the diffraction problem, at which homoge-neous solutions exist, and their existence depends on the number

of cylinders and the ratio of cylinder diameter and separation

distance between adjàcent cylinders. Since the waves amplified at these frequencies do not propagate outward, these waves aré called "trapped waves'.

It is known that the reflection and trans mi ssion of incident

waves are reláted closely to the wave drift force through the

momentuth-comemtion principle. In the limit of very short

wavelength, the drift force Will be of the same value irrespective of the types of structure, because almost all of the wave will be re-

-flected. Near trapped-mode frequencies, however, no information is given concerning how the wave drift force varies. Furthermore,

few studies have been made on the drift yaw moment acting on a column-supported type VLFS.

The steady drift force can be computed by integrating the

pressure over the wetted surface of a structure and taking, time average over a peribd This pressure-integration method enables us to evaluate the force on each column but is not effective for a VLFS, because the number of columns will be of order more than several thousands.

In the meantime, as Maruo has proved by means of the

momentum-conservation principle, the wave drift force can also be evaluated using the velocity potential at a large distance from the structure. In this method, contributions of the amplitude

function of waves generated by a structure may be numerically integrated in the direction of the azimuth angle. it is doubtful, however, that réliable results are given by a method using

nu-merical integrations particularly when the number of bodies is very large, like a column-supported type VLFS.

Recently, a hierarchical interaction theory was de eloped to compute hydrodynamic interactions among a great number of floating bodies. With this theory, the distúÈbance potential valid at a far field may be given in a simpler form in terms of a cylin drical coordinate system. in this case, aU necessary integrations can be analytically performed and thus accurate results can be

expected, provided that enough numbers of terms are retained in

a series expansion.

In the présent paper, the. diffraction and radiation problems are solved by employing the hièrarchical interaction theory' and an analytical expression is derived for the disturbance velocity.

potential valid at a far field. The effects of elastic motions of

the upper, deck are also taken into accoûnt as the generalized radlation problems. Applying orthogonal relations of the Fourier series to integrals in the circumferential direction and Wronskian

formulae to some prodùcts of Bessel functions, simple calculation formulae aie derived for the drift fòrce in the horizontal plane and

the drift yaw moment:

Experiments are also coñducted in the present study using 64 truncated circulär cylinders arranged in a rectangular array of 4 rows and 16 columns with equal separation distance. Measured results in head waves are compared with corresponding results of the computation. In particular, cautious measurements and computatiòns are conducted at frequencies around near-resônant modes, and discussions are made on variation characteristics of the wave drift force and moment.

FORMULATION

We consider a VLFS, comprising a thin deck and a great num-ber of buoyancy columns which are identical and equally spaced

As shown in Fig. 1 the deck.is rectangular in plan, with length L and width B. The geometry of an elementary column considered here is a truncated circular cylinder with radius a and draft d. The centerlines of adjacent cylinders are separated by a distance 2s in both n- and y-axes of a Cartesian coordinate system, where z = O is the plane of the undisturbed free surface and the water depth is constant at z = h. radius:a draft : d incident wave amplitude: A) - B,'?

Fig. i Coordinate system and notations

Under the assumption of incornpressive and inviscid flow with irrotational motion, we introduce the velocity potential satisfying

the Laplace equation The boundary conditions are linearized and all oscillatory quantities are assumed to be time-harmonic

with circular frequency w. Then, we express the velocity potential

in the form

'I' = Re[cb(x,t,z)e't]

(1)

(2)

where g, A, and K are the gravitational àcceleratiön, the ampli-tude of an incident wave, and the wavenumber given by w2/g,

respectively. cbi and ç6s are the incident-wave and scattering po-tentials, respectively, and the sum, cbx + çbs es , is referred to as the diffractioù potential.

Incident plane waves propagate in the direction with angle ¡3 relative to the positive x-axis, and hence cbi is given by

coshko(z - h)

'

coshkoh

where k0 tanh koh K.

Xk in (2) denotes the complex amplitude of the k-th modé

of motion in the radiation problem, in which not only rigid-body

motiOns but also a set of "generalized" modes to represent elastic

deflections of a deck are included. cbk is the velocity potential of a single body oscillating in the k-th mode (with no interactions)

and çQk represents the remaining part of the potential due to

hydrodynarnic interactiOns with radiated and scattered waves by the other cylinders.

HIERARCHICAL INTERACTION THEORY

The number of columns of a realistic VLFS will be of order more than several thousands. Hydrodynamic interactions among the columns of this order may be taken into account by a hier-archical interaction theory' developed recently by the present aüthor. In this hierarchical interaction theory, a number of ac-tual bodies (labeléd as level one) are grouped to form a fictitious body (level two), and several fictitious bodies are grouped fur-ther to form a bigger fictitious body(level three). This procedure can be repeated theoretically up to any hierarchical level. Then, the interactions are computed at each level and information on interactions can be transmitted upward or downward as required. At the highest level, the number of fictitious bodies may be of order of several tens, to which Kagemoto & Yue's interaction

theory5 can be applied. it is a premise in their theory that the

diffraction characteristics of a fictitious body in response to a set of "generalized" incident waves are known. For details of the analysis, we refer the reader to reference 1). In the present pa-per, with the assumption that the diffraction characteristics of a body at the highest level (denoted as L) are already obtained, the calculation method of hydrodynamic interactions at the highest

level is summarized.

In the analysis to follow, we will use a local cylindrical coor-dinate system (ri, _ei, z), with the origin placed at (xi, y,, O),

i.e. the center of the j-th fictitious body. The number of bodies at level £ is denoted as N1, which is, needless to say, identical to NB (the number of actual columns) in the case of no hierarchical

level.

Diffraction Problem

The incident waves impinging upon a fictitious body i consist of not only the wave expressed by (3) coming from the outside but also scattered waves due to other fictitious bodies. Thus we can write N1

=

{a1}T{4.e}

₊

i=1 N, = ₍₄₎ j=1

ii

(3)

Here {} arid {} are the vectors of the "generalized"

inëident-wave and scattering potentials, respectively, which are

expressed as

-

f

Zo(z) Jp(kori),eOi

'

_{-) Z,(z) Ip(knr) e0}

-

f

Zo(z)H$(korj)e_im8i

{}

_-

_i

Zn(Z)Km(kni)emej

where

coshko(zh)

cosk(zh) Zo(z) -

, Z,(zj

-coshkoh cosJch

and k, is a solution of k tan kh

=

K (n

= 1, 2, ), giving

the wavenumber of evanescent wave modes. The number of terms

in the O-direction,.p and m, must be taken as O,±l, ±2,

The coefficient vector of the incident wave, { at }, is known and may be explicitly given by expressing (3) in terms of a cylindrical

coordinate system of the j-th body. Meanwhile, [Tj] is the

coordinate transformation matrix, relating {'} with {&};

a concrete expression of which can be given by Graf's addition

theOrem for Bessel functions.

Let the diffraction characteristics matrix corresponding to

{i}} of a fictitious body be expressed as Then the

scat-tering potential due to (4) can be obtained in the form m

({at} +

{A}T[T.])

i i

=

{A}T{,.}

(8)

From this relatión, the unknown coefficient vector of the

scat-tering potential, {A}, can be determined. Resulting

simulta-neous equations are N1

{A'9e}

-

[B]

[T}T{A}

=

[Bi,]{at},

i1

ii

_i=1Nt

₍₉₎

Solving (9) completes the flow field at the highest level. By

transmitting the information on hydrodynamic interactions down to lower, levels, the flow field around aclual bodies may be

deter-mined and thus the wave forces on flóating cohrnrns of a VLFS

can, be computed.

where n denotes the k-th component of normai vector on the j.th body.

Therefôre, as already described, Ø* is a solution of the radia-tion problem fOr a single body andcok is a sólution of a sort of

the diffraction problem dùe to radiated and scattered waves by the other bodies.

As a result of forced oscillations of each body and hydrody-namic interactions among other bodies at the same hiérarchical

lével, the radiation potential of a fictitious body i at lével £ can be given in the form

=

{1}'

(11)

where {7Z} may be explicitly given by transmitting the infor-mation on hydrodynamic interactions upward to a fictitious body

at level L.

On the other hand, a sohition of , can bedetermined in the same way as the diffraction problem and given as

=

_{AL1}T{1}

₍₁₂₎

where {A1} is the unknown coefficient vector.

When viewed from the z-th body, (11) and (12) may be re-garded as incident waves. Therefore contributions from all other bodies at level £ can be written as

$({}T±

_{{A}T) [TJ]}

₍₁₃₎

i=i

_it

This velocity potential corresponds to (4) in the diffraction problem, and thus in the same way as in obtaining (9), one can

obtain a linear system of simultaneous equations for {A1} in

the following form

N1 {A,1}[B,1}

[.}T{A2}

i:=i

ii

N1 =

[B,l][TflT{l1},

_i=1'N1

₍₁₄₎

ii

It is noteworthy that the left-hand sides of (9) and (14) are exactly the same and hence (9) and (14) can be solved at the

same time.

-Using above results for the radiation problem, the added-mass and damping coefficients for' ail modes of motion, will be

com-puted. Then, by solving the motiOn equation of a thin upper

deck, the complex amplitude Xk appearing in (2) will be

deter-mined.

Velocity Potential at Far Field

Substituting (8), (11) and (12) into (2), all but in brackets

of (2) (which are denoted as B) may be expressed in the form

As a next step to obtain the wave drift force and moment

by means of the momentum-conservation principle, we need to rewrite the above expression with the global coordinate system

Q-xyz (or equivalently0-rOz).

At a large distance from the structure, evanescent wave com-ponents decay, and thus we can consider only the progressive wave components (i.e. the Hankel function of the 2nd kind) in the vector expressed as

Radiation Problem

_N1

in the following form

The body boundary condition for the j-th actual body is given

=

_{A}Tfr}

i

072

(is)

{A}

=

{A}

-

K({R1}

+ {,1})

_}

Fig 2 Symbols in the coordinate transforrñation Equations expiessed with a cylindrical coordinate system of the j-th body must be rewritten in terms of the global coordinate. system O-iOz. For that purpose, using notations hown in Fig. 2

and noting that r » L0 at a far field from a structure, Graf's

addition theorem gives the following:

H,(kor1) e_mei

=

_p=

.J,1...(koLo) e_i(m_j0

x {H2)(kor) e°}

This relatión can be expressed in a matrix form as follows:

{9t} = [M:oJ{s}

Substituting the above into (15), one can obtaiñ an expression

fôr ØB:

=

{AT[,M;o]{S}

{ } {il's} (18) where

{A} =>2 [MJO]T[{A}

:jf. ({i.} +{A})]

(19)

Let us express the components of the above vector {A} as

Am (m = 0, ±1, ±2, ...). Then, as clear from (16), the vector

{il-'s} comprises only the Hànkel function and hence q in (18)

can be written in a simpler form

=

>2 Am{Z0(z) H$,(kor)e_me}

On the othèr hand, with the global cylindrical coordinate sys-tem, the incidentwave potential expressed by (3) may be written

as

= >2

am{Zo(z)Jm(kor)e_imo} (21)

- r

where am

im(ß)

(22)

Noting that the sum of (20) and (21) gives the total velocity potential in brackets of (2), we can write Ø(x, y, z) valid at a far.

field in the following fòrm:

= _{Zo(z) {rimJm(kor) ± AmH)(koT)} e_ime (23)}

WAVE DRIFT FORCE AND MOMENT

Following Maruo4 and Newman61, let us derive calculation formulae for the wave drift forces m the horizontal plane and the drift yaw moment on the basis of the conservation principle of

linear and angular momènta. A notable feature in the present

paper is to perform all necessary integrations analytically using

(23)

Firstly, let us consider the time-averaged steady force acting in the x-axis. Retaining quadratic terms in the velocity potential and taking time average over one period, an expression for the steady force can be obtained in the form

-, az,

.tte

P

f

f

r..

(5Ô}

_{VVcosO] rdO}

2i

J0 L

i.ôxOr

-

Kfcb'rcosOd9

(24)

where

where çi denotes the complex conjugate of q5.

Since the integrals in (24) are to be evaluated for large values of r, we can discard the terms decaying as r - . Taking this

mto account and performing the integral with respect to z, it

follows that

=

±kØ]rcos8dO

(25)

k2

Co=K+h(k°2K2)

(26)

The wave drift force in the y-axis can be analyzed in the same manner, and the expression corresponding to (25) is given by

= J2,T [ _ + k q5 r sin O dO (27)

As shown by Newman61, the wave drift moment about the z-axis can be evaluated by applying the conservation principie of

angular momentum; which gives the following expression

=

__ReJrdo

(28)

To perform integrations with respect to O in (25), (27) and (28), we substitute (23) and use orthogonal relations in

trigono-metric functions given as

p2

I imO mO

(20)

Jo e e dO=2irömn

2ir

eTmO e'0 cosO dO= ir àm,n±i (29)

L

2,r,

e-imB ¡nOe sinOdO=7rzömn±j

Here 6m;fl is Kroenecker's delta, equal to I when m = n and zero

Noting thatZo(z)= i atz = 0, the result after applying (29)

to (25) takes the form

Re irk0r

x{{amJ'n +AmH'}{c4+iJni A+1H1}

{amJm + AmHS,2}{crn+iJm+i + (30)

The above equation can be transformedfUrther using the

for-mulae of Wronskians given by

7! 7! 7 7

,nm-f1 +JmJrn+1

J,Ç,H1 + JmH2j

_{= ;}

ilTI1 +

_{= 7rkor} (31)

H'J,Ç,1 +HS,)Jm+i =

_7rkor

Then the final result for the wave drift force along the x-axis can be obtained in the form

=?[m

-X j2AmAn+i +

+ Amcxi

1 (32)

With almost the same transformation, the calculation formula for the drift force along the y-axis may be given in the form

pgA2 k0

Re

2 C0K

moo

Concerning the steady yaw moment given by (28), substitution

of (23) and implementation of necessary calculations using (29)

gives the following:

M

pgA2 it/car 2 C0K

> [AmAn + Re(amÄn)] (JmY,,i J'nYm) (34) Here denotes the second kind of Bessel function of order

m, and the following formula of Wronskian exists:

T T

f

Jm'm Jm'm _7rkor

Substituting this formula in (34) gives the final expression for the steady yaw moment in the form

=

_{_pgA2jj >}

_m[itm

₊

₍₃₆₎

It should be noted that (32), (33) and (36) includes only the coefficients of the disturbance potential due to floating columns, Am, and of the incident-wave potential, Qm explicitly given by

(22) These formulae are one of the important results in the

present paper. We can see that the steady drift force and moment

consist Of quadratic terms in the disturbance and cross ternis between the incident wave and the disturbance.

OUTLINE OF EXPERIMENTS

Although the calculàtion method in the previous sectiôn is

intended for a VLFS supported by à great number of floating

columns, experiments were conducted with 64 equally-spaced cylinders because of limitations in the tank size and other

fa-cilities.

As shown in Fig. 3, an elementary body is a circular cylin-der with horizontal base and its diameter (D = 2a) is 114 mm This cylinder was placed in a periodic array with 4 rows times 16 columns The separation distance between the centerlines of adjacent cylinders is denoted as 2s, and the experimental setting was such that s =

D

in both x= and y-axes andß= 00 (i.e. head

waves only).

-In this experiment, motions were completely fixed and, as

shown in Fig.3, the x- and z-componénts of the wave force were measured by dynamometers at two different positions. The draft of the cylinders was set to d = 2D, considering the capacity of the dynamometers used.

The experiments have been carried out at Ocean Engineering Model Basin (length 65 m, breadth 5 m, water depth 7 m) of Re-search Institute for Applied Mechaniics, Kyushu University. The steepness of regular waves (the ratio of wave height with wave

length, H/)) was Set approximately equal to 1/50. The circular

wave frequency ú =

/R')

was changed iii the range of Ks

0.2-1.6, considering that an important parameter in

hydrody-namicmteractions is Ks (= 2irs/A) Measured data were

ana-lyzed using an ordinary Fourier-analysis technique, from which the wave drift force in the x-axis was obtained in addition to the linear wave-exciting forces in the z- and z-axes oscillating with circular frequency c.

RESULTS AND DISCUSSION

Computations corresponding to the experiments with 64

columns can be- perfornaed 'withOut introducing hierarchical

lev-eis. That is, L = I and thus N = NB = 64.

In actual computations, the number of Fourier series in the 9-direction (M) and of evanescent wave modes (N) must be

fi-nite. In the present paper, M = 4 and N = 3 are chosen after

convergence checks for Ks := 1.0,_ß= 00 and h = 3d, for which

five decimals absolute accuracy has been achieved.

The number of t tal unknowns for M = 4, N 3, and N8 =

64 is (2M +1) X (N + -1) X N8 = 2304. The computation time in this case may be very long, if detailed computations must be carried out at many frequencies. Therefore, double symmetries with respect to the z- and y-axes are explOited, which canreduce the number of unknowns to 1/4 (ie. 2304/4=576).

When computing the coefficients of disturbance potential due to floating cylinders given by (19), we must compute the

trans-formation matrix _{[MJ0 J} defined by (17). As shown by (16), the

elements in this matrix comprise Bessel functions. The

conver-gence rate in the series expansion of (16) is very slow. It is found

by pilot computations that the value of p (terms on the

right-hand side) must be more than six times the value of m (terms

-on the left-hand side) for Obtaining sufficiently c-onverged results.

For example, in the case of M = 4 (2M + i 9), the number of

terms in {'s} and thus {A} will be P 6M = 24 (2P+1 = 49).

In the present paper, all computations have been perfOrmed with

P = 8M, i:e. 2F + i = 65, to ensure accurate results even in

high frequencies.

e 114 Unit:mm

4-Eb-4-E

OO -c--

Ô

G-8 E?-G-+ !E

t

Wave (fi =0) 4 5 6 7 8

' 9

10 11 12 13 14 15 16

z

Fig. 3 Arrangement of 64 truncated cylinders fixed in head waves and measurement system

Linear Wave-Exciting Force

To confirm validity of the present calculation method, the re-salts of liiear wave-exciting forces acting on 64 cylinders are

shown in Figs. 4 and 5.

Figure 4 is concerned with the surge exciting force. In the

frequency range less than Ks 1.2, we can see regular fluctu-ation due to hydrodynamic interactions. On the other hand, at frequencies higher than Ks 1.2, the variation pattern changes,

which is also clear from the phase difference.

In fact, measurements of the wave elevation along the center-line of the present model reveals that approximately Ks = 1.24 corresponds to a trapped-mode frequency of Neumann type, dis. cussed by Maniar & Newman3 (although the results supporting this fact are not shown here). Wave forces on each cylinder also change drastically near this critical frequency.

Figuré 5 shows the heave exéiting force. As the frequency increases, the amplitude of the force becomes very small,

be-cause variatiOn in the pressure may be confined to the vicinity of

the free surface, not contributing to the vertical force. However, variation in the phase tells us that a rapid change occurs near Ks = 1.24, corresponding to a trapped-mode frequency.

At any rate, it can be aid that the overall agreement is satis-factory between experiments and computations.

Drift Fórce in Head Waves

Figure 6 shows the drift force in thex-axis in head waves (ß = 00), which is nondimensionalized in terms of the wave-energy density of a regular wave (0.5pgA2) and the projected length of cylinders along the y-axis (the diameter times the number of cylinders in the y-axis = D x NEY = 0.456 m).

Measured results somewhat scatter, probably because of

diffi-culty in measuring small physical quantities with the apparatus shown in Fig. 3. However, at least in order and qualitatively, the results agree favorably with computed results

In lower frequencies, measured results are obviously higher than the calculation, which may be attributed to viscous effects not mcluded in the present theory In the frequencies less than

dla=4, s/a=2, ß=O deg. D Fz(2) -1 : I Total Force on W =64 I B Calculation

-

Experiment o

A

1'

t !'

r

9OSI

1 /

80 -

s _

. -02 04 0.6 0.8 10 12 14 1.6 02 04 06 08 10 1.2 14 16 Ks

Fig. 4 Wave exciting force in surge on 64 cylinders

0.4 "4 0.3 0.2 e u il

f

0.0

0.20 0.15 0.05 180 s 0.ò0 0.2 04 -go -180 e 3.0 2.0 1.0 u) 0.0 o 06

Fig. 5 Wave exciting force in heave on 64 cylinders

7x10-5pgA2DN

4.0

08 1.0

d/a=4, ala=2, ß0 deg.

d/a4, ß=O deg.

12 14

Fig. 6 Steady drift force in surge on 64 cylinders

Ks 1.24 corresponding to the trapped mode, we can see regular

variation of hump and hollow with increasing amplitude.

In cntrast, for frequencies higher tian Ks 1.24, the

varia-tion pattern changes and relativèly large drift fOrce cad be seen. This is because the drift force is related closely to the reflection

16

of incident waves, and the reflection in high frequencies becomes

large due to sheltering effects by .a large number of cylinders, especially at frequencies higher than a trapped mode.

Drift Force and Moment in Oblique Waves

Although there are no measured results for comparison, nu-merical computations have been performed for oblique waves to understand variation tendency of the steady drift force and yaw

moment as a function of incidence angle.

To increase the resolution, computatiOns were made at 270

frequencies in the range of Ks 0.2 2.0. Computed values are in fact shown byo (open circle) for_fi 30°, and by x (cross)

for_fi.= 60° in Figs..7'9.

Firstly, looking at the surge drift force (F4 for fi 3Q0, we

dla=4, s/a=2

IxIO.5pgA2DN

5.0 --- BY

Fig. 7 Surge drift force on 64 cylinders in oblique waves

?'io.

5pgA2DN

2.0

Steady Sway Force on N =64 (N =16)

B

-

Caläulation

Ks

Fig. 8 Sway drift force on 64 cylinders in oblique waves

d/a=4, s,'a=2 o Total Force on N =64 B calculation Experiment -

--- Steady onN=64

BEY

Surge Force (Ñ=4)

111

t

iUIII

-LjT

s Steady Force Calculation on N=64

-:_t

!

f

-. 5U0

iS

.4. ..l

I iT -04 06 08 1.0 1.2 14 1.6

Ks

0.2 04 06 08 1.0 12 14 16 1.8 2.0

Ks

08 10 Ks 0.6 0.2 04 12 14 16 0.0 02 04 0.6 08 10 12 14 1.6 18 20 4.0 e o Il o 3.0 4.) '14 14 2.0 1.0 0.0

43

J

i±,o.5pg/2DN DN_Bk

BY

d/a=4, s/a=2

velocity potential valid at a far field and necessary integrations are anàlytically carriéd out; whith greatly contributes to higher accuracy in numerical results.

A defect in the theory is that no information is given

con-cerning the steady force on each column; that is, only the total force and moment can be computed. However, no matter how many cohimns are used, computation burden does not increase so much, because a hierarchical interaction theory1 can. be

ap-plied and only the coefficients of disturbance potential at highest

hierarchical level are réquired in the calculation formulae derived

in this paper.

Experiments were also conducted in head waves, using 64 cir-cular cylinders arranged in 4 rows and 16 columns with equal separation distance. Although somewhat experimental scatter exists, the overall agreement with computed results was good. Computations in oblique waves were performed and the depen-dence of the wave incidepen-dence angle on the steady force and yaw

moment was discussed.

Numerical results in this paper were just for the diffraction problem. However, it s easy to include the effects of motions of a structure, because necessary modification is to superimpose additional terms due to body motions onto the diffraction terms in evaluating the coefficient vector of the body disturbance

po-tential.

Acknowledgments

Experiments in this study have been conducted as a paìt of the master thesis of Mr S Yoshida, in the Interdesciplinary Gradu-ate School of Engineering Sciénces, Kyushu University. The au-thor is thankful for his contribution. Thañks are also extended to Mr. M. Liada for his help in preparation of experiments

-REFERENCES

Kashiwagi, M (1998). "Hychodynamic Interaátións among a Great Number of Columns Supporting a Very Large Flexible Structure", Proc 2nd Intl Conf on Hydroelasticity, Fukuoka,

pp 165-176

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::

.

..

. ._..._.-,-..-__-4-_. . . L ' - Steady N =64 B

-Yaw Moment on (N =16,.N 4) BZ BI Ca1ci1ation ß=3Ode

IP

AI&

iumii

60d

02 04 06 08 10 1.2 14 1.6 18 20 Ks

Fig. 9 Yaw drift morient on 64 cylinders in oblique waves can see that the amplitude of hump and hollow in lower frequen-cies decreasès compared to the head-wave case and a large drift force is predicted in higher frequencies. For

_fi

= 60°, occurrence of hump and hollow cannot be seen and the value itself is smáll; this is obviOusly because the waves reflected in the x-axis are

very small.

Secondly, looking at the sway drift force (F5), relatively slOW

variation can be seen forfi = 600 in lower frequencies. This is because the number of cylinders along the y-axis is 1/4 of that along the xaxis and thus the number of peaks due to hydrody-naxnic interáctions may be small (approximately equal tó 1/4).

In higher frequencies, a large drift force is predicted over a

some-what wide. range of the frequency around Ks = 1.6.

Lastly, in the steady yaw moment (M), we note that the mo-ment becomes positive or negative for both /3 = 300 and 60°, depending on the value of Ks. Variation tendency for /3= 30°

is similar tó that of F, and rapid changes are observed near

Ks = 1.47 and 1.85. Meanwhile, for/3 60°, rapid variatión is

not observed, which is of the same feature as that of F5. These results may be understood by considering that the cylinders are arranged in 4 rows and 16 columns in the present case and there-fore hydrodynamic interactions along the r axis are more corn-plicated than those along the y-axis.

CONCLUDING REMARKS

A calculation method and numerical results have been

demon-strated for the wave-induced steady drift force and moment on a great number of columns. The present theory is based on the conservatiOn principle of linear and angular momenta. There-fore, calculations of the forée and moment can be done using the