The hydrodynamic behaviour of long floating structures in directional seas

E L S E V I E R 0 1 4 1 - 1 1 8 7 ( 9 5 ) 0 0 0 1 1 - 9

Applied Ocean Research 17 (1995) 233-243

Copyright © 1996 Elsevier Science Liinited Printed in Great Britain. All rights reserved 0141-1187/95/$09.50

The hydrodynamic behaviour of long floatin

structures in directional seas

S. A. Sannasiraj, V . Sundar & R. Sundaravadivelu

(Received 15 May 1995; accepted 27 July 1995)

The diffraction/radiation boundary value problem arising from the interaction of oblique waves with freely floating long structures is studied using finite-element techniques. Further, the hydrodynamic behaviour of two-dimensional horizontal floating structures under the action of muhi-directional waves has been studied. The hnear transfer function approach is used to determine the wave exciting forces and motion responses of a structure of finite length in short-crested seas. The directional spectrum is obtained from a unidirectional spectrum with an associated frequency-dependent or frequency-independent cosine power-type energy spreading function. Based on the numerical predictions, the motions and forces on a rectangular floating structure experiencing unidirectional and multi-directional wave fields are computed.

1 I N T R O D U C T I O N

Due to the rapid depletion of resources on land, exploitation o f ocean resources is expanding rapidly. I n order to explore and exploit these ocean resources, a variety o f floating structures, like floating bridges, floating dry docks, floating breakwaters, etc. are being developed. The study of the hydrodynamic behaviour o f floating structures, numerically as well as through experiments with monochromatic waves, was i n vogue f o r several years until the importance o f random sea states was reahsed. By the late 1970s, researchers had started scientific work on the behaviour of floating bodies i n multi-directional sea states. As testing structures i n short-crested waves i n laboratories is quite expensive, the task o f numerical studies is easier, due to the rapid development of computing facilities.

Induced forces due to waves, as well as the transmis-sion behaviour concerning the motion o f floating structures, can be decisive factors i n determining their dimensions and thus lead to their practical reahsation. The importance o f studying structures i n three-dimen-sional waves has been discussed by several investigators. Notable are the contributions of Isaacson and N w o g u , ' in which the reduction o f forces by about 40% on long floating cylinders i n sway and roll has been reported.

The first step i n the study of the motion response o f long floating structures i n directional seas is to solve the problem associated with the interaction of regular oblique waves with the structure. Bolton and UrselF

considered the generalised heave problem i n infinite depth water and presented the hydrodynamic coefla-cients for larger wave numbers i n asymptotic form. Bai and Leonard et al^ used a finite-element technique to solve the oblique wave-structure interaction problem for finite depth water, whereas Garrison^ apphed the Green's function procedure f o r infinite depth water.

Studies of the effect o f directional waves on long structures have been carried out by several investigators. Battjes'' presented load reduction factors (ratio o f the directionally averaged load i n a directional sea to the load i n a unidirectional beam sea) for a vertical wah and a pipeline. Georgiadis' used a Monte-Carlo simulation process to determine the nodal forces on floating bridges. Hutchison^ analysed floating bridges and breakwaters i n short-crested seas by two principal methods—hnear superposition of responses to long-crested components o f the directional spectrum, and beam sea responses modified by a scalar coherency function. Battjes,^ Georgiadis'' and Hutchison* assumed the hydrodynamic coefficients and wave exciting forces for obhque waves to be the same as that for beam seas.

A systematic set o f experiments was carried out by Scheffer and Fittschen^ to describe the effects of water depth and wave approach angle on the motion behaviour o f ships. A study of the effects of wave direction on the first-order and second-order low-frequency motions o f the floating body i n multi-directional irregular waves has been done theoretically and experimentally by Maeda et al}° Based on

234 S. A. Sannasiraj et al. 4^

i

Fig. 1. Definition sketch.

experimental observations, it was reported that the first-order motions i n two-dimensional irregular waves were nearly equal to the summation of the first-order motions i n each unidirectional irregular wave. Isaacson and N w o g u ' developed a generalised numerical procedure based on Green's theorem to compute the exciting forces and hydrodynamic coefficients due to the interaction o f a regular oblique wave train w i t h an infinitely long, semi-immersed floating cylinder of arbitrary shape in the frequency domain. The wave loads and motions o f a rigid structure with a frequency-independent cosine power spreading function were computed using the linear transfer function technique.

The present study deals with the computation of the exciting forces and hydrodynamic coefficients o f an infinite semi-immersed cylinder of arbitrary shape i n oblique seas based on the finite-element method. The responses of the structure in the three modes, viz. sway, heave and roll, i n directional seas have been evaluated using a hnear transfer function approach i n the frequency domain. The force ratio (the ratio o f the directionally averaged load i n a directional sea to the load i n a unidirectional beam sea) and the response ratio [the ratio of root-mean-square (rms) response i n directional waves to the rms response i n long-crested normally incident waves] in the multi-directional sea state condition have been presented i n terms of the spreading parameter, mean direction and the non-dimensionahsed frequency. T w o types o f directional spreading function—a cosine power type spreading function with the spreading index both inde-pendent and deinde-pendent of frequency are considered here.

2 T H E O R E T I C A L C O N S I D E R A T I O N S 2.1 General formulation

Fluid is assumed to be ideal, flow is considered as irrotational and the application of linear wave theory is valid. The body is assumed to be rigid. I t is assumed that no flow o f energy takes place through the bottom surface or the free surface. Energy is gained or lost by the system only through waves arriving or departing at infinity or the external forces acting on the body. The motions are assumed to be small, so that the body's boundary conditions are satisfied very close to the equilibrium position o f the body. A Cartesian coordi-nate system is employed, with the origin in the mean free surface, Qz directed positive upwards and O x directed w i t h positive i n the direction o f propagation o f the waves. The state o f t h e fluid can be completely described by the velocity potential, $(x, y, z, /) satisfying Laplace's equation.

v2$(x, y, z, t)=0 (1) The general configuration of an infinitely long floating structure interacting with a monochromatic oblique hnear wave o f height H, and wave angular frequency,

Lü = 2n/T is shown i n Fig. 1. Waves approach the

structure at an angle 9 with respect to the normal to the body axis. I t is generally convenient to separate the total velocity potential into incident potential, $ i ; radiation potentials, j = l , 2, 3, i n three modes viz. sway, heave and roll; and scattered potential, $ 4 . This is mathematically represented as

<f>. $4 (2)

where $y = Xj(j)j in which t^^ is the radiation potential per unit body velocity, Xj.

2.2 The diffraction problem

For the present wave structure interaction problem, the incident velocity potential is represented by

$ i ( x , y, z, t) = Re AHg cosh /c(z + d)

2LU cosh led

X exp{i(/cA' cos 9 + ley sin 9 — wt)} (3) where g is the gravitational constant, d is the water depth and /c is the wave number satisfying the dispersion relation

= g/rtanh iid (4) The linear wave diffraction problem is described by a sinusoidally varying diffracted velocity potential i n time

Long floating stnictures in directional seas 2 3 5

as well as along the axis of the structure and given by

*D('^-, y, z, 0 = ^i{x, y, z, I) + $ 4 ( x , y, z, t)

= Re[{,^i(x, z) + ^4(x, z)}

X exp{i(A:^ sin 0 - ul)}] ( 5 )

The three-dimensional Laplace equation reduces to a two-dimensional Helmholtz equation f o r the sinusoid-ally varying potential along the axis of the structure. The boundary value problem for the diffracted potential can be defined by the governing Helmholtz equation and the boundary conditions as defined below:

7^04 - (/csin 61)^4 = 0 in the fluid domain Ü;

dc/>, u?

04 = 0

dz g

at the free surface, Lp, z = 0;

d4>^

( 6 )

(7)

dz 0

at the sea bed, L B , Z = - d ; T i/c cos 9(j)^ = 0

dz

( 8 )

( 9 )

at the radiation boundary, T^, x ^ ± o o ; and

dn dn ^^"^

on the body surface, T^, where n is the unit outward normal f r o m the fluid domain. The infinite boundary Too is fixed at a finite distance, x = XR. A detailed discussion of the distance o f the radiation boundary is given by Bai.^

2.3 The radiation problem

The wave radiation problem can also be described by a sinusoidahy varying radiated potential both in time and along the axis of the structure which is given by

$y(x, y, z) = RQ[-iojXj(l)j{x, z ) e x p { i ( / c j s i n 6 ' - w ? ) }

( 1 1 )

The linear radiation boundary value problem is defined by the Helmholtz equation as a governing equation and the boundary conditions as given below:

V in n (/c sin ( 0 d^ dz at T F , Z = 0; -4>i = 0 dz 0

(12)

at Too > oo; and ~ =F ik cos = 0

dn

( 1 5 )

( 1 6 )

on Ta, where ni and «2 are the x - and z-components of the unit inward normal to the body, and

«3 = {X - X j « 2 - (Z - Z c ) " l ( 1 7 )

in which ( x c , z^) are the coordinates o f the center o f rotation.

The Helmholtz equation and the boundary conditions stated above f o r the radiation problem describe a flexural wave which travels along the longitudinal axis of the structure and generates an oblique wave i n the water.

For 0 = 0°, the above problem reduces to a beam sea condition. I t is applicable only for 9 < 90°. This is due to the fact that the radiation boundary condition becomes a rigid wall boundary condition when 6» = 9 0 ° . Thus, the special case o f a head sea condition can only be described by using three-dimensional wave diffraction/ radiation theory where the end effects can be taken into account.

2.4 Hydrodynamic forces

The hydrodynamic pressure at any point in the fluid can be expressed as

P{x, y, z, t) - p — = iwp$ 9 $ . ( 1 8 )

where p is the mass density of fluid, The hydrodynamic forces can be determined by integrating the pressure over the wetted body surface TQ.

FJ pnj AT ( 1 9 )

where 7 = 1, 2, 3, corresponding to sway, heave and roh modes, respectively.

The hydrodynamic forces thus evaluated can be separated into wave exciting forces governed by the diffraction problem and hydrodynamic restoring forces governed by the radiation problem.

The wave exciting force, F / , due to the diffracted potential can be expressed as

F j = iijp ^ ( $ , + $4)7?; d r = Re [/;exp{i(/c^ sin 9 - ujt)}

(20)

where f j is the complex force amplitude. From the radiation potential, the hydrodynamic restoring forces,

Fj^ can be evaluated as

2 3 6 S. A. Sannasiraj et al.

where is the added mass coefficient proportional to the body acceleration and Xjk is the damping coefficient proportional to the body velocity, iijk and A^^t are evaluated f r o m the real and imaginary parts of the complex radiation potential, respectively.

The relative length of the structure (kL) plays a vital role i n the force reduction on the structure in oblique waves which results in lesser motions. The total wave exciting force, Fj, on the structure is evaluated by integrating the two-dimensional sectional force along its length, neglecting the end effects,

r i / 2 Fj{t) - L / 2 Fj{y, t) Ay 2sin((/cL/2)sing) '^^ /csinö (22) is the complex amplitude of the total exciting force exerted on the f u l l length of the structure. The above expression can be written in terms of a force multiplier 7(/cL, 6) for a particular relative length of structure.

_ 2{sin[(/cL/2)sin6']}

kL sin 6 (23)

For a particular value of kL, the force multiplier -yQcL,

6) indicates the ratio of the wave force due to obhque

wave attack to that due to normally incident waves (0 = 0°). The added mass and damping coefficients for a two-dimensional rigid structure of finite length can be obtained by simply multiplying the corresponding coefficients calculated across a cross-section of the structure i n a beam sea by the length of the structure.

2.5 The motion response problem

The response of the structure i n waves with amplitudes varying sinusoidally in time, as weU as along the axis of the structure, can be evaluated f r o m the basic equations of motion given as

[Mjk + Ujk] + iw [Xjk] + Cjk] Sk = (24)

i n which j, k = 1, 2, 3 refers to the sway, heave and roh modes, respectively, Mjk and Cj^ are the body mass and hydrostatic restoring force matrices as described by J o h n " and Sf. is the complex amphtude of the motion response, X j . = (5^e""^'. The response of the body can generahy be expressed in terms of the response amphtude operator, RAO, defined as the response of the structure in a unit amplitude wave.

RAOkiu,,

-H/2' 1, 2, 3 (25)

3 N U M E R I C A L PROCEDURE

I n order to solve the above diffraction/radiation boundary value problems, solutions are sought using

the standard finite-element method. The infinite fluid domain is made finite by incorporating either a plane or a higher-order boundary damper at the radiation boundary which is at a finite distance, X = ± X R f r o m the structure. The use of a plane boundary damper to absorb the outgoing waves at the radiation boundary requires a larger fluid domain. To limit the size of the fluid domain, matching analytical boundary series solutions (Yue et al}^) or matching boundary integral equation solutions (Zienkiewicz et a/.") can be con-sidered at the radiation boundary. This would result i n a larger bandwidth and hence become tedious to solve. One has to, thus, model the radiation boundary according to the requirement of the solutions sought and the availability of computing facihties. For the present study, a plane damper is used to model the radiation boundary.

Bai^ has formulated the finite-element systems of equations for the diftraction problem using the varia-tional principle. For the present study, finite-element formulations for the diffraction/radiation boundary value problem i n the two-dimensional vertical plane have been done based on the Galerkin approximation. The fluid domain is divided into discrete elements with a total of M nodes. The velocity potential, is approximated by a linear combination of interpolating functions and may be represented as

"e

% = Y,N,.(j)jrexp{i{kysm9 - uJt)}, j = 1, 2, 3, 4 (•=1

(26) where Nr{x, z) are the shape functions and n^ denotes the number of element nodes. Using the Galerkin approx-imation, should satisfy

0 (27)

Application of the divergence theorem and using eqns (6)-(10) and eqns (12)-(16), the finite-element formula-tion of a diffi-acformula-tion/radiaformula-tion problem leads to linear algebraic equations of the f o r m

[K'+ + K']{<Pj} = {Pj} (28) where

Cl'

'{VN,){VN,) + {k sine f N , N ,

(29) where e denotes a finite-element. The summation implies assembly of the element property matrices over the entire domain Ü. Further, the contributions by the free

Loiig floating stnictures in directional seas 237 surface and radiation conditions are as follows

.2 ^ W- "

N,N,

d r

Kl = J2Tik cose

The load vector is given by

7 = 1 , 2, 3, 4

(30)

(31)

(32) where 7 = 4 corresponds to the diffraction problem and 7 = 1 , 2, 3 correspond to the radiation problem in three modes viz. sway, heave and roll. For the radiation problem, the load vector becomes

N,nj dV J=h 2, 3 (33) The load vector f o r the diffraction problem is given by

i v , ^ d r

N,"-£ér- (34)

The assembly of the element matrices has been done in the usual manner (Zienkiewicz and Taylor''') and the resulting simuhaneous equations can be solved for <pjs,

s=l, 2, . . . , M, taking advantage of the symmetric and

banded nature of the tnatrix. A n eight-noded isopara-metric element has been used i n this study to formulate the system matrices.

4 R E P R E S E N T A T I O N O F A D I R E C T I O N A L S E A

The sea surface elevation is assumed to be a zero-mean, stationary, random, Gaussian process. The multi-direc-tional wave climate can often be modelled by a linear superposition of long-crested waves of all possible frequencies f r o m all possible directions, represented as

7?(x, y, t) = Re

+ kjy sin öj _{w,/ + e y ) }} (35) where /c,-, w,- and ay denote the wave number, frequency and amplitude of the ith wave component approaching a given location (x, z) f r o m the direction 9j. e,y are random phases with a uniform probability distribution and

^/2S{uj, 9) Aoj A6 ( 3 6 )

S(uj, 9) is the specified directional wave spectrum and

can be expressed as a product of an energy spreading function D{uj, 9) and the specified unidirectional spectrum, S{(x)).

S{<JO, 9) = S{LÜ)D{W, 9) ( 3 7 )

I n the present study, the well-established frequency-independent cosine power type spreading function is used which is defined by,

f Nis)co^^'i9 - 9o) f o r \9 - önl < 7r/2

D{9) = (38)

[ 0 otherwise

where 9o is the mean wave direction, N(s) is the normalising coefficient defined in terms of the gamma function F, and s is the spreading index describing the degree of wave short-crestedness with ^ ^ oo represent-ing a long-crested wave field.

Nis) = 1 r ( ^ + i )

A r ( ^ + o-5) (39) Since the above formulation considers the spreading function to be independent of frequency, representation of a directional spectrum close to reality is questionable. ReaUsing this problem, Goda and Suzuki'^ defined a frequency-dependent spreading function which basically has the same f o r m as the cos^''(6'/2) distribution except that the spreading index, s, is defined as

^ p ( / / / p ) ' f o r ƒ < / p ^ p ( / / / p ) - ' - ' f o r / > / p

(40)

where /p is the peak frequency, and the spreading parameter, Sp, belongs to a universal set of constants applicable to different types of sea state: 5 p = 10 f o r wind waves; 25 f o r sweh f r o m a near distance; and 75 f o r sweU f r o m a far distance.

I n the present context, the spreading index s ( f ) has been used with the spreading funcdon defined by eqn (38). The directional sea state described by the spreading index independent of frequency, as well as that represented by eqn (40) have been employed.

5 T H E E F F E C T O F D I R E C T I O N A L W A V E S

The exciting force on the rigid structure under the action of an oblique wave can be expressed as

Fj{t) = Hjiuj, 9)^{t) (41)

where Hj{LO, 9) is a complex-valued linear transfer

function defined as, f r o m eqn (22),

Hjiuj, 9) = C,(^, 0)

HJ2 (42)

Assuming that linear wave theory is valid, and that the transfer function between the wave and the structure's motion is linear, the force spectrum f r o m the incident wave spectrum is estimated as

\Hj{w, 9)YSr,{u, 9) d9 (43) Substituting eqn (37), the above expression f o r the force

238 S. A. Sannasiraj et al. 5.00 1.50 2.00 1.00 H > o (a O ^ 0.50 0.00

(b)

/ '

* \

Fig. 2. Response amplitude operators as a function of frequency in oblique waves

spectrum takes the f o r m

S^j{u) \Hj{uj, e)\^D{uj, 6) d9 ( 4 4 )

The rms value o f the force generally represents a characteristic force. The ratio o f the frequency-depen-dent, directionally averaged transfer function i n short-crested seas to the transfer function f o r long-short-crested, normally incident waves is given as

The body response ratio, RM can also be defined as the ratio of the rms value o f the response i n short-crested waves to the rms value of the response i n long-crested normally incident waves.

R

\RAOj{u), 9)fD{uj, 9)S^{uj) d9 dw

MJ ( 4 6 )

\RAOj{uj, 'S,{u) duj

\HJ{LÜ, 9)\^D(ÜJ, 0) d9

( 4 5 )

The authors feel that defining the above as a force reduction factor, as defined by Battjes,* is misleading since i t simply relates the force on the structure in short-crested seas to that i n long-short-crested seas. Hence, i n the present study, the above parameter is defined as a force ratio.

I n the present context, i t is assumed that the waves approach the structure f r o m one side only.

6 RESULTS A N D DISCUSSIONS

The numerical procedure described above has been solved using an eight-noded isoparametric element. This element can best represent the variation o f the poteiUial within the element and ensure inter-element

Long floating structures in directional seas 239

Table 1. Comparison offeree ratios in directional waves (0o = O° and s = l )

Sway Heave Roll

lea Isaacson and Present Isaacson and Present Isaacson and Present Nwogu' Nwogu' Nwogu'

0 0-866 0-866 1-00 1-00 0-866 0-866 2-0 0-41 0-43 0-40 0-41 0-41 0-43

compatibility. The radiation boundary is fixed at a distance of 5-5fl (Leonard et al^) f r o m the body's surface irrespective of the angle of wave attack.

6.1 The response amplitude operator

The response amplitude operator, RAO, as a function of

ka, f o r the three modes of motion in oblique waves, 0 = 0, 30 and 60° is presented in Fig. 2(a)-(c). A freely

floating rigid rectangular structure of breadth £=l5m, draft T= 3 m and length L = 75 m as studied by Isaacson and N w o g u ' is analysed with the centre of mass 1 m below the still water level. Since the structure is assumed to be rigid, the hydrodynamic added mass and damping coefficients used for all oblique angles o f wave attack correspond to the beam sea condition (0 = 0°). The agreement between the present results and that o f Isaacson and N w o g u ' for sway RAO is found to be good except for shght deviations noticed in a narrow range o f ka between 0-8 and 1-2. The discrepancy in the results is attributed to the occurrence of roll resonance in this regime. I t is also noticed f r o m Fig. 2(c) that the wave approach direction essentially reduces the roh resonance maximum amplitude while the roll resonant frequency remains unchanged. The comparison o f heave

RAO [Fig. 2(b)] is found to be good for ah the obhque

waves tested. The sway, heave and roh RAOs for oblique wave attack have zero amphtudes at certain ka values. This would occur when the encountering wave length is an integer multiple of the body length.

6.2 The force ratio

The variation of the force ratio, i^p, for the three modes with the scattering parameter f o r the various frequency-independent spreading indices in a directional sea associated with a mean wave direction, 0o = 0°, is presented in Fig. 3(a)-(c). The numerical results of Isaacson and N w o g u ' are also incorporated in the respective plots. The agreement found among the resuhs is good. The-force ratios at l<:a = 0 and /c<7 = 2-0 are tabulated in Table 1 with the results of Isaacson and N w o g u . '

Battjes^ derived a common asymptote f o r force ratio as /cL ^ oo .

Rl=^N{s)cos'% (47)

This is due to the fact that with increasing kL, there is an

increasing phase cancellation for all oblique compo-nents. Hence, i t is observed that the contribution to the force ratio is mainly due to the higher frequency spectral components closer to the normal angle of wave incidence. Hence, the force ratio is independent of the term cos^ö. The variation of RAOs being very smah f o r increasing angle of wave attack, as discussed earlier, supports the above observation. For the reasons stated above, the asymptotic function is certainly valid i f i s also frequency dependent. Further, at /c(7= 3-0 and ^ = 1, the force ratio attains a value of 0-359 i n sway and roh modes, and 0-346 in the heave mode against the asymptotic value of 0-365. From the plots, it is seen that the spreading index does not influence the force ratio for heave at lower frequencies, whereas f o r the other two modes, it influences the variation of Rp throughout the entire range of frequencies, being significant in the case of the higher frequency compo-nents.

The force ratio f o r the frequency-dependent spreading function, defined by / p = 0-2Hz, 5p = 10, 25 and 75, is plotted as a function of ka i n Fig. 4(a)-(c) in the sway, heave and roh modes, respectively. The wide range of s^, chosen covers a wider range of wave directionality i.e. f r o m short-crested waves to nearly long-crested waves, indicating swch. The results have been obtained for a mean wave direction of zero. From the plots, it is seen that the sway and roll force ratios approach a value of 0-707, while the heave force ratio approach a value o f 0-971 at extremely small ka values (ka^O), where the variation of does not affect Rp. For the entire frequency range considered, the sway and roll force ratios are of same order of magnitude for all the spreading parameters.

I t is interesting to note that the force ratio increases up to ka= 1-25, which corresponds to the selected peak frequency after a slight decreasing trend near the lower frequency range. This is attributed to the fact that waves closer to the dominant frequency are expected to be long crested, whereas waves with frequencies away f r o m the dominant frequency are expected to be short crested. This has been modelled by the frequency-dependent spreading function defined by Goda and Suzuki.'^ Beyond the peak frequency, the force ratios show a steady decreasing trend up to 0-454 in the sway mode and 0-443 i n the heave mode at ka = 3 f o r 5 p = 1 0 , as against the asymptote of 0-469. As ka increases further,

240 S. A. Sannasiraj et al.

force ratios for the three modes considered are found to increase with increase of ^p.

The sway and heave force ratios for the different mean wave directions (ÖQ = 0, 3 0 and 6 0 ° ) for a constant Sp of

1 0 as a function of ka are shown in Fig. 5(a) and (b), respectively. The variation o f the roll force ratio is not depicted i n the hgure as it is close to the variation of the sway force ratio. The force ratio shows a decreasing trend with increase in lea f o r the short-crested seas described by öo = 30 and 60°. The rate of its decrease is observed to be drastic up to /ca= 1 - 25 corresponding to the peak frequency. This is due to the fact that the waves are restricted to approaching the structure f r o m one side only. This would mean that only the pordon of the spreading function that spreads the energy in the range of direction - 7 r / 2 <{9- 9Q) < wjl is considered. As

ka-^0, the sway force ratio tends to 0 - 6 9 1 and 0 - 6 1 7

for 00 = 3 0 and 6 0 ° , respectively, while in the heave mode, it tends to 0 - 8 8 1 and 0 - 7 8 1 f o r 00 = 30 and 6 0 °

respectively. I t is noticed that at frequencies beyond the peak frequency, that is, at /ca = 3-0, Rp for the sway force approaches a value o f 0-297 f o r 9o = 30° and 0-084 for 0o = 60°. The Rp values for the heave force approach a value of 0-283 for 6io = 30° and 0-059 f o r 9o = 60° as against the asymptotic values of 0-295 and 0-051, respectively.

6.3 The response ratio

The response ratio as dehned earher is computed f o r a long floating structure subjected to a Bretschneider spectrum with both of the spreading functions discussed above.

The Bretschneider wave spectrum is defined as exp (48) where Hs is the significant wave height.

Long floating structures in directional seas 241 1.00 0 . 8 0 0 . 6 0 D 3 0 0 ' 0 . 4 0 0 . 2 0 0 . 0 0 0.00 1.00 2.00 3.00 1.00 0 . 8 0 H > O OJ 0.60 ' 0 . 4 0 H 0.20 0.00 ko 0.00 3.00 1.00 0.80 , 0 . 6 0 O ' 0 . 4 0 H Ü1 0 . 2 0 0 . 0 0 0.00

Fig. 4. Force ratios for the frequency dependent spreading function. öo = 0°

I n the present study, a peak frequency of /p = 0 - 2 H z and a significant wave height of jys = 2 m are used. The response ratios are plotted as a function o f s i n Fig. 6 and as a function of the mean wave direction, ÖQ, i n Fig. 7 f o r the frequency-independent spreading function. Response ratios of 58-5, 58-6 and 63-0% i n the sway, heave and roh modes are observed as against response ratios o f 57, 57-5 and 58-5% found by Isaacson and N w o g u ' f o r spreading index 5 = 1 and f o r öo = 0°.

From Fig. 7, the response ratio decreases as the mean wave direction increases. This is due to two factors: (i) the motion response of floating structures is at a maximum when exposed to a beam sea; and (ii) the waves are assumed to act f r o m one side of the structure and to ensure this the spreading function is cut off.

The response ratios for the frequency-dependent spreading function as a function of spreading index, are given i n Fig. 8 and as a function of mean wave

direction are given i n Fig. 9. For 5'p= 10 and 6*0 = 0°, the response rados in the sway, heave and roh modes are 74-2, 76-3 and 75-5% respectively. I n general, it is nodced that the reducdon of responses is greater while using the frequency-independent spreading function.

The effect of the mean wave direction on motion responses is shown i n Fig. 9 for i'p=10. As discussed earlier, the response ratio decreases as the mean angle o f wave attack increases.

7 C O N C L U S I O N S

The hydrodynamic behaviour of long floating structures i n short-crested seas has been studied. A numerical scheme based on the finite-element method was sought to solve the hydrodynamic coefficients and wave exciting forces due to the obhquely incident waves on a structure

242 S. A. Sannasiraj et al. 1.00 0.80 0.60 s 0 0 ' 0 . 4 0 0.20 0.00 1.00 0.00 1.00 0.80 QJ > D 0) I 0.60 '0.40 -\ 0.20 0.00 0.00 3.00 1.00 2.00 3.00 ka

Fig. 5. Force reduction factors as a function of mean wave

direction for the frequency-dependent spreading function.

2.50 5.00 7.50 spreoding index, s

10.00

Fig. 6. Variation of response ratio with spreading index. 00 = 0°.

0-806 at the peak frequency of 0-2 Hz. I n the heave mode, the force ratio varies f r o m 0-971 as /ca—>0 to 0-443 at /cfl = 3-0. Even though no direct comparison is possible between the two spreading functions considered above, since the frequency-dependent spreading func-tion represents more closely the realistic sea state, the use o f the frequency-independent spreading function underestimates the force near the peak frequency and over estimates h i n the low- and high-frequency ranges. I n a short-crested sea state, i n which the spreading is independent of frequency (s=l, 9o = 0°), the rms value of the motions i n the sway, heave and roh modes reduces by 41-5, 41-4 and 37-0% respectively. The reductions i n responses i n the corresponding modes are 25-8, 23-7 and 24-5% w i t h the frequency-dependent spreading function ( 5 p = 1 0 , 6lo = 0°). As the spreading index, s, or the spreading parameter, s^, increases, as the of arbitrary shape. The numerical model has been

vahdated by comparing the sway and heave responses of a freely floating rectangular structure i n oblique seas with the results of earher investigators.

The wave exciting forces on a rectangular floating structure, and hence the response induced i n multi-directional seas, has been evaluated using the linear transfer-function approach. The force ratio and the response ratio were evaluated numerically for frequency-independent and frequency-dependent cosine power type directional spreading functions.

For the structure considered, the sway force ratio varied f r o m 0-866 as / c a ^ 0 to 0-359 at /ca= 3-0 f o r the frequency-independent spreading function with s = 1 and 00 = 0°. The variation i n the force ratio i n the heave mode was f r o m 1-0 as lea—>0 to 0-346 at/cfl = 3-0. I f the spreading function is dependent on frequency ( j p = 10, öo = 0°), the sway force ratio varies f r o m 0-707 as ka-^0 to 0-454 at ka = 3-0, with a maximum value of

0.00 30.00 60.00

Mean direction, 0° 90.00

Fig. 7. Variation of response ratio with mean wave direction.

Long floating stnictures in directional seas

case may be, the rms value of the responses i n short-crested seas approaches the rms response i n the long-crested sea state.

I f the mean direction of wave approach is other than normal incidence, there are significant reductions in the motion responses, which are quite useful in design in the case of long relative structure lengths.

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