Indian Journal of Engineering & Material s Sciences Vol. 8, April 200 I, pp. 59-65

Experimental investigation of wave forces on submerged horizontal cylinders

Gazi Md. Khalil

Department of Naval Architecture and Marine Engineering. Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh

Received 27 March 2000; accepted 4 January 2001

This paper presents the results of an experimental investigation of the wave-induced forces acting on hori zontal cy linders submerged beneath water waves with axis parallel to the wave crests. The experiments are can'ied out in a towing tank of the University of Tokyo. A nap-type wave generator is used to produce regul ar si nusoidal waves of length 1.57 and 3.14 m. The experiments are performed with a circular cylinder and a rectangular cylinder separately with a view to predicting the effect of geometry of the cylinder on the wave-induced forces. Each cylindrical model is tested at eight different depths of submergence. A mUlti-component load cell is used for the measurement of forces acting on the submerged cylinder. The measured forces are instantaneously digitized and stored in a transient memory. A personal computer , viz., PC 980 I is installed on the carriage of the towing tank in order to compute the mean forces right at the time of performing the experiments . The time-averaged mean hori zontal as well as vertical forces ac ting on the cy lindrical model at variolls depths of submergence are plotted and then physically interpreted . The results are compared with the avail able data in the literature. The principal conclusion of thi s investi gation is that the breaking of waves behind a shallowly submerged cy linder is primarily responsible for the generation of non-linear wave forces. The negative drifting force which acts on a shallowly submerged cylinder is a direct consequence of wave-breaking. This drifting force , however, tends to vani sh when the cylinder is deeply submerged. The findings of this investigation are expected to be useful in the accurate assessment hydrodynamic loads ac ting on various types of offshore structures.

At present a wide variety of offshore structures are being used, often under severe environmental conditions. These are predominantly related to the exploration, recovery and production of oil and gas, but they are also used in ocean energy extraction, harbour engineering etc. Difficulties in the design and construction of such structures are enormous, particularly as they are being installed in ever­increasing depths and are subject to extremely hostile environmental conditions. The potential of major catastrophic failures, both in terms of human safety as well as financial loss, underlines the paramount importance of reliable design of these structures.

It is well known that, according to linear theory, surface waves in deep water suffer a phase shift in passing over a submerged cylinder with its axis parallel to the crests. This was first proved by Dean I ,

using the conformal mapping technique. He showed also that to the same approximation there are no waves reflected from the cylinder. This result was confirmed by Urse1l 2

, using a series of multi-pole potentials, from which complete details of the flow may be inferred .

From the first-order velocity potential for the flow around a horizontal cylinder submerged beneath waves in deep water, Ogilvie3 derived expressions for

the first-order force on the cylinder (i.e. the force component that is proportional to the wave amplitude a when other independent parameters remain constant) and the time-independent part of the second­order force (proportional to a\ The horizontal and vertical components of the first-order force have equal magnitudes, and oscillate in quadrature with the wave

period 2rr/w. Besides the time-independent part of the second-order force, which acts vertically upwards, there is a second-order unsteady force with frequen cy 2w. Furthermore, Ogilvie showed that the first-order force on the cylinder lags the local acceleration in the undisturbed flow at the cylinder's axis by a phase angle which is one-half that suffered by the waves in pass ing over the cylinder. It is clear from the non­linearity of the free-surface boundary condition that, unless the cylinder is deep!y submerged, there are also higher-order forces whos'e frequencies are the sums and differences of those at lower orders. Without a solution for the non-linear velocity potential, however, it is not clear at what frequencies the higher­order forces will appear. The invi scid flow around a cylinder that is submerged at sufficient depth for its effect at the free surface to be neglected can be analysed by means of Milne-Thomson' s circle theorem4

•

60 INDIAN J. ENG. MATER. sel., APRIL 2001

In reality, whether the cy linder is deeply submerged or not, increasing the wave amplitude may ultimately cause the flow around it to be dramatically changed by the occurrence of separation, and the force to be dominated by form drag.

More recently, alternative and more general solutions to the lineari zed irrotational flow problem of the effects of the cylinder on the waves have been reported5

.7

.. For the case of the circular cylinder, their conclusions are identical with those of Dean, and, in developing the velocity potential only to first order, they are unable to make any advance in calculating the non-linear forces beyond those derived by Ogilvie. A numerical solution to the non-linear problem by Stansby & Slaouti 8 for waves of more than 50% limiting height yielded for the fluctuating force values which agree with Ogilvie's theory to within 3% for Keulegan-Carpenter numbers up to 1.2. This suggests that the errors arising from the assumptions of the linearized analyses of the inviscid flow in this range are not severe.

However, in addition to the non-linearities of the inviscid flow, there remain those that arise from the action of viscosity . In a series of papers, Chaplin has studied by both experimental and theoretical means the flow of a real fluid around a horizontal circular cylinder submerged beneath waves with its axis parallel to the wave crests. In the first paper, Chaplin9

investigates the mass transport flow around the cylinder. For conditions in which the wave amplitude (or the Keulegan-Carpenter number) is sufficiently small for the effects of separation to be unimportant, and with the cylinder submerged at a depth about 2.5 diameters below mean water level , the results are in good agreement with predictions. The theoretical analysis is based on the assumptions that the waves are of small amplitude and, at the free surface, are unaffected by the presence of the cy linder. In a second paper, Chaplin 10 has observed that the forces experienced by the cy linder reveal non-l inear components with frequencies up to three times the fundamental wave frequency. The dominant non­linear contribution to the loadi ng is at the third order in the wave amplitude, and, for Keulegan-Carpenter numbers approaching 2, its magnitude is found to be as much as one-half that of the inertia force . Chaplin suggests that the third-order force is associated with circulation generated by steady streaming in the oscillatory boundary layer on the cy linder. At higher Keulegan-Carpenter numbers, form drag becomes increasingly importan t, i1l1d velocity measurements

close to the cy linder show the rapid development of the wake. More recently Chaplin I I has considered the general orbital flow about a circu lar cylinder, usi ng boundary-layer techniques introduced by Stuare 2

. In his most recent paper, Chaplin l3 attacks the problem of uniform circular orbital flow in the presence of a circular cylinder at finite Reynolds number using the fu ll Navier-Stokes equation. The same problem is addressed by Stansby & Smithl4, us ing the random vortex method. Several other researchers 15·21 have also made significant contributions to this field.

The aim of this study is to determine experimentally the wave-induced forces acting on submerged horizontal cylinders located at different depths of submergence. Experiments are performed with a circular cylinder and a rectangular cylinder separately with a view to predicting the effect of geometry of the cylinder on the wave forces . The study relates to the practical areas of flow about the horizontal pontoons of semi-submersibles and tension-leg platforms. It may be mentioned that pontoons are hollow metal cylinders used to support a platform. These may be circular or rectangular in shape.

Description of Model and Experimeutai Set-up The experiments are conducted in a towing tank of

the University of Tokyo. The length, width and depth of the tank are 86, 3.5 and 2.5 m respectively. A flap­type regular wave generator is used to produce regul ar sinusoidal waves. A wave-board of size 3.455 m x 1.530 m is installed at the end of the towing tank. A rectangular channel having two side walls made of transparent plexiglas with sharp leading edges is used for generating a two-dimensional flow field . The length, width and depth of the channel are 3.0, 0.53 and 0.80 m respectively (see Fig. 1).

Two cylindrical models, one being circular and the other rectangular, are separately used for the experiments. Both of the models are made of plexiglas of thickness 10 mm, and four circular holes of diameter 10 mm each are bored at their top and bottom ends wh ich let water in and out of the models and thereby neutralize the buoyancy effects. Each model is 0.52 m long which leaves a clearance of 5 mm at each end in order to avoid any contact with the wall of the channel. This ensures unhindered motion of th e cylinder under the ac ti on of wave forces. The cross-sect ional area of the rectangu lar model is 0.4 mxO.4 m whereas the diameter of the circular model is 0.4 m. The model is igidly fixed to a carri age

KHALIL: WAVE FORCES ON SUBM ERGED HORIZONTAL CYLINDERS 61

Fi g. I- Arrangement of the cylindrical modcl in thc channel

through a strut and a box of gauge system (a multi­component load cell) for the measurement of forces .

Each model is tested at eight different depths of submergence viz. , DIS = 1.000, 1.125, 1.250, 1.375, 1.500, 1.750, 2.000 and 2 .250. It is of significance to note that the top surface of the cy linder just coincides with the still water surface when DIS = 1. Regular waves of length 1.57 and 3.14 m are generated by the wave maker. The amplitude a is 6 cm for both of the waves.

One of the important parameters of the flow is the amplitude of fluid motion relative to the cy linder s ize, which is usually expressed as the Keulegan-Carpenter number

K =UII/T c 2B

where

U - kD 11/ =wae

2n w= -

T

k = 2n ).

In the present in vestigation , the values of Kc at the shallowest immersion of the models are 0.42 and 0.63 when the wavelength is 1.57 m and 3.14 m respectively. Again, Kc attai ns a value of 0.16 and 0.38 corresponding to wavelengths of 1.57 m and 3.14 m respectively at the deepest submergence. Since the Keu legan-Carpenter number never exceeds unity in thi s investigation, the viscous effects are assumed to be very small. Table I summari zes all the experimental conditions of the investi gation.

Items

Tab le I-Principal particulars of the expcriments

Dimcnsions I Magnitude

Towing

Channcl

Wavelength

Amplitude of thc wave

Wavc period

Wave slope

Keulegan-Carpcnter nu mber

Length of each model

Diamcter of the circular model

Cross-sect ional area of the rectangular model

dlA (or 2BIA)

DIB

Tank 86 Illx3 .5 mx2.5 m

3.0 m x 0.53 111 x 0.80 III

1.57 m and 3. 14 m

0.06111

1.003 scc and 10419 scc

0. 153 and 0.0765

0.16,0.38, 0042 and 0.63

0.52 m

0040 m

0040 mX 0040 m

0.2548 and 0.1274

1.000, 1.1 25, 1.250, 1.375,

1.50, 1.75, 2.00 and 2.25

The horizontal and vert ical components of wave fo rces acting on the submerged cy linder are measured with the multi-component load cell. It may be noted that the down wave and downward forces are considered positi ve as shown in Fig. 2. The forces are made dimensionless with respect to the length and width of the model as follows:

FX pgaBL

F _ FZ

z -pgaBL

Fig. 3 shows a schematic diag ram of data acqu isition and analysis system. The measured forces are instantaneously dig iti zed and stored in a transient memory. Then they are copied onto a floppy magnetic diskette which is subsequently conveyed to the Computer Centre of the Univers ity of Tokyo where the analysis of the measured data is executed on the supercomputer HIT AC M 680/682 H. Moreover, a personal computer viz., PC 9801 (made by NEC, Japan) is install ed on the caITiage of the towing tank in order to compute the mean forces right at the time of performing the experiments.

Analysis of Experimental Results During the course of this investi gation, two distinct

cases are studied viz., ( i) a circular cylinder horizontally submerged beneath waves with its ax is parallel to the wave-crests; and (i i) a rectangular

62 INDIAN J. ENG. MATER. SCI. , APRIL 2001

~..LJ"'---!I c

l lL-...j' :~ X.fX

7.F Z l.fZ

Fig. 2-Definition sketch for notations used in the study

Fig. 3-Schematic diagram of data acquisition and analysis system

cylinder horizontally submerged beneath waves with its axis parallel to the wave-crests. The diameter of the circular model is the same as the width of the rectangular model so that the wave forces acting on the two models can be compared with each other.

Fig. 4 shows the variation of the time-averaged mean horizontal force acting on the circular cylinder as well as the rectangular cylinder at various depths of submergence under a wave of length 1.57 m. The drifting force is found to be zero when DIB = 1.5 in case of the circular cylinder, and DIB = 1.75 in case of the rectangular cylinder and DIB= 1.75 in case of rectangular cylinder. At larger depths also, this drifting force is more or less zero for both the models. (Slight departures from the zero value may be due to the experimental errors which are inherent in the measuring system). However, the force curves clearly indicate that the non-linear effects are absent in case of deeply submerged cylinders irrespective of their shape. But as the depth of submergence decreases, a negative drifting force is developed on both of the models. This negative drifting force (i.e., the upwave force) attains a peak value when DIB = 1.25 in case of the rectangular cylinder, and DIB = 1.125 in case of the circular cylinder. This is physically significant and

STiLL IJATER SUR fACE

-0.16 -0.12 - 0.08 - 0.04

--- _ F,

1.6 <D

1.8

2.0

2. 2

2. 4

, o

0 08 0.12 0.16

o CIRCULAR CYLUH>ER

o RECTANCULAR cnINDER

Fig. 4-Measured mean horizontal force acting on the cylinders at various depths of submergence under a wave of length 1.57 m

merits explanation. The model is completely submerged beneath the wave only if DIB exceeds 1.30. But the top surface of the model pierces the free water surface and a part of it is exposed to air when DIB < 1.30. It is but easy to imagine th at the depth of SUbmergence, which corresponds to a value of DIB = 1.125 or 1.25, lies at such a critical juncture when the top surface of the model is about to pierce the free water surface and thereby induces strong non-linear interactions with the incident water waves. However, as the depth of submergence decreases further, the negative drifting force rapidly decreases. This negative drifting force is a direct consequence of the non-linear interactions of waves wi th the cylinder, and cannot be explained in the light of classical theoretical hydrodynamics. The present author would like to attribute this phenomenon to the fact that the waves pass over the shallowly submerged cylinder and then break behind it. As a consequence, the fluid particles in that region are decelerated . Such decelerated water particles give rise to a high static pressure on the rear side of the model. Hence it experiences an upwave force .

Fig. 5 shows the variation of the measured mean horizontal force acting on the circular cylinder as well as the rectangular cylinder submerged under a wave of length 3.14 m. The circular cylinder is found to experience a zero drifting force when DIB = 2.15 whereas the rectangular cylinder experiences the same when DIB =1.80. Both of the models are subjected to a maximum negative drifting force at a depth of

KHALIL: WAVE FORCES ON SUBMERGED HORIZONTAL CY LINDERS 63

submergence, DIB = 1.25 . Apart from these, the force curves exhibit simi lar characteri stics to those corresponding to a wave of length 1.57 m. It is clearly observed from both Figs 4 and 5 that the maximum negative drifting force on the rectangular cy linder is greater than that on the circul ar cylinder. It may, therefore, be concluded that the non-linear effect is less severe in the case of a circul ar cy linder than that in the case of a rectangular cy linder.

The time-averaged mean vertical forces acting on the circular model as well as the rectangular model

-0.16 STlLL W'ATi:R SURFAcE

- 0.12 -O.OB -0.04

----Fx

I. B

2.0

2.2

2.4

., , a

0.04 O.OB 0.12

o CIRCULAR CYLINDER

o RfCTANGUl..AR CYL 1NDER

0.16

Fig. S-Measured mean horizontal force acting on the cy linders at various depths of submergence under a wave of length 3.14 m

STILL WATER SUiFACE ----. Fz - 0~.B_~-~O.~6_~-0~.4~~-0~.2_~r-_~0.~2 __ 0~. 4~~0~. 6~~0. B

Ql , 2.0 0

2.2

2.4

o CUCULAR CYLINDEJ.

a lECTAHCUI..AR CYLlNO£1l

Fig. 6--Measured mean vertical force acti ng on the cylinders at various depths or submergence under a wave of length 1.57 m

under waves of length 1.57 m and 3.1 4 m are shown in Figs 6 and 7 respectively. The force curves exhibit approx imately si milar characteristics . . The general trend of the variation of the force due to the change of the depth of submergence is that. the upward mean force steadily increases with the decreasing depth of submergence, and then abruptly decreases when a part of the model emerges above the free water surface. However, the upward forces tend to vanish when DIB > 2. This remark holds good for both of the models and under both of the waves.

Fig. 8 shows the temporal fluctuation of the horizontal wave force acting on the rectangular model submerged at a depth of DIB = 1 under a wave of length 1.57 m whereas Fig. 9 deal s with the same fluctuation of forces when the same model is submerged at a depth of DIB = 2.25 under a wave of length 3.14 m. It has been already stated that the top surface of the cylinder coincides with the still water level when DIB = 1 and the case of deepest immersion is obtained (in this investigation) when DIB = 2.25. These two typically extreme cases have been selected for discussion here with a view to illustrating clearly the salient features of the non-linear wave forces acting on a shallowly submerged structure. Since the wave-maker generates approximately sinusoidal waves, the records of measured forces also show sinusoidal fluctuations when the model is deeply submerged. However, the record of horizontal force in the shallowly submerged condition (i .e., when DIB = 1) shows very peaky wave profiles which are simply the manifestation of non-linear wave effects.

STILL WATER. SUiU'ACE F I

-0~iB __ -,~~6_~-OTi4~ __ -~Or·2_~r-_,0.~2 __ ~0~. 4~~0~.6_~0. 8

o ClRCUUJiI. CYLINDER

IJ CfL1HDER

1.4

1.6

I.B Ql , a

2.0

2.4

Fig. 7-Measured mean vertical force acting on the cy li nders at various depths of submergence under a wave of length 3.1 4 m

64 INDIA J. ENG. MATER . SCI., APRIL 200 1

F, 10.,----------------·~

-30+------,~~~._~~-~~~~~~~~C'"::' o 10 12 " 16 1B 20

TIME (SEC)

>lEAN FOOCE • 0.01530

Fig. 8-Temporal variat ion of thc horizontal wave fo rce acti ng on thc rcctangular cylinder submcrged at a depth of DIS = 1.0 undcr a wave of length 1.57 m

It is necessary to mention here that the current work is not the first of its kind . Inoue & Kyozuka 15 have carried out simi lar experimental in vest igation of non­linear wave loads acting on submerged cylinders by using fai rl y large models. They have studi ed the effects of the incident wave amplitude, the cylinder submergence and the cross-sectional geometry. The experimental results have been compared with the numerical calculations to the second-order. The calculation method is based on the regular perturbation theory and the wave loads are obtai ned by the integrat ion of hydrodynamic pressure over the body surface. The Iluid is assumed in viscid, incompressib le, homogeneo us and infin itely deep . Consequently, when the cylinder is placed appropriately deep, the second-order theory agrees well with the experiments. However, when the cylinder is close to the free-s urface, the theory would overest imate the wave loads obta ined by the experiments. The resul ts of their invest iga tion are found to be in agreement with the present experimental results. The characteristics of wave loading on cylinders located at different depths of submergence are fo und to be similar to those observed in the present study. And fin all y, their conclusions on the drifting forces on hori zontal cyli nders are identi cal with those of the present au thor.

It is, however, admitted that any general conclusions on the drifting forces on submerged horizontal cylinders req uire more experimental study usi ng different wavelengths and heights. Wave kinematics and forces under the given conditions are too complex phenomena to deri ve conclusions based on limited experiments conducted.

Before closing the discussion of experimental results, it is appropriate to make some comments about the se lection of model dimensions and the experimental conditions. Offshore structures of large

F,3.0,.------------------,

-3. 0+--,---,--,--,-,-,.--,---,--, .-,.-,-~--..-..-,.-,-~~.-,-J o e 10 12 " 15 18 10

TIME ISEC)

MEAN FORCE: 0.0020)

Fig. 9-Tcmporal variat ion of thc horizontal wavc forcc acting on thc rcctangular cylinder submcrged at •. Jcpth of DIS = 2.25 under a wave of length 3.14 m

horizontal dimensions have now found extensive app li cations. When a body spans a significant fraction of a wavelength, the incident waves genera ll y undergo signi ficant scattering or diffraction and wave-force calculations should then take such scattering into accou nt. This situation characterizes the diffraction regime of wave-structure interaction and is generally considered to occur when the structure spans more than about 20% of the incident wavelength22 .This is in contrast to the interact ion of waves with a slender structural elemen t, in which case flow separation dominates the loading behaviour but beyond the immediate vicinity of the element the wave train remai ns re latively unaffected. These two regimes of wave-structure interaction give ri se to two disti nct approaches by which wave- force prob lems are treated. In the present investigati on, the diameter of the cylin.Jrical model is 0.4 m and the wavelengths are 1.57 and 3.14 m. Therefore, selection of the model dimension is justified in the case of wavelength 1.57 m since the diameter of the cyl inder is 25.48% of the wavelength. But when the wavelength is 3.14 m, the diameter of the cy linder is on ly 12.74% of the wavelength. It is a shortcoming of this work.

In this investigation, the wave amplitudes are restricted to 0.06 m. It may be noted that the wave­maker is capable of generating a range of wave heights. Higher wave heights must have shown the non-lineari ty more explicitly . More results involving a range of wavelengths and wave heights and representation of the resu lts in non-dimensional quantiti es would have been useful.

Conclusions The study of two-di mensional surface waves is of

importance to the case of wave- induced forces on cylindrical structures, in that it is the first step to understanding the complex three-di mensional wave­structure interaction. The fi ndings of the present

KHALIL: WAVE FORCES ON SUBMERGED HORIZONTAL CYLINDERS 65

investigation are, therefore, expected to be useful in the accurate assessment of hydrodynamic loads acting on various types of offshore structures.

The principal conclusion of this experimental investigation is that the breaking of waves behind a shallowly submerged cylinder is primarily responsible for the generation of non-linear wave forces. The negative drifting force (or the upwave force) which acts on the submerged cylinder is a direct consequence of wave-breaking. When a wave passes over a shallowly submerged cylinder and breaks behind it, the water particles in that region are naturally decelerated. These decelerated fluid particles exert a high static pressure on the rear side of the cylinder, and as a consequence, it experiences a negative drifting force . This force attains a peak value when the top surface of the cylindrical model pierces the free water surface. On the other hand, this drifting force tends to vanish when the cylindrical model is deeply submerged (viz., at depths given by DIB > 2).

The geometry of the cylindrical structure also plays a vital role in the wave-structure interaction. The negative drifting force acting on the rectangular model is greater than that on the circular one. It is primarily because of the difference in their shape. It is only the horizontal component of the static pressure acting on the rear side of the circular model which contributes to the negative drifting force . On the other hand, the entire static pressure (and not merely the horizontal component of it) acting normal to the flat vertical rear side of the rectangular model is responsible for the large upwave force.

Acknowledgement This paper is the cherished outcome of the research

carried out by the author when he was a visiting professor in the Department of Naval Architecture and Ocean Engineering in the University of Tokyo (Japan). The author is grateful to Professor Dr Hisashi Kajitani, the Superintendent of the Towing Tank Laboratory of the University of Tokyo, for his valuable suggestions which have improved the quality of this research.

Nomenclature a = amplitude of the wave B = half-width of the rectangular cy linder,

= radius of the circular cylinder D = depth of the centre of the model from the still water level d = diameter of the circular cy linder FX = horizontal component of wave forces Fx = non-dimensional horizontal component of wave forces FZ = vertical component of wave forces Fz = non-dimensional vertical component of wave fo rces g = acceleration due to gravity k = wave number (= 27t1A) Kc = Keulegan-Carpenter number L = length of the cylinder T = time period of the wave VIII = velocity amplitude of the flow w = circular frequency of the wave (= 27t17) A = wavelength p = density of water

References I Dean W R, Proc Camb Phil Soc, 44 (1948) 483-491. 2 Ursell F, Proc Camb Phil Soc, 46 (1950) 141-158. 3 OgilvieTF, JFluidMech, 16(1963) 451-472. 4 Milne-Thomson L M, Theoretical Hydrodynamics, 5th Ed,

(The Macmillan Company, USA), 1968, 157- 158. 5 Leppington F G & Siew P F, Appl Ocean Res, 2 (1980) 129-

137. 6 Mehlum E, Appl Ocean Res, 2 (1980) 171-177. 7 Grue J & Palm E, Appl Ocean Res, 6 (1984) 54-60. 8 Stansby P K & Siaouti A, Appl Ocean Res, 6 (1984) 108-

115. 9 Chaplin J R, J Fluid Mech, 140 (1984) 175-187.

10 Chaplin J R, J Fluid Mech, 147 (1984) 449-464. II Chaplin J R, J Fluid Mech, 237 (1992) 395-411. 12 Stuart J T, J Fluid Mech, 24 (1966) 673-687. 13 Chaplin J R, J Fluid Mech, 246 (1993) 397-418. 14 Stansby P K & Smith P A, J Fluid Mech, 229 (1991) 159-

171. 15 Inoue R & Kyozuka Y, J Soc Naval Architects Japan , 156

(1984) 115-127. 16 Miyata H, Kajitani H, Zhu M, Kawano T, & Takai M, J

Kansai Soc Naval Architects, Japan, 207 (1987) 11-23. 17 Vada T A, J Fluid Mech, 174 (1987) 23-37. 18 McIver M & McIver P, J Fluid Mech, 219 (1990) 519-529. 19 Wu G X, Appl Ocean Res, 13 (1991) 58-62. 20 Yan 8 & Riley N, J Fluid Mech, 316 (1996) 241-257. 21 Arena F, J Fluid Mech, 394 (1999) 355-356. 22 Sarpkaya T & Isaacson M, Mechanics of Wa ve Forces all

Offshore Structures, (Van Nostrand Reinhold Company, New York), 1981,381-385.