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J. Appl. Math. & Computing Vol. 18(2005), No. 1 - 2, pp. 171 - 182

NONLINEAR FREE SURFACE CONDITION DUE TO SECOND

ORDER DIFFRACTION BY A PAIR OF CYLINDERS

DAMBARU D. BHATTA

Abstract.An analysis of the non-homogeneous term involved in the freesurface condition for second order wave diffraction on a pair of cylinders

is presented. In the computations of the nonlinear loads on offshore struc-tures, the most challenging task is the computation of the free surface

integral. The main contribution to this integrand is due to the non-homogeneous term present in the free surface condition for second order

scattered potential. In this paper, the free surface condition for the sec-ond order scattered potential is derived. Under the assumption of largespacing between the two cylinders, waves scattered by one cylinder may

be replaced in the vicinity of the other cylinder by equivalent plane wavestogether with non-planner correction terms. Then solving a complex ma-

trix equation, the first order scattered potential is derived and since thefree surface term for second order scattered potential can be expressed interms of the first order potentials, the free surface term can be obtained

using the knowledge of first order potentials only.

AMS Mathematics Subject Classification: 76B15, 35Q35, 35J05.

Key words and phrases : Water wave, free surface condition, nonlinear,wave diffraction, pair of cylinders.

1. Introduction

The prediction of the wave loads on an offshore structure immersed in waterin the presence of a free surface is one of the most important tasks for oceanengineers. The forces exerted by surface waves on offshore structures such as off-shore drilling rigs or submerged oil storage tanks are of important considerationsin the design of large submerged or semimerged structures. The nonlinearity for

the wave-structure interaction problem arises from the free surface boundarycondition. When the structure is large compared to wavelength, the incidentwaves upon arriving at the structure undergo significant scattering, and hence

Received December 13, 2003. Revised May 20, 2004.

c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

171

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172 Dambaru D. Bhatta

the diffraction theory is to be used to account for the scattering. Most of the off-shore structures are designed in the form of cylindrical structures. The problemof hydrodynamic interactions within a group of large multi-component offshorestructures has become an area of interest among the researchers in recent years.From practical consideration it is extremely important to be able to predict thewave loads on multicolumn structures in waves. When many offshore structuresare placed together in a configuration, the hydrodynamic loads on individualcomponents may be significantly different from the loading they each wouldexperience in isolation. This is simply because of the hydrodynamic couplingbetween the bodies. Simon (1982) proposed an approximate theory for comput-ing wave forces on an array of wave energy devices. According to this theory, adiverging wave scattering from one cylinder is replaced by a plane wave of ap-propriate amplitude in the vicinity of another cylinder. Once the amplitude and

phase of the equivalent plane wave have been determined, the problem reducesto summing the effects of the plane waves on any given cylinder. The effect ofthis equivalent plane wave on the given cylinder is then computed. The solutioninherently assumes that the spacing between two cylinders is fairly large relativeto the incident wavelength. The evaluation of the free surface integral is the mostcritical part in the computation of the wave loads and the main contribution tothis integral is from the free surface boundary condition due to the second orderscattered potential. Dean and Dalrymple (1984) presented a review of potentialflow hydrodynamics. Solutions for standing and progressive small amplitude wa-ter waves provide the basis for application to numerous problems of engineeringinterest. The water particle kinematics and pressure field within the waves aredirectly related to the calculation of forces on fixed or floating bodies. They dis-cussed the formulation of the linear water ware theory and development of thesimplest two-dimensional solution for standing and progressive waves. EatockTaylor and Hung (1987) analyzed the convergence of the free surface integral fora single vertical cylinder. By dividing the whole integral into two integrals, onefor the near field and the other for the far field. They presented numerical resultsfor the near field by using Gaussian quadrature and for the far field by usingasymptotic expansions. An explicit computation of the free surface term forsecond order diffraction on a single cylinder was discussed by Chau and EatockTaylor (1992). Rahman and Bhatta (1993) presented second order wave loadingon a pair of cylinders. They derived the quadratic forces for a pair of bottommounted, surface piercing circular cylinders in waves of arbitrary depth.

Debnath (1994) presented theoretical studies of nonlinear water waves as theyappeared in the literature over the last three decades. His work is primarily de-

voted to the mathematical theory of nonlinear water waves with applications. Asin-depth study of nonlinearity requires some linearized theory first, a substantialpart of the first few chapters is concerned with the linearized theory of surfacegravity waves of water. He studied the theory of nonlinear shallow water waves

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Nonlinear free surface condition due to second order diffraction 173

and solitons, with emphasis on methods and solutions of several evolution equa-tions that are originated in the theory of water waves. The nonlinear shallowwater equations were used to discuss the breaking of waves on beaches. Specialattention was given to the Boussinesq equation, the Korteweg-de Vries equationand the inverse scattering transform. Rahman (1994) presented an introductionto the mathematical and physical aspects of the theory of water waves. He dis-cussed the wave theory of Airy, nonlinear wave theory of Stokes, tidal dynamicsin shallow water. Developments of cnoidal waves, solitary waves and Korteweg-de Vries equation were presented in the last chapter. Johnson (1997) presentedan introduction to mathematical ideas and techniques that are directly relevantto water wave theory. Beginning with the introduction of the appropriate equa-tions of fluid mechanics, together with the relevant boundary conditions, theideas of nondimensionalisation, scaling and asymptotic expansions are briefly

explored. He presented the nonlinear problems including Stokes wave, nonlinearlong waves, waves on a sloping beach and the solitary wave. Based on previousworks, there have been remarkably exciting development in nonlinear dispersivewaves in general, nonlinear water waves in particular. Indeed, the theory ofnonlinear water waves and solitons has experienced a revolution over the pastfour decades. Here we present an analysis to evaluate the free surface term forsecond order diffraction by a pair of cylinders.

In the present paper, at first, we formulate the first-order wave diffractionproblem for a pair of bottom mounted, surface piercing cylinders in water ofarbitrary uniform depth. Then we present the formulation for the second-orderdiffraction problem for those cylinders. These formulations are based on the as-sumption of a homogeneous, ideal, incompressible, irrotational and inviscid fluid.The free surface condition for the second order scattered potential is derived.Then solving a complex matrix equation, the first order scattered potential isderived and since the free surface term for second order scattered potential canbe expressed in terms of the first order potentials, the free surface term can beobtained using the knowledge of first order potentials only. Here we have incor-porated the idea initially presented by Simon where a diverging wave scatteredfrom one cylinder is replaced by equivalent plane waves of appropriate amplitudeand phase in the neighbourhood of another cylinder.

2. Mathematical formulation

We consider two right circular cylinders of radius a separated by a distance s.We assume that these cylinders extend from the ocean bed to the free surface, asshown in Figure 1. A fixed coordinate system Oxyz is employed with the x andy axes in the horizontal plane with the z-axis pointing vertically upwards froman origin on the free surface. The corresponding cylindrical polar coordinatesystem (r,,z) is assumed. The water depth is h.

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Here we consider a regular wave of frequency and amplitude A propagatingin the positive x direction. The formulation of the wave structure interaction isbased on the assumption of a homogeneous, ideal, incompressible and inviscidfluid. Also the inclusion of the irrotationality condition yields us to write thefluid velocity qas(r , ,z ,t). Using the Stokes expansion the velocity potential(r , ,z ,t) and the surface elevation (r,,t) can be written as

(r , ,z ,t) = (1)(r , ,z ,t) + 2(2)(r , ,z ,t) + .... (1)

(r,,t) = (1)(r,,t) + 2(2)(r,,t) + .... (2)

where is the wave steepness, (1)(r , ,z ,t) is the first order potential and(2)(r , ,z ,t) is the second order potential.

2.1. First order diffraction problem

For an incompressible and inviscid fluid, the velocity potential satifies Laplaceequation. Thus the governing differential equation for the first potential is

2(1) = 0 in the fluid region. (3)

Bottom boundary condition satisfies the following

(1)

z= 0 on z = h for rj a. (4)

Body surface boundary condtion is

(1)

rj= 0 on rj = a for h z 0. (5)

Free surface condition becomes

2(1)

t2+ g

(1)

z= 0 on z = 0 for rj a. (6)

Here g is the acceleration due to gravity and we will take rj = r1 for the firstcylinder and rj = r2 for the second cylinder. Also the far field radiation conditionis to be satisfied by the complex scattered potentials.

Considering the first order velocity potential (1) to be time harmonic, wecan write (1) as

(1) = Re

(1)eit

(7)

where Re stands for real part. The potential (1) is conveniently decomposed

into contributions (1)i , (1)s1 and (1)s2 , where (1)i is the incident potential, (1)s1 is

the scattered potential due to the cylinder 1 and (1)s2 is the scattered potential

due to the cylinder 2. The centers of the two cylinders, O1 and O2, lie on Ox.Let O1 and O2 have the local polar coordinates (r1, 1) and (r2, 2) respectively.In the subsequent analysis, without loss of generality, (r, ) will be used in place

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of (r1, 1). In terms of cylindrical coordinates, we can express (1)i relative to

the center of cylinder 1 as

(1)i (r,,z) =

gA

cosh k(z + h)

cosh kh

n=

(i)nJn(kr)ein . (8)

The scattered potential due to cylinder 1 may be written

(1)s1 (r,,z) =

gA

cosh k(z + h)

cosh kh

n=

an(i)nH(2)n (kr)e

in (9)

where

2

= gk tanh kh, (10)k is the wavenumber,Jn is a Bessel function of the first kind,

H(2)n is a Hankel function of the second kind satisfying the radiation condition,

andthe coefficients an are to be determined from the boundary condition on the

cylinder.The dispersion relation (10) can be derived from the combined free surface

boundary condition (6). The properties of Bessel and Hankel functions aredescribed by McLachlan (1961).

In view of the symmetry about Ox, the scattered potential due to cylinder 2must be expressible in terms of cylindrical coordinates r2, 2, z in the same form:

(1)s2 (r2, 2, z) =

gA

cosh k(z + h)

cosh kh

n=

an(i)nH(2)n (kr2)e

in2 . (11)

Equation (11) can be conveniently transferred into coordinates r, and z us-ing the Graf addition theorem for Bessel functions (Equation 9.1.79 given byAbramowitz and Stegun (1972)) as

H(2)n (kr2)ein(

22) =

l=

H(2)l+n(ks)Jl(kr)e

il(2). (12)

Thus Equation (11) is transferred into

(1)s2 (r,,z) =

gA

cosh k(z + h)cosh kh

n=

an

l=

(i)lH(2)l+n(ks)Jl(kr)eil. (13)

Thus the total potential is (1) = (1)i +

(1)s1 +

(1)s2 =

(1)i +

(1)s . Here

(1)s is the

scattered potential due to the combined cylinders. The body surface boundary

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condition at the cylinder 1 yields

n=

(i)nJn(ka)ein +

n=

an(i)nH(2)n (ka)e

in

+

n=

an

l=

(i)lH(2)l+n(ks)J

l (ka)e

il = 0 (14)

which holds for all in the range (0, 2).Rearranging this equation and interchanging the order of the two convergentsummations, we obtain equations for an:

anH

(2)n (ka)

Jn(ka)+

l=

alH(2)l+n(ks) = 1, < n < (15)

where n is an integer. Therefore the combined velocity potential (1) = (1)i +

(1)s1 +

(1)s2 expressed in the coordinates r, and z is then given by

(1)(r,,z) =gA

cosh k(z + h)

cosh kh

n=

(i)nein [Jn(kr)

+anH(2)n (kr) +

l=

alH(2)l+n(ks)Jn(kr)

. (16)

This corresponds to the incident wave plus the total effect of scattering by thetwo cylinders. But with the aid of (15), this may be written in the simple form:

(1)(r,,z) = gA cosh k(z + h)cosh kh

n=

an

H(2)n (kr) H

(2)

n (ka)Jn(ka)Jn(kr)

ein(

2). (17)

3. Second order diffraction problem

For the second order potential, the governing differential equation is

2(2) = 0 in the fluid region. (18)

Bottom boundary condition is

(2)

z= 0 on z = h for rj a. (19)

Body surface boundary condtion satisfies the following

(2)

rj= 0 on rj = a for h z 0. (20)

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Free surface condition becomes2(2)

t2+ g

(2)

z=

t

(1)

2+

1

g

(1)

t

z

2(1)

t2+ g

(1)

z

on z = 0 for rj a. (21)

3.1. Derivation of the free surface term for a pair of cylinders

Here we discuss the combined free surface condition (21) due to the secondorder diffraction. Since

(1) = Re

(1)eit

=(1)eit + conjugate of

(1)eit

2

,

this yields

t

(1)

r

2= Re

i

(1)

r

2e2it

.

Thus we can write

t

(1)

2= Re

i(1)

2e2it

. (22)

Also

(1)

t

z

2(1)

t2+ g

(1)

z

=

1

2Re

ig(1)

2(1)

z 2

2

g

(1)

z

e2it

ig(1)

2(1)

z

2

2

g

(1)

z

where denotes the complex conjugate. Since (1)takes the form (1)(r,,z) =cosh k(z + h) (r, ), the above result can further be simplified using the factthat the second term in the right hand side is purely imaginary as

(1)

t

z

2(1)

t2+ g

(1)

z

=

1

2Re

ig(1)

2(1)

z 2

2

g

(1)

z

e2it

.(23)

Assuming (2) takes the form

(2)(r , ,z ,t) = Re

(2)(r,,z)e2it

. (24)

Here we consider the oscillatory second order potential with frequency 2. Sofrom (21), we get

(2)z

42g

(2) = i2g

(1)

2(1)z 2

2g

(1)z

i

g

(1)

2 (25)

which yields

(2)s

z

42

g(2)s = F(r, ) (26)

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where

F(r, ) =i

2g(1)

2(1)

z 2

2

g

(1)

z

(2)i

z

42

g

(2)i

i

g

(1)

2(27)

at z = 0. Here (2)i and

(2)s are second order incident and scattered potentials

respectively. Our interest is to evaluate free surface term due to the second orderscattered potential, i.e., F(r, ).

Since the incident wave also satisfies the free surface condition, we have

F(r, ) = i

2g

(1)i

2(1)i

z 2

2

g

(1)i

z +i

g (1)i

2

+i

2g(1)

2(1)

z 2

2

g

(1)

z

i

g

(1)

2(28)

which can be written as

F(r, ) =i

2g

(1)

2

(1)s

z 2

2

g

(1)s

z

4

(1)i

r

(1)s

r+

(1)i

r

(1)s

r

+

(1)i

z

(1)s

z

+ (1)s

2

(1)i

z 2

2

g

(1)i

z

2

(1)s

r

2+

(1)s

r

2+

(1)s

z

2. (29)

Now assuming that (1)i and

(1)s are of the forms

(1)i (r,,z) =

n=

Pn(r, z)ein, (30)

(1)s (r,,z) =

n=

Qn(r, z)ein (31)

where Pn and Qn respectively are given by

Pn(r, z) =gA

cosh k(z + h)

cosh kh(i)nJn(kr), (32)

Qn(r, z) =gA

cosh k(z + h)

cosh kh(i)nanH(2)n (kr) +

l=

alH(2)l+n(ks)Jn(kr) (33)

and using the property of the product of two infinite series, we can write

(1)i

(1)s =

n=

m=

PnmQm

ein. (34)

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Hence F(r, ) can be expressed as

F(r, ) =

n=

Fn(r)ein . (35)

Now, at z = 0, we get

(1)

2

(1)s

z 2

2

g

(1)s

z

+ (1)s

2

(1)i

z 2

2

g

(1)i

z

=

n=

m=

k2

1 tanh2 kh

(2PnmQm + QnmQm)

ein , (36)

(1)sr

2

+(1)s

r

2

+(1)s

z

2

=

n=

m=

k2 tanh2 kh

(nm)m

r2

QnmQm

+Qnm

r

Qm

r

ein (37)

and

(1)i

r

(1)s

r+

(1)i

r

(1)s

r+

(1)i

z

(1)s

z

=

n=

m=

k2 tanh2 kh (nm)m

r2 PnmQm+

Pnm

r

Qm

r

ein (38)

So finally, at z = 0, we have

Fn(r) =

m=

i

2g

k2

13tanh2 kh

+ 2

(nm)m

r2

QnmQm + 2PnmQm

2

Qnm

r

Qm

r+ 2

Pnm

r

Qm

r

. (39)

This is the expression which contributes to the free surface term due to the

second order scattered potential for a pair of cylinders.

3.2. Derivation of the free surface term for a single cylinder

To derive the results for a single cylinder from the results for a pair of cylin-ders, we assume that (1) is of the form Re{(1)eit}, which yields

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(1)i (r,,z) =

gA

cosh k(z + h)

cosh kh

n=

(i)nJn(kr)ein

and for a single cylinder, the scattered potential is given by

(1)s (r,,z) =gA

cosh k(z + h)

cosh kh

n=

J

n(ka)

H(2)

n (ka)(i)nH(2)n (kr).

For an isolated cylinder, equation (15) reduces to

anH

(2)

n (ka)

J

n(ka)= 1 (40)

from which it is clear that the total velocity potential function becomes

(1)(r,,z) =gA

cosh k(z + h)

cosh kh

n=

(i)nein

Jn(kr) + anH(2)n (kr)

(41)

and equation (33) becomes

Qn(r, z) =gA

cosh k(z + h)

cosh kh(i)nanH

(2)n (kr). (42)

Now the free surface term in this case can be written as equation (39) where Qnis given by the equation (42).

4. Conclusion

Here we presented an analysis of the non-homogeneous term involved in thefree surface condition due to second order wave diffraction by a pair of cylinders.Here we derived the free surface condition for the second order scattered poten-tial. Solving a complex matrix equation, the first order scattered potential hasbeen derived and since the free surface term for second order scattered potentialcan be expressed in terms of the first order potentials, the free surface term isobtained using the knowledge of first order potentials only. We also derived thefree surface term for a single cylinder.

The following studies can be pursued in future regarding the behaviour of thenonlinear free surface condition:

(a) for a single cylinder due to diffraction and radiation,(b) for a pair of cylinders due to radiation,(c) for a pair of cylinders due to diffraction and radiation, and(d) similar analysis for multuiple cylinders (more than two).

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O

1 2

x

y

x

hs

z

O

Figure 1. Definition Sketch

5. Acknowledgement

Author would like to thank the referee for his constructive comments andsuggestions. Author also thanks his former colleagues for their suggestions.

References

1. M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover Publications,1972.

2. F. P. Chau and R. Eatock Taylor, Second order wave diffraction by a vertical cylinder, J.

Fluid Mech. 240 (1992), 571-599.3. R. G. Dean and R. A. Dalrymple, Water wave mechanics for engineers and scientists.

Prentice-Hall Inc, New Jersey (1984).4. L. Debnath, Nonlinear water waves. Academic Press, London, England (1994).5. R. S. Johnson, A modern introduction to the mathematical theory of water waves . Cam-

bridge University Press, Cambridge (1997).6. M. Rahman, WATER WAVES: Relating modern theory to advanced engineering practice.

Oxford University Press, Oxford (1994).7. M. Rahman and D. D. Bhatta, Second order wave forces on a pair of cylinders, Canadian

Applied Mathematics Quarterly 1(3) (1993), 343-382.

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8. M. J. Simon, Multiple scattering in arrays of axisymmetric wave energy devices, Part 1:A matrix method using a plane wave approximation, J. Fluid Mech. 18 (1982), 273-285.

9. N. W. McLachlan, Bessel functions for engineers. Clarendon Press, Oxford (1961).10. R. Eatock Taylor and S. M. Hung, Second order diffraction forces on a vertical cylinder

in regular waves, Applied Ocean Research 9 (1987), 19-30.

Dambaru D. Bhatta received his Ph.D from Dalhousie University, Halifax, Canada.Currently he is a faculty in the Department of Mathematics, University of Texas-Pan

American, TX, USA. His research interests include applied mathematics, water wave,wave-structure interactions, fractional differential equations, digital signal processing.

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539,USA

e-mail: [email protected]

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