Keywords: potential theory, waves, current, inhomogeneous condition, semi-analytical formulation, Sturm-Liouville problem.

The scope of the present paper is the derivation of a semi-analytical formulation for the first-order diffraction potential due to the presence of a vertical surface-piercing cylinder in a flow field in which waves and currents coexist. The specific subject is very important for offshore applications in which floating or fixed structures usually consist of cylindrical elements. In addition the wave-current interaction with a floating structure may be regarded as the opposite problem of the structure moving with a slow speed in a wave field. Usually the methods implemented for treating this problem are numerical [1,2]. This is due to the fact that the presence of the current is accommodated in the free surface boundary condition which in turn becomes inhomogeneous. The inhomogeneous form of the associated equation makes the analytical approach of the solution more difficult for the diffraction potential. On the other hand the derivation of a closed-form solution for the scattered velocity potential due to the presence of a structure within the wave-current flow field provides a robust and reliable tool for calculating the induced hydrodynamic forces, the wave drift damping and the wave run-up on the structural elements. In the present work this is performed by seeking a semi-analytical formulation for the diffraction potential of the resulting complex flow field.

As mentioned before, the main difficulty arises from the boundary condition on the free surface for which it can be shown that it is given by the following relation.

where is the uniform current velocity, r, and z are the coordinates of the cylindrical coordinate system, is the heading angle of the current, is the total potential of the wave field and finally is the steady state potential that describes the perturbation of the current due to the presence of the body. For treating the resulting hydrodynamic problem a perturbation method is followed according to which the diffraction potential that participates in the total potential and is influenced by the velocity of the current is analyzed in a perturbation series with respect to the powers of the normalized velocity . Then the diffraction potential is decomposed into a sufficient number of components which are used for the proper satisfaction of all relevant boundary conditions. Thus, a Sturm-Liouville problem is derived which is treated using the method of integral equations. The solution of the associated problem requires the construction of a one-dimensional Green's function and the manipulation of the inhomogeneous term in order to derive the radial dependent effective pressure distribution on the free surface. Finally the implementation of the above described derivation procedure results in the following form for the current dependent diffraction component:

where and H/2 are the frequency and the amplitude of the undisturbed incident wave, h is the water depth, q_m is the radial dependent effective pressure distribution on the free surface, is the one dimensional Green's function and denotes the vertical eigenfunctions.

The results presented in the paper depict the variation of the complex effective pressure distribution for various eigenmodes. It is shown that although this term is convergent for the integrand of the above equation does not reach a bound limit. This is reflected also on the variation of the diffraction potential and consequently for the total velocity potential of the combined wave-current field. This remark that was highlighted by other authors in the past [3,4], is discussed in the present work using the asymptotic forms of the Bessel functions.

1

B. Buchmann, J. Skourup and K.F. Cheung, K.F, "Run-up on a structure due to second-order waves and current in a numerical wave tank", Applied Ocean Research, 20, 297-308, 1998. doi:10.1016/S0141-1187(98)00022-4

2

M.H. Kim, M.S. Celebi and D.J. Kim, "Fully nonlinear interactions of waves with a three-dimensional body in uniform currents", Applied Ocean Research, 20, 309-321, 1998. doi:10.1016/S0141-1187(98)00025-X

3

G. Bannister, "Wave/current forces on axisymmetrical bodies with vertical axes", Ship Technology Research, 40, 138-156, 1993.

4

S. Malenica, P.J. Clark and B. Molin, "Wave and current forces on a vertical cylinder free to surge and sway", Applied Ocean Research, 17, 79-90, 1995. doi:10.1016/0141-1187(95)00002-I

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