Integral representation of linear free-surface potential flows for bottom pressure calculation

A numerical method for evaluating the steady, finite-depth Green function and a new boundary-integral representation of velocity in a potential flow region are presented in this paper. Unlike the classical formulation using Green identity where the velocity potential ¿ in the fluid domain is given in terms of boundary values of ¿ and its normal derivative, the new representation defines the velocity components u and v directly in terms of the velocity distribution at the boundary control surface. This difference is very useful when the near-field flow is computed using a RANS solver, and far-field flow parameter such as bottom pressure is desired. Typical RANS solution does not involve the potential ¿ and is not very accurate in the far field because of numerical damping, coarse grid, and inexact radiation conditions. However, RANS solution can be used to prescribe the velocity distribution at a control surface enclosing the ship. This control surface is large enough so that the flow outside can be considered linear and potential. The new integral representation can be applied to this outer potential flow region to quickly compute the velocity components and pressure on the seafloor. The new representation is similar to that given in [1] for infinite fluid depth, but it differs in one important aspect. Instead of using a Fourier-Kochin formulation, the new representation expresses the velocity explicitly in terms of the derivatives of the Green function. This was done intentionally to take advantage of the new numerical method for the Green function.