On the Integration of the Diffraction-Radiation with Forward Speed Green Function

Abstract

The article summarizes the difficulties involved in calculating the various integrals required to develop a first-order panel method to calculate the behavior of a ship in a seaway using the diffraction-radiation with forward speed Green function. This function automatically satisfies both a linearized form of the free surface boundary condition and the radiation condition. Use of the third Green identity leads, for lifting flows, to the calculation of boundary integrals of this Green function and its first and second derivatives on ship-hull panels, on waterline segments, and on semi-infinite strips extending from the trailing edges of the hull’s lifting parts to downstream infinity. These integrals are computed after having interchanged the boundary and Fourier integrals. The first integrals are calculated analytically, using the Stokes theorem. The last integrals are computed numerically using an Adaptive Simpson method of integration, which controls the numerical error. The level of difficulties decreases from the calculation of the function, to the integration on semi-infinite strips, on waterline segments or on panels respectively. The level of difficulty increases with the order of the derivatives. Difficulties also increase close to the free surface, particularly for the waterline integrals. A technique to calculate these various integrations is presented.