Practical evaluation of flows due to arbitrary singularity distributions in the 3D theory of ship motions in regular waves at τ<1∕4

Abstract The 3D frequency-domain analysis of the flow around a ship advancing in regular waves, in deep water, via the Green-function and boundary-integral method requires a practical and reliable way of evaluating the flow due to an arbitrary distribution of singularities (sources and dipoles) over the (flat or curved, quadrilateral or triangular) panels that are commonly used to approximate a ship hull-surface. This crucial element of the Green-function method, also widely called panel method, in a 3D analysis of ship motions is classically performed via a two-step procedure that involves evaluation of the Green function G and its gradient ∇ G as a first step, and their subsequent integration over hull-surface panels. This usual direct method, widely studied and applied in the literature, involves notorious well-documented mathematical and numerical complexities. A novel alternative method, based on the Fourier–Kochin approach, is expounded in the study. In this method, evaluation of G and ∇ G is bypassed, i.e.  G and ∇ G are not evaluated, and the flow due to a distribution of singularities is evaluated directly, in the manner already expounded for the particular cases of steady flow around a ship advancing in calm water and diffraction-radiation of regular waves by an offshore structure or a ship at zero forward speed. The method expounded in these previous studies and extended here is based on an analytical representation of the flow due to a distribution of singularities (including the special case of a Green function and its gradient) that yields a formal decomposition into a wave component, given by single Fourier integrals along the dispersion curves associated with the dispersion relation, and a non-oscillatory local-flow component given by a double integral that has a smooth integrand mostly dominant within a compact region of the Fourier plane. An important aspect of this analytical flow decomposition is that the wave and local-flow components are smooth, unlike the wave and local-flow components in the expressions for G and ∇ G given in the literature. Moreover, the analytical representation of the flow due to a distribution of singularities given in the study provides a practical mathematical basis that is well suited for accurate and efficient numerical evaluation, as is demonstrated via the illustrative applications reported previously for steady flow around a ship advancing in calm deep water and diffraction-radiation of regular waves by an offshore structure in deep water, and reported here for the more general case of a ship advancing in deep water at a constant speed V through regular waves of frequency ω in the regime τ ≡ V ω ∕ g 1 ∕ 4 where g denotes the acceleration of gravity.