Practical evaluation of flows due to arbitrary singularity distributions in the 3D theory of ship motions in regular waves at [formula omitted

Abstract

The Green-function and boundary-element method, widely used in ship and offshore hydrodynamics, requires accurate and efficient numerical evaluation of flows created by (typically polynomial) distributions of singularities (sources and dipoles) over (flat or curved) panels of various shapes (notably quadrilateral or triangular) that approximate the surface of a ship or offshore structure. This crucial core-element of the Green-function and boundary-element method is considered for the 3D theory of ship motions in regular waves in the regime 0.3≤τ≡Vω∕g where V and ω denote the ship speed and the (encounter) wave frequency, and g is the acceleration of gravity. In this regime, the dispersion relation yields two dispersion curves in the Fourier plane (α,β) that are conveniently defined in the Cartesian form α=α∗(β) with −∞<β<∞, whereas the polar representation k=k∗(γ) where k≡α2+β2 and −π≤γ≤π is convenient and used in a related study to represent the three dispersion curves associated with ship motions in the regime τ<1∕4. The complementary analytical representations given in this study for 0.3≤τ and previously for τ<1∕4 provide simple and practical expressions for the flow due to an arbitrary compact distribution of singularities that are suited for accurate and efficient numerical evaluation for all values of τ outside the relatively narrow range 0.25≤τ<0.3.