Practical representation of flows due to general singularity distributions for ships steadily advancing in calm water of finite depth

Abstract

Flow around a ship that advances at a constant speed V in calm water of uniform finite depth D is considered within the practical, realistic and commonly-used framework of the Green-function and boundary-integral method in conjunction with potential-flow theory. This framework entails accurate and efficient numerical evaluation of flows due to singularities (sources, dipoles) distributed over flat or curved panels of diverse geometries (quadrilaterals, triangles) that are employed to approximate the ship hull surface. This basic core element of the Green-function and boundary-integral method is considered for steady ship waves in the subcritical flow regime gD / V² > 1 and the supercritical flow regime gD / V² < 1, where g is the acceleration of gravity. The special case of deep water is also considered. An analytical representation of flows due to general distributions of singularities over hull-surface panels is given. This flow-representation adopts the Fourier-Kochin method, which prioritizes spatial integration over the panel followed by Fourier integration, in contrast to the conventional method in which the Green function (defined via a Fourier integration) is initially evaluated and subsequently integrated over the panel. The mathematical and numerical complexities associated with the numerical evaluation and subsequent panel integration of the Green function for steady ship waves in finite water depth are then circumvented in the Fourier-Kochin method. A major advantage of this method is that panel integration merely amounts to integration of an exponential-trigonometric function, a straightforward task that can be accurately and efficiently performed. The analytical flow-representation proposed in the study offers a smooth decomposition of free-surface effects into waves, defined by a regular single Fourier integral, and a non-oscillatory local flow, characterized by a double Fourier integral featuring a smooth integrand that primarily dominates within a compact region near the origin of the Fourier plane. Illustrative numerical applications to the flow potentials and velocities associated with a typical distribution of sources over a panel show that the flow-representation given in the study yields a practical method well suited for accurate and efficient numerical evaluation.