An alternative linear flow model and boundary integral flow representation for ship motions in regular waves

Abstract

The basic issue of formulating a linear flow model for the analysis of potential flow around a ship advancing at a constant speed in regular waves is considered. Specifically, the ‘rigid-waterplane flow model’, previously considered for diffraction–radiation of regular waves by offshore structures, is applied to analyze flows around ships advancing in regular waves. A straightforward application of Green’s fundamental identity to the rigid-waterplane flow model yields a remarkable new boundary integral flow representation. Indeed, this flow representation does not involve a line integral around the mean waterline of the ship hull surface, is weakly singular, does not involve a distribution of dipoles over the ship-hull surface, and is free from irregular frequencies. In the particular case of flow around a ship steadily advancing in calm water, the boundary integral flow representation obtained in the study via an analysis based on the rigid-waterplane flow model is identical to the flow representation associated with the Neumann–Michell theory, which is based on a different linear flow model that only considers the flow region strictly outside the ship but consistently accounts for the linear contribution of the thin band of water between the wave profile along the ship hull surface and the undisturbed free surface. This result strongly suggests that the classical application of Green’s basic identity to analyze linear potential flow around a ship advancing in regular waves or in calm water, which yields a notoriously troublesome line integral around the ship waterline, may be questionable due to possible incompatibilities between the Neumann boundary condition at the ship hull surface and the linear free surface boundary condition associated with flows around ships advancing in calm water or in regular waves. This fundamental difficulty may be alleviated in the rigid-waterplane flow model, which imposes a compatibility condition between the Neumann boundary condition at the rigid lid that artificially closes the open ship hull surface and the linear boundary condition at the free surface above the waterplane-lid. The boundary integral flow representation given in the study also holds for diffraction–radiation of regular waves by offshore structures. Thus, this new flow representation provides a common mathematical basis, which fundamentally differs from the basis steadfastly adopted over the past fifty years, for the analysis of three basic classes of flows in ship and offshore hydrodynamics, and indeed can be applied more generally to other boundary-value problems such as those associated with flexural-gravity waves for ice sheets or very large floating structures.