Practical flow-representations for arbitrary singularity-distributions in ship and offshore hydrodynamics, with applications to steady ship waves and wave diffraction-radiation by offshore structures

Abstract Diffraction-radiation of regular waves by offshore structures and flows around ships advancing in calm water or in regular waves are commonly analyzed via potential-flow methods based on the Green functions that satisfy the corresponding free-surface boundary conditions. This realistic, practical, and widely-used approach requires evaluation of free-surface flows due to arbitrary (notably constant, linear or quadratic) distributions of singularities (sources and dipoles) over (flat or curved) panels of various shapes (notably rectangles and triangles). Indeed, reliable, efficient and practical methods to evaluate the flows due to arbitrary singularity-distributions over hull-surface panels is a crucial core-element of the Green-function method in ship and offshore hydrodynamics. This core-issue is the object of the study. Free-surface flows due to singularity-distributions over panels (and related ‘influence coefficients’) are ordinarily evaluated via a two-step procedure that involves evaluation of a Green function G and its gradient ∇ G (a Fourier integration) and subsequent integration of G and ∇ G over a panel (a space integration). This common approach involves notorious analytical and numerical complexities related to the complicated singularities of the Green functions in ship and offshore hydrodynamics. An alternative general approach, applicable to generic dispersion relations and arbitrary distributions of singularities, is expounded. This alternative approach is based on a Fourier-Kochin representation of free-surface effects, in which the space integration over hull-panels is performed first and the Fourier integration is performed subsequently. Thus, the Green function and its gradient are not evaluated in this approach. Indeed, this usual first step is bypassed, and the flow due to a singularity distribution is evaluated directly. A major advantage of the Fourier-Kochin method is that the panel-integration is a trivial task as it merely involves integration of an exponential-trigonometric function. A crucial element of the approach expounded in the study is a general analytical decomposition of free-surface effects into waves and a local flow. The waves in this fundamental flow decomposition are expressed as single integrals along the dispersion curves defined by the dispersion relation in the Fourier plane, and the local flow is given by a double Fourier integral that has a smooth integrand dominant within a compact region of the Fourier plane. This analytical flow representation does not involve approximations, i.e. is mathematically exact, as is verified via numerical applications for two main classes of flows in ship and offshore hydrodynamics: ships advancing in calm water, and diffraction-radiation of regular waves by offshore structures. These applications also demonstrate that the general approach expounded in the study provides a practical, remarkably simple, basis that is well suited for accurate and efficient evaluation of flows due to arbitrary singularity-distributions. Indeed, the approach yields a smooth wave and local-flow decomposition that avoids the complexities related to the evaluation and subsequent hull-panel integration of the singular wave and local-flow components in the classical Green functions of ship and offshore hydrodynamics. The general approach expounded in the study is applicable to other classes of flows, notably wave diffraction-radiation by ships advancing through waves in deep water and by offshore structures in finite water-depth, and these important particular applications will be considered in sequels to the study.