Optimal Fourier–Kochin flow representations in ship and offshore hydrodynamics: Theory

Abstract

This study considers the core issue of evaluating flows created by general distributions of sources and dipoles over panels (typically flat/curved triangles/quadrilaterals) used to approximate the surface of a body (ship or offshore structure) in usual implementations of the Green function and boundary integral method in marine hydrodynamics. This crucial basic issue is considered – within the framework of the Fourier–Kochin (FK) method – for four classes of flows in deep-water ship and offshore hydrodynamics: diffraction-radiation of regular waves by an offshore structure, and flow around a ship that advances at a constant speed V in calm water or in regular waves of (encounter) frequency ω in the regimes τ≡ωV/g<1/4 or 0.3≤τ where g is the acceleration of gravity; diffraction-radiation of regular waves by an offshore structure in water of uniform finite depth is also considered. Two notable features of the Green functions G used in the study of these five classes of flows are that (i) they are based on optimal decompositions into Rankine and Fourier components GR and GF, and that (ii) these Green functions are consistent. In particular, a new representation of the Green function for flows around ships advancing in regular waves at τ<1/4 that is consistent with the Green functions for the special cases V=0 or ω=0 is given. The study also provides a complete and largely self-contained account of the FK method for evaluating the Fourier component ϕF in the Rankine–Fourier decomposition ϕR+ϕF of the velocity potential ϕ (and the related velocity) of the flow created by a general distribution of singularities. An essential element of this FK theory is an optimal waves/local-effects (WL) decomposition ϕF=ϕW+ϕL, given for a general dispersion relation associated with a broad class of dispersive waves, in which the component ϕW represents the (far-field and near-field) waves contained in ϕF and the component ϕL corresponds to a non-oscillatory local disturbance that is mostly significant in a small near-field region. Applications of this general WL decomposition to the five classes of flows in marine hydrodynamics of primary interest in the study yield simple analytical flow representations – for general compact distributions of singularities – that offer two notable advantages over the classical direct Green function method: (i) Integration over hull-surface panels within the FK method only involves smooth ordinary functions, which are incomparably simpler than the highly singular functions GF and ∇GF, and (ii) The computations related to GF and ∇GF in the FK theory – implemented in the manner expounded in the study – are proportional to the number N of panels that approximate the body surface, whereas common panel methods require O(N2) computations. Thus, the Fourier–Kochin method and the optimal Rankine–Fourier decompositions and optimal waves/local-flow decompositions given in the study lay the foundation of a new type of computational methods, which offers two compelling advantages over the classical ‘direct Green function method’ steadfastly applied in the past fifty years.