Practical Analytical Approximations For Free-surface Green Functions And Ship Waves

Free-surface Green functions are the fundamental elements in the linear potentialflow theory of ship and offshore-structure hydrodynamics.Specifically,the Green function for steady ship waves,and the Green function of the theory of diffraction radiation of water waves by a body,are of main interest.With the help of a freesurface Green function,a flow in an unbounded 3D flow domain can be formulated on a finite 2D body surface,although this simplification comes at the price of a relatively complicated Green function,of which the practical evaluation is regarded as the main difficulty in performing computations of the flow,as is well known.The analysis of ship waves is a classical topic that has been extensively considered in a broad literature.Three classical analytical approximations to farfield ship waves given by Kelvin,Havelock and Peters are only valid for points located inside,at or outside the cusps of the Kelvin wake.A relatively complicated uniform approximation,given by Ursell,however involves the Airy function and its derivative,and does not provide a simple explicit decomposition of ship waves into transverse and divergent waves.The classical Fourier–Kochin representation of ship waves expresses ship waves as a linear Fourier superposition of elementary waves.The amplitudes of these elementary waves are determined by an integral over the mean wetted ship hull surface.This surface integral is ill suited for analysis,and cannot be evaluated efficiently in the short wave limit.Practical analytical approximations and their applications for free-surface Green functions and ship waves are considered.The present study consists of three parts:(i)Analytical approximations for the practical evaluation of free-surface Green functions.Errors that stem from a practical analytical approximation to the Green function associated with steady linear potential flow around a ship hull are considered.The flow potential,the sinkage,the trim angle and the wave drag evaluated using the integral representation or the analytical approximation of the Green function are in very close agreement.And global approximations,valid within the entire flow region,to the local-flow and wave components in the Green function and its gradient for diffraction radiation of water waves are given.These global approximations provide a particularly simple and highly efficient way of numerically evaluating the free-surface Green functions.(ii)Analytical approximations for farfield ship waves.A simple approximation to farfield ship waves that is uniformly valid for points located inside,outside or at the cusps of the Kelvin wake is given.This approximation,named the Kelvin–Havelock–Peters(KHP)approximation,is then applied to illustrate the influence of the Froude number and the submergence depth on wave patterns created by a submerged point source,a surface-piercing monohull ship,or a fully-submerged body.Analysis shows that appearance of actual wave patterns greatly differs from Kelvin’s classical wave pattern in many cases,notably at high Froude numbers for which a ship wave pattern mostly contains divergent waves that are most apparent well inside the cusp of the Kelvin wake,and at low Froude numbers for which the dominant waves can be found outside the cusps of the Kelvin wake.(iii)Analytical approximations for short ship waves.Short-wave approximations,valid for short(transverse and divergent)waves created by ships that travel at low Froude numbers,and for short divergent waves created by ships traveling at any Froude number,are obtained within the framework of the Neumann–Michell(NM)linear potential-flow theory for steady ship waves.These approximations provide useful insight into the influence of the Froude number and the hull geometry on short ship waves.The short-wave approximations are then coupled with nonlinear analytical relations for inviscid flow along the wave profile at the ship hull surface.The analysis of steepness of divergent ship waves shows that,for a full-scale Wigley hull,divergent waves are too steep to exist inside a broad inner Kelvin wake.Thus,the analytical approximations proposed here provide(i)simple and efficient computational tools that are well suited for routine applications to ship and offshorestructure design,and(ii)theoretical insight into physics of farfield or short ship waves.