An analytical approach for the three-dimensional water entry problem of solids creating elliptical contact regions

Abstract

The purpose of the present study is to approach analytically the water entry problem of solids creating elliptical contact regions with the free surface during impact. The governing boundary value problem has been linearized using asymptotic analysis and considering the early stages of the phenomenon. The fundamental assumption is made that the contact region between the solid and the free surface is always an ellipse. This allows to consider the underlying boundary value problem in elliptical coordinates exploiting the properties of elliptical harmonics. It is shown that for elliptical contact regions the solution sought can be detached from the angular dependence. This finding is extremely beneficial as it allows the derivation of a perturbation sequence of mathematical systems of mixed type, involving Fredholm integral equations, which can be treated using rigorous methods of solution for mixed boundary value problems in potential theory. The theory developed for elliptical contact regions during water entry situations, is accordingly applied for anticipating the details of the slamming phenomenon during the free-surface impact of a nonaxisymmetric prolate spheroid assuming that the contact line between the liquid and the solid is always an ellipse. The solid is assumed to descend steadily, without rotating, towards an initially undisturbed free surface. The perturbation technique requires that the slenderness parameter is small compared to unity, while slenderness affects significantly the hydrodynamic load exerted on the solid during water entry. Numerical results are presented and discussed using both the von Karman and the Wagner models of water impact. Finally, the study was extended to consider the 3D Wagner water impact problem of an elliptic paraboloid and to compare computations against available data reported in the past.