Wedge entry into initially calm water

Abstract

This paper presents results of calculations based on the Cauchy's theorem method of Vinje and Brevig 1 for the two-dimensional entry of wedges of various angles into initially calm water. The problem has a long history which is briefly reviewed in the introduction, and significant progress has been made with both linear theories (valid for low entry speed) and with theories which treat the free surface conditions exactly but with the assumptions of zero gravity and constant speed of entry. This simplifies the problem to one which is self-similar in dimensionless space variables ξ = x/vt and η = y/vt and this has a number of consequences. For wedges with half-angles up to about 45° and with high entry speeds, the numerical approach, which includes gravity, validates these assumptions and the agreement between both free surface displacements and pressure distributions on the wetted wedge surface is excellent except in the region of the jet of fluid which rises up the side of the wedge. Because the potential flow initial value problem is singular at the intersection of the free surface and wedge surface, exact numerical resolution of the jet is not possible. Nevertheless, the rest of the fluid motion is insensitive to the treatment of the jet, which itself may be calculated quite realistically. Of particular interest (but little practical relevance) is the pressure on the upper part of the wedge surface (in the jet region) which according to self-similar theories is very small but positive, but which is calculated to be small but negative by the numerical scheme. This effect, which is enhanced when gravity is included, is insensitive to the numerical resolution of the jet and suggests that the jet may separate from the wedge surface, the new intersection point being where the pressure vanishes on the wedge surface. A modified numerical scheme allows this to happen and the results are in qualitative agreement with the experiments by Greenhow and Lin. 2 The numerical method presented here is extremely versatile and a number of other effects may be explored. Examples of transient motion, non-constant speed of entry, oblique entry and complete penetration of the surface so that a cavity is formed behind the wedge are presented.