Three-dimensional theory of water impact. Part 2. Linearized Wagner problem

Abstract

The three-dimensional problem of blunt-body impact onto a free surface of an ideal and incompressible liquid is considered within the Wagner approximation. This approximation is formally valid during an initial stage, when the depth of penetration is small, the wetted part of the body can be approximately replaced with a flat disk and the boundary conditions can be linearized and imposed on the undisturbed liquid surface. In the present context this problem will be referred to as the classical Wagner problem. However the classical Wagner problem of impact is nonlinear despite the fact that the equations of liquid motion and boundary conditions are linearized. The reason is that the contact region between the liquid and the entering body is unknown in advance and has to be determined together with the liquid flow. Several exact solutions of the three-dimensional Wagner problem are known as detailed in Part 1 (J. Fluid Mech. vol. 440, 2001, p. 293). Among these solutions the axisymmetric one is the simplest. In this paper, an additional linearization of the Wagner problem is considered. This linearization is performed on the basis of an axisymmetric solution via a perturbation technique. The small parameter $\epsilon$ is a measure of the discrepancy of the actual shape with respect to the closest axisymmetric shape. The method of solution of this problem is detailed here. The resulting solutions are compared to available exact solutions. Three shapes are studied: elliptic paraboloid; inclined cone; and pyramid. These shapes must be blunt in the vicinity of the initial contact point and hence only small deadrise angles can be considered. The stability of the obtained solutions is analysed. The second-order solution of the present Wagner problem with respect to $\epsilon$ is considered. That yields the leading-order correction to the hydrodynamic force which acts on an almost axisymmetric body entering liquid vertically. Other nonlinearities are not accounted for. Among them, there are the nonlinear terms in the boundary conditions and the actual geometry of the wetted body surface. Both the vertical and the horizontal components of the hydrodynamic force are obtained. For the inclined cone, comparisons with available experimental data are shown. The method developed can be helpful in testing other numerical approaches and optimizing the shape of the entering body accounting for three-dimensional effects. This paper appears as a necessary intermediate step before solving the general three-dimensional classical Wagner problem in Part 3.