Mathematical models for the prediction of the hydrodynamic pressure distribution and the force on a body entering liquid are investigated. Particular attention is paid to analytical models which are based on the velocity potential given by the classical Wagner theory. Formal use of the Wagner theory provides the loads on an entering body, which are higher than the measured ones. To improve the predictions, the higher order terms in the Bernoulli equation are taken into account within the generalized Wagner model and the Logvinovich model. It is shown that the Logvinovich model corresponds better to the experimental data than the generalized Wagner model. A rational derivation of the Logvinovich model is given in the paper for the two-dimensional case. The analytical models are tested against both numerical and experimental results.

The three-dimensional problem of blunt-body impact onto the free surface of an ideal incompressible liquid is considered within the Wagner theory. The theory is formally valid during an initial stage of the impact. The problem has been extensively studied in both two-dimensional and axisymmetric cases. However, there are no exact truly three-dimensional solutions of the problem even within the Wagner theory. At present, three-dimensional effects in impact problems are mainly handled approximately by using a sequence of two-dimensional solutions and/or aspect-ratio correction factor. In this paper we present exact analytical rather than approximate solutions to the three-dimensional Wagner problem. The solutions are obtained by the inverse method. In this method the body velocity and the projection on the horizontal plane of the contact line between the liquid free surface and the surface of the entering body are assumed to be given at any time instant. The shape of the impacting body is determined from the Wagner condition. It is proved that an elliptic paraboloid entering calm water at a constant velocity has an elliptic contact line with the free surface. Most of the results are presented for elliptic contact lines, for which analytical solutions of the inverse Wagner problem are available. The results obtained can be helpful in testing other numerical approaches and studying the influence of three-dimensional effects on the liquid flow and the hydrodynamic loads.

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.

Hydroelasticity brings together hydrodynamics and elastic theories. It is concerned with deformations of elastic bodies responding to hydrodynamic excitations, which themselves depend on elastic deformation. This Theme Issue is intended to identify and to outline mathematical problems of modern hydroelasticity and to review recent developments in this area, including physically and mathematically elaborated models and the techniques used in their analysis.