The self-similar solution that characterizes the water impact, with a constant vertical velocity, of a wedge entering the free surface with an arbitrary orientation is derived analytically. The study is carried out by assuming the fluid to be ideal, weightless and with negligible surface tension effects. The solution is based on the complex analysis of nonlinear two-dimensional problems of unsteady free boundary flows and is written in terms of two governing functions, which are the complex velocity and the derivative of the complex potential defined in a parameter domain. The boundary value problem is reduced to the system of an integral and an integro-differential equation in terms of the velocity modulus and of the velocity angle to the free surface, both written as functions of a parameter variable. The system of equations is solved through a numerical procedure which is validated in the case of symmetric wedges. Comparisons with data available in literature are established for this purpose. Results are presented in terms of free surface shape, contact angles at the intersection with the wedge boundary, pressure distribution, force and moment coefficients. For a given wedge angle, the changes induced by the heel angle on the above quantities are discussed. A criterion is proposed to determine the limit conditions beyond which flow separation from the wedge apex occurs. Comparisons with experimental results available in literature are presented.

A nonlinear analysis has been made to determine the effects of the free surface and transverse gravity field on the steady cavity flow past a shaped hydrofoil beneath the free surface. A closed cavity wake model has been proposed, and a method for the determination of an analytical function from its modulus and argument on the region boundary has been employed to derive the complex flow potential in a parameter plane. The boundary-value problem is reduced to a system of integral and integro-differential equations in the velocity modulus along the free boundaries and the velocity angle along the hydrofoil surface, both written as a function of parametric variables. The system of equations is solved through a numerical procedure, which is validated in the cases of a cavitating flat plate and non-cavitating shaped hydrofoils by comparison with data available in the literature. The results are presented in a wide range of Froude numbers and depths of submergence in terms of the cavity and free-surface shapes and force coefficients. The influences of the free surface and gravity on the aforementioned quantities are discussed. The limiting cavity size corresponding to zero cavitation number in the presence of gravity is found for various initial flow parameters.

For the oblique impact of a wedge on a liquid half space, the limiting angles of the entry velocity and the wedge orientation corresponding to flow separation from the wedge vertex during the initial stage of the impact are investigated on the basis of an analytical solution of the problem. The liquid is assumed to be ideal and incompressible; gravity, surface tension, and air cushioning effects are ignored. The flow generated by the impact is two dimensional and potential. The solution is presented in terms of two governing expressions, which are the complex velocity and the derivative of the complex potential defined in a parameter region. These expressions are obtained using generalized integral formulas for solving mixed and uniform boundary-value problems for the first quadrant. They include two unknown functions, which are the velocity magnitude and angle to the free surface determined from the dynamic and kinematic boundary conditions. The obtained system of integral equations is solved by using the method of successive approximations. The effect of the horizontal component of the entry velocity is studied for various wedge orientations. The analysis of the computations revealed configurations of the impact such that the pressure along the whole length of one side of the wedge becomes less than the pressure on the free surface. Although air effects are not included in the analysis, such a pressure distribution provides conditions for the ventilation of the wedge side, which, in the presence of the air, starts from the contact point on the free surface and extends suddenly along the whole length of the wedge side, thus leading to flow separation from the wedge vertex. The theoretical predictions of flow separation and the experimental data on flow separation by Judge et al. [“Initial water impact of a wedge at vertical and oblique angles,” J. Eng. Math. 48, 279 (2004)] are remarkably close to each other.