Initial stage of flat plate impact onto liquid free surface

The liquid flow and the free surface shape during the initial stage of flat plate impact onto liquid half-space are investigated. Method of matched asymptotic expansions is used to derive equations of motion and boundary conditions in the main flow region and in small vicinities of the plate edges. Asymptotic analysis is performed within the ideal and incompressible liquid model. The liquid flow is assumed potential and two dimensional. The ratio of the plate displacement to the plate width plays the role of a small parameter. In the main region the flow is given in the leading order by the pressure-impulse theory. This theory provides the flow field around the plate after a short acoustic stage and predicts unbounded velocity of the liquid at the plate edges. In order to resolve the singular flow caused by the normal impact of a flat plate, the fine pattern of the flow in small vicinities of the plate edges is studied. It is shown that the initial flow close to the plate edges is self-similar in the leading order and is governed by nonlinear boundary-value problem with unknown shape of the free surface. The Kutta conditions are imposed at the plate edges, in order to obtain a nonsingular inner solution. This boundary-value problem is solved numerically by iterations. At each step of iterations the “inner” velocity potential is calculated by the boundary-element method. The asymptotics of the inner solution in both the far field and the jet region are obtained to make the numerical algorithm more efficient. The numerical procedure is carefully verified. Agreement of the computed free surface shape with available experimental data is fairly good. Stability of the numerical solution and its convergence are discussed.
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