Hydrodynamic loads on flat plate entering water

We consider a two-dimensional problem of floating plate which starts suddenly to penetrate water. The analysis is focused on early stage, during which the hydrodynamic loads are high. The liquid is assumed ideal and incompressible, gravity and surface tension effects are not taken into account. Method of matched asymptotic expansions is used to derive second-order uniformly valid solution of the problem. Non-dimensional plate displacement plays the role of the small parameter of the problem. The initial flow close to the plate edges is approximately self-similar and is governed by non-linear boundary-value problem with unknown shape of the free surface. The non-linear self-similar inner solution is matched to the second-order outer solution and is obtained numerically by the boundary-element method. The pressure distribution along the plate is obtained with the help of the nonlinear Bernoulli equation. The hydrodynamic pressure is integrated asymptotically with the aim to derive evolution of the hydrodynamic force acting on the plate during the early stage of impact. It is shown that the inner solution provides important contribution to the hydrodynamic force. We obtained that the initial asymptotics of the loads involve negative non-integer powers of the plate displacement and the log-term.