Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth

Abstract

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number F and the depth d (draft) of the depressed plate. Previous analytical studies have been restricted to small values of d and subcritical flows F<1, when linearized theory can be used. Here, our main concern here is with the critical regime, F≈1, where we use a weakly nonlinear long-wave analysis, For finite values of d, we solve the fully nonlinear problem numerically using a boundary integral equation method. Our results confirm previous studies and extend these to the critical regime. As d increases, our numerical results demonstrate that the downstream waves approach the highest possible wave. We also find some wave-free solutions, which when compared to the weakly nonlinear results are essentially just one-half of a solitary wave.