Liquid flow close to intersection point

The plane unsteady problem of water impact is considered with focusing the attention on the flow near the intersection point between a liquid free surface and a rigid contour. The correct description of the flow field in this region is highly expensive in spite of its small influence on the total hydrodynamic load. On the other hand, in numerical study of the slamming problem accurate treatment of jet flow can improve numerical algorithms in use and increase accuracy of the numerical solutions. We stay within the non-linear potential theory of ideal incompressible liquid flow generated by a floating wedge impact. The liquid initially occupies a lower half-plane (y < 0) and is at rest. Initial draft of the wedge is h_0 and the deadrise angle of the wedge is gamma. The parts of the liquid boundary y = 0, x < x_c and x>x_c, where x_c = h_0 cot gamma, correspond to the initial position of the liquid free surface. At some instant of time which is taken as the initial one, the wedge begins to move down at a constant velocity V . We shall determine the liquid flow, position of the free surface and the pressure distribution at each instant of time t > 0. The numerical method proposed by Longuet-Higgins and Cokelet for free-surface flows with non-linear boundary conditions is employed. The free surface position is updated at each time step with the help of numerical solution of the corresponding mixed boundary-value problem. This solution may be singular at the intersection points, where the type of the boundary condition changes, and which are usually corner points of the flow domain. The singularities at the intersection points influence the numerical solution at all subsequent time steps, and thus can have disastrous cumulative effects. In order to avoid the difficulties with the time-stepping numerical method, it is suggested to distinguish small vicinities of the intersection points at each time step and to build there approximate analytical solutions matching them with the numerical solution in the main flow region (section 2). The numerical solution is obtained by the boundary-element method. Initial conditions for the numerical solution are obtained by the method of matched asymptotic expansions in section 3. Several models have been suggested in the past, which are based on the idea to cut off the jet and to replace it with a suitable boundary condition to be applied on the jet cut. Zhao & Faltinsen [1] suggested to cut the jet there, where the angle between the tangential to the free surface and that to the body contour drops below a small given value. The flow in the jet region is not considered. The cut is orthogonal to the body contour and a linear variation of the velocity potential along the cut is assumed. The normal derivative of the velocity potential obtained by solving the discretized boundary integral equation is used to move the cut. A slightly different approach has been suggested by Fontaine & Cointe [2]. Within this approach it is suggested to cut off the jet there, where the jet thickness becomes smaller than a given limiting value. The normal velocity on the jet cut is assumed to be equal to the tangential velocity on the body contour. Both these models have been found to work well and in good agreement with similarity solutions [3]. It is expected that both models are approximately equivalent to each other for small deadrise angle, which follows from the asymptotic analysis of the wedge-entry problem. On the other hand, both models are not easy to justify for moderate and large deadrise angles, where the "cut-off" technique is also attractive to use. It should be noted that the reliabilities of these models are highly dependent on the limiting values for the jet angle or the jet thickness. The values cannot be t