STARTING FLOW GENERATED BY A FLOATING WEDGE IMPACT

The small time analysis of the starting flow close to the intersection between the liquid surface and a floating wedge is considered. The liquid and the body are initially at rest. At some instant of time taken as the initial one (t = 0), the floating wedge with deadrise angle gamma impulsively starts to move down with a constant velocity V . The liquid is assumed ideal and incompressible, and its flow potential. We shall determine both the liquid flow and position of the free surface, which are uniformly valid during the initial stage, where V t/h0 << 1, h0 is the initial draft of the floating wedge. The pressure-impulse theory was used by Sedov (1935) to obtain the liquid flow just after the impact instant. Within this theory the boundary conditions are linearised and imposed on the initially undisturbed liquid boundary. The theory predicts the flow singularity at the intersection points. The right-hand side intersection point, x = xc, y = 0, where xc = h0 cot gamma, is considered below. In a small vicinity of this point, r/h0 << 1, the velocity potential phi(x,y,t) behaves like (Iafrati & Korobkin 2000) phi simeq -A r^{sigma_0} cos(sigma_0 theta) where sigma_0=pi/(2beta), beta=pi-gamma and (r,theta) are the cylindrical coordinates with the origin at the intersection point (see figure 1). The coefficient A depends on the entry velocity V , the deadrise angle gamma and the initial draft h0. Here 1/2 < sigma_0 < 1, which implies that the solution by Sedov predicts non-physical behavior of the flow close to the intersection point and has to be considered as the first-order 'outer' solution. The higher-order 'outer' solution can be derived using the small-time expansion procedure. In order to obtain uniformly valid description of the flow during the initial stage, an 'inner' solution must be considered within stretched variables. It is shown that in the leading order as V t/h0 to 0 the inner solution is non-linear, self-similar and depends only on the wedge deadrise angle gamma. Deflection of the inner free surface cannot be neglected even in the leading order in contrast to the Sedov's solution. The solution of the inner problem is achieved by decomposing the fluid domain in three parts: the far-field region, the intermediate region and the jet region. Asymptotic methods are used to evaluate both the shape of the free surface and the velocity potential in the far-field region. The flow in the jet region is described within the shallow-water approximation. The solution in the intermediate region is determined numerically by iterations. A boundary integral representation is used for the velocity potential. Shape of the free surface in the intermediate region is obtained using a pseudo-time stepping procedure developed. The shallow-water solution in the jet region is updated at each step of the iterations and is directly incorporated into the solution of the boundary-value problem in the intermediate region. It is shown that the developed procedure is stable and provides the combined numerical-asymptotic solution of the inner problem with a given accuracy.