An ideal fluid jet impinging on an uneven wall

Abstract

The two-dimensional free-surface problem of an ideal jet impinging on an uneven wall is studied using complex-variable and transform techniques. A relation between the flow angle on the free surface and the wall angle is first obtained. Then, by using a Hilbert transform and the generalized Schwarz–Christoffel transformation technique, a system of nonlinear integro-differential equations for the flow angle and the wall angle is formulated. For the case of symmetric flow, a compatibility condition for the system is automatically satisfied. In some special cases, for instance when the wall is a wedge, the problem reduces to the evaluation of several integrals. Moreover, in the case of a jet impinging normally on a flat wall, the classical result is recovered. For the asymmetric case, a relation is obtained between the point in the reference ζ-plane which corresponds to the position of the stagnation point in the physical plane, the flow speed and the shape of the wall. The solution to a linearized problem is given, for comparison. Some numerical solutions are presented, showing the shape of the free surface corresponding to a number of different wall shapes.