Title:

Free surface problems for jet impact on solid boundaries

A nonlinear twodimensional 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 generalised SchwartzChristoffel transformation technique, a system of nonlinear integraldifferential equations for the flow angle and the wall angle is formulated. For the case in which the wall geometry is symmetric, a compatibility condition for the system is automatically satisfied. Some numerical solutions are presented, showing the shape of the free surface corresponding to a number of different wall shapes. For the case of in which the wall geometry is asymmetric, a pair of conditions which determine the position of the stagnation point are revealed, using the integral form of the momentum equation. Thus the shapes of the free surface of the jet impinging on a few asymmetric uneven walls are shown. The stagnation point is located for each of the different cases. A flow passing through a porous film and then impinging on a flat solid boundary is analysed. Since the pressure field and fluid velocity are discontinuous along the film, the flow region is divided into two parts. At the film by analogy with Darcy'law the pressure difference is taken proportional to the flow rate. The free surface problem for this transpiration flow is formulated as a system of three coupled integral equations using a boundary integral method. As the system is solved numerically the normal speeds at the film and the shapes of the free surface for different permeability coefficients are presented.
