Unsteady interaction of uniformly vortex flows

Abstract

The problem of the decay of an arbitrary discontinuity (the Riemann problem) for the system of equations describing vortex plane-parallel flows of an ideal incompressible liquid with a free boundary is studied in a long-wave approximation. A class of particular solutions that correspond to flows with piecewise-constant vorticity is considered. Under certain restrictions on the initial data of the problem, it is proved that this class contains self-similar solutions that describe the propagation of strong and weak discontinuities and the simple waves resulting from the nonlinear interaction of the specified vortex flows. An algorithm for determining the type of resulting wave configurations from initial data is proposed. It extends the known approaches of the theory of one-dimensional gas flows to the case of substantially two-dimensional flows.