Interaction of shear flows of an ideal incompressible fluid in a channel

Abstract

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established.Introduction. Many mathematical models describing the propagation of long-wave perturbations in shear (vortex) fluid flows reduce to nonlinear integrodifferential equations. A qualitative analysis of various models of long-wave theory was performed in [1] using the generalization of the hyperbolicity concept developed by Teshukov and the method of characteristics for systems of equations with operator coefficients [2]. The results of these studies show both differences and similarities between these models and hyperbolic differential systems (in particular, the presence of a continuous range of characteristic velocities). The evolution of solutions of generalized hyperbolic nonlinear integrodifferential equations can lead to the occurrence of strong discontinuities, which makes it necessary to correctly formulate the equations of motion in the form of conservation laws and to analyze the problem of the decay of an arbitrary discontinuity (Riemann problem).In the present paper, the problem of the decay of an arbitrary discontinuity is considered for nonlinear equations describing shear flows with piecewise constant vorticity in a narrow channel. In this class, a self-similar solution is obtained and studied. In the region of shear flow interaction, the fluid flow is shown to have a substantially two-dimensional unsteady nature. This is manifested in the formation of jet flow along the interface between the flows, which is directed to the upper or lower boundary of the channel, depending on the vorticity ratio. A similar formulation was studied by Teshukov for a free-boundary model [3] with certain constraints imposed on the initial data to satisfy the conditions of strong nonlinearity of the characteristics. In the present paper, a solution of the shear flow interaction problem is constructed without constraints on their vorticities and the corresponding wave configurations that include a simple wave or a simple wave and a shock are analyzed. In the case of interaction of flows with arbitrary monotonic (in depth) velocity profiles, a discretization of the integrodifferential equations is proposed and differential conservation laws are derived.1. Formulation of the Problem. The equations of plane-parallel motion of an ideal incompressible fluid in a channel are written as

Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

“…In this case, the relations at the discontinuity can be used only to determine the thickness of the zone of return flow. Similar configurations for plane-parallel flows were studied in [3,5]. …”

“…A similar analysis for a model of plane-parallel flows of barotropic fluids was performed in [4]. New approaches to the description of the interaction of shear flows of an incompressible ideal fluid were used in [5]. Problems of conjugation of various filtration and channel flows of a viscous incompressible fluid and various mathematical models of two-phase fluids were studied in [6,7].…”