Strong discontinuities in spatial stationary long-wave flows of an ideal incompressible fluid

Abstract

Spatial stationary flows over an even bottom of a heavy ideal fluid with a free surface are considered. Jump relations for flows with a strong discontinuity are studied. It is shown that the flow parameters behind the jump are defined by a certain curve which is an analog of the (θ, p) diagram in gas dynamics. A shock polar and examples of flows with a hydraulic jump are constructed for a particular class of solutions.Introduction. The long-wave approximation is widely used in the theoretical analysis of wave processes because, for real fluids, the asymptotics of solutions at large times is determined by long waves, due to the presence of viscosity. In addition, this approximation provides new results in analytical studies of nonlinear wave processes. The integrodifferential model of stationary ideal fluid flows studied in the present paper considers velocity shear along the vertical and is more complex than the classical shallow-water model.Shear plane-parallel flows with discontinuities were studied in [1], and barotropic fluid flows in [2]. Generalized characteristics and hyperbolicity conditions for systems of integrodifferential shallow-water equations describing spatial stationary shear flows of a free-boundary ideal fluid layer were obtained in [3], and theory of spatial simple waves was constructed in [4].The system of equations describing stationary long waves in spatial ideal fluid flows are not reduced to divergent form. It has been noted [5] that the flow equations can be rearranged so that the nondivergent terms are regular discontinuous functions. This allows one to consider solutions in the class of functions with a strong discontinuity and to derive jump relations from these equations. In the present paper, the indicated approach is used to study stationary hydraulic jumps. Relations linking the flow parameters behind the jump are found, and analogs of (θ, p) diagrams of two-dimensional gas dynamics are proposed. The problem of interaction of spatial shear flows is solved for a definite class.1. Formulation of the Problem. We consider the long-wave equations describing spatial stationary flows of an ideal incompressible fluid in a gravity field: