The Riemann problem for shallow-water equations on a step: stationary solutions, the quasi-two-layer model and numerics

For the problem of a steady-state flow over a fixed step, in the shallow-water approximation the dependence of the flow regime on the flow direction and the ratio of the fluid depth on the right of the step to the step height are found. The range of admissible values of the fluid flow as a function of the fluid mass flow and the step height is determined. We propose an arbitrary discontinuity decay model for shallow-water equations on a step boundary, which is based on the two-layer representation of fluid flows taking into account the flow peculiarities in the vicinity of a step. A distinctive feature of the suggested model is the separation of a studied flow into two layers in calculating flow quantities near each step, and thus improving approximation of depth-averaged solutions to the initial three-dimensional Euler equations. We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Thus, the solution to the problem reduces to an arbitrary discontinuity decay problem for classical shallow-water equations on a plane boundary. The proposed model realizes all analytically possible configurations in the case of presence of the step boundary, qualitatively takes into account the dissipation of translational mechanical energy due to turbulence. A finite-volume numerical method for studying shallow-water flows over the step is proposed. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with an advanced consideration of the flow features near the step. The results of computations showing the efficiency of the algorithm developed are discussed. (paper)