Novel Numerical Approach for 1D Variable Density Shallow Flows over Uneven Rigid and Erodible Beds

Abstract

The numerical modeling of hyperconcentrated shallow flows is a challenging task because they exhibit special features, such as propagation over dry beds, profound bed elevation modifications owing to erosion or deposition phenomena, and flow discontinuities. In this paper, a novel depth-positivity preserving Harten, Lax, and van Leer—contact (HLLC) Riemann solver is devised in order to approximate the solution of the Riemann problem for the 1D (one-dimensional) hyperconcentrated shallow flows equations over horizontal beds. The solver is used as a building block for the construction of hyperconcentrated shallow flows (HCSF), a well-balanced finite-volume scheme for the solution of the hyperconcentrated shallow flows equations with variable elevation. HCSF is able to handle the case of dry beds, to take into account the variability of the topography also in the presence of bed discontinuities, considering the flow resistance and the mass exchange between the flowing mixture and the mobile bed. The numerical tests carried out confirm the well-balancing property of the scheme proposed, the robustness in the presence of dry beds, the ability to approximate the analytic solution of problems with smooth or discontinuous beds, and the ability to reproduce reasonably the results of a laboratory experiment.