A numerical model for three-dimensional shallow water flows with sharp gradients over mobile topography

Abstract

Abstract This study aims to develop a three-dimensional (3-D) numerical model for shallow water flows over mobile topography, which is capable of simulating morphological evolution under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations (SWEs) using a finite volume method on prismatic cells in σ-coordinates. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. Using a relaxation approach, a hyperbolic system is built for hydrodynamic system, which allows for using a Godunov-type central-upwind method to capture the shocks and approximate the numerical fluxes. Consequently, the 3D-SWEs-Exner model proposed in the present study can stably and accurately solve the dam-break flows over mobile beds. A spatially and temporally second-order “prismatic” central-upwind method is used to approximate the numerical fluxes through cell interfaces. The Exner equation is solved using an upwind method. Using spatially linear reconstruction and explicit two-stage Runge–Kutta time discretization, second order accuracy is achieved in space and time. The proposed model can preserve the well-balanced property due to the special discretization of bed-slope source terms. The proposed model is validated by several tests with experimental measurements, and is compared with the simulated results using reported two-dimensional (2-D) models.