关键词: 三维浅水方程, 近似黎曼求解器, 静水平衡, 高分辨率, σ坐标系

Abstract: The Godunov finite volume method for solving planar 2D shallow water equations is extended to solve 3D shallow water equations, so as to establish a three-dimensional mathematical model with shock capture characteristic, the application of 3D shallow water equations will be expanded. In this paper, the 3D shallow-water equations in σ-coordinates were solved based on the Godunov-type finite-volume method. The nonlinear k-? model was employed for turbulent flow closure, the HLLC approximate Riemann solver was involved to calculate the horizontal numerical fluxes. To improve the numerical stability, the vertical diffusion-term was implicitly discretized, and the 3D model was locally switched to a horizontally depth-averaged 2D model in which the water depth was sufficiently small. The developed model was form second-order form in both space and time. The model was verified by classical tests including hydraulic jump and dam-break flood propagating in a dry river bed, and the results showed that the developed model is stable, well-balanced, capable of predicting high-resolution solution around discontinuities, and simulates wetting and drying processes well.

Key words: 3D shallow water equations, approximate Riemann solver, balance in calm water, high-resolution, sigma-coordinates