Relationship Between Water Surface Elevation and Mean Water Depth for a Partially Wetted Rectangular Grid Cell in Numerical Simulations of Shallow Water Flows with Complex Topography

We propose a relationship between the water surface elevation and the mean water depth for partially wetted rectangular grid cells in the numerical simulation of shallow water flow with complex topography, and apply it to the numerical simulation of shallow water flow to verify the correctness of the relationship. Many natural terrains have complicated surface topography. It is very important to accurately predict the steep-fronted flows that occur after heavy rainfall flash floods or as inundation from dyke breaches. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. In order to achieve a numerical balance of flux gradient and the source terms and to avoid numerical instability, a relationship between the water surface elevation and the mean water depth was derived if wetted and dry parts coexist within a rectangular grid cell. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. Next, these relationships are applied to numerical simulations of three-dimensional shallow water flows with three humps and the validity of the proposed relationship is discussed. The relationship presented in this paper accurately reflects time-varying water regimes and boundaries in flow problems with moving wetting and drying zone interfaces and can be used for numerical simulations of three dimensional shallow water flows with arbitrary topography.