A novel well-balanced scheme for spatial and temporal bed evolution in rapidly varying flow

Abstract

This paper presents a novel method of preserving well-balanced conditions for dam-break flows on irregular beds. The research on preservation of well-balanced conditions commonly deals only with spatial variation of beds. It is challenging to design a well-balanced scheme that can model both spatial and temporal variation of beds particularly on domains discretized with unstructured meshes. This challenge arises due to the non-trivial nature of the governing equations modeling hydro-morpho dynamics. This study proposes a new and simple well-balanced method applicable for both spatial and temporal bed variations without requiring additional treatment and complicated discretization of slope source terms. The work relies on a cell-centered Godunov-type finite volume method for solving coupled Saint Venant and sediment continuity equations. The two most important issues, i.e., the 'lake at rest condition' and 'water depth positivity' of shallow-water flows over irregular beds, are tested and verified in the proposed numerical scheme. The authors employed Rieman solvers to compute fluxes through the control volume surfaces. The proposed scheme is evaluated for various benchmark cases of dam-break flows on rigid and erodible beds. Comparisons between the results of the proposed numerical scheme, analytical solution, experimental data, and previous numerical observations show significant improvement for some cases and a good agreement in the rest.