Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography

Abstract

Many natural terrains have complicated surface topography. The simulation of steep-fronted flows that occur after heavy rainfall flash floods or as inundation from dyke breaches is usually based on the non-linear shallow water equations in hyperbolic conservation form. Particular challenges to numerical modellers are posed by the need to balance correctly the flux gradient and source terms in Godunov-type finite volume shock-capturing schemes and by the moving wet–dry boundary as the flood rises or falls. This paper presents a Godunov-type shallow flow solver on adaptive quadtree grids aimed at simulating flood flows as they travel over natural terrain. By choosing the stage and discharge as dependent variables in the hyperbolic non-linear shallow water equations, a new deviatoric formulation is derived that mathematically balances the flux gradient and source terms in cases where there are wet–dry fronts. The new formulation is more general in application than previous a priori approaches. Three benchmark tests are used to validate the solver, and include steady flow over a submerged hump, flow disturbances propagating over an elliptical-shaped hump, and free surface sloshing motions in a vessel with a parabolic bed. The model is also used to simulate the propagation of a flood due to a dam break over an initially dry floodplain containing three humps.