Unsteady Boussinesq-type flow equations for gradually-eroded beds: application to dike breaches

Abstract

Dike breach due to overtopping involves unsteady flow over a curved bed continuously deforming in time due to erosion. The curvatures and slopes on the water surface and the bed profiles indicate that the shallow-water-equations may be inappropriate for modelling purposes. The next degree of mathematical approximation is given by the Boussinesq equations accounting for vertical inertia effects. Herein novel unsteady flow equations for a curved erodible bed are developed. Using a Boussinesq-type closure, expressions for the vertical velocity, the pressure distribution and the momentum function are presented. These depend on two parameters relating to the temporal and spatial derivatives of the flow depth, namely bottom elevation and average flow velocity. The model equations are compared with previous unsteady Boussinesq-type models, indicating that the novel formulation is a generalization for erodible beds. The Boussinesq coefficients are further investigated using selected experiments conducted at VAW, ETH.