Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy

Abstract

In this work we present a deduction of the Saint-Venant-Exner model through an asymptotic analysis of the Navier-Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the uid layer. This leads us to consider a shallow water type system for the uid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classical Saint- Venant-Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classical models do not have an associated energy and how to modify them in order to recover a model with this property. (ii) The model incorporates naturally a necessary modi cation that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that this modi cation is di erent of the ones considered classically, and that it coincides with a classical one only if the solution has a constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, what allows to ensure sediment mass conservation. Moreover, we include a simpli ed version of the model for the case of quasi-stationary regimes. Some of these simpli ed models correspond to the generalization of classical ones such as Meyer- Peter&M uller and Ashida-Michiue models. Three numerical tests are presented to study the evolution of a dune for several de nition of the repose angle, to see the in uence of the proposed de nition of the e ective shear stress in comparison with the classical one, and by comparing with experimental data.