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

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 fluid layer. This leads us to consider a shallow water type system for the fluid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classic Saint-Venant–Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classic models do not have an associated energy and how they can be modified in order to recover a model with this property. (ii) The model incorporates naturally a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that in general this modification is different from the ones considered classically. Nevertheless, it coincides with a classic one in the case of constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, which allows to ensure sediment mass conservation. Moreover, we include a simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to a generalization of classic ones such as Meyer-Peter and Müller and Ashida–Michiue models. Three numerical tests are presented to study the evolution of a dune for several definition of the repose angle, to see the influence of the proposed definition of the effective shear stress in comparison with the classic one, and by comparing with experimental data