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 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 classical SaintVenant-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 modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that this modification is different 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 simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to the generalization of classical ones such as MeyerPeter&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 classical one, and by comparing with experimental data.

A simple three‐wave approximate Riemann solver for the Saint‐Venant‐Exner equations

“…Even for the simpler case of the classical SW system, the way to ensure a fully discrete energy inequality is not obvious (see the work of Bouchut). It is then much more tricky for the SWExner model, where issues already occur at the continuous level (see the works of Fernández‐Nieto and Zabsonré et al). One notes that the nonconservative product in the momentum Equation (1b) is not well defined when both bottom topography and water depth are discontinuous at the same point .…”

“…Both the separation between 2 distinct types of transport and the SWExner model itself present important limitations and drawbacks that have been discussed in a series of recent papers,() where the authors exhibit different limitations of the classical SWExner model and introduce some improvements by different ways. In this work, we do not discuss this point and focus on the numerical strategy to compute the approximate solutions of system (1).…”

Comparison of models for the simulation of landslide generated Tsunamis

“…These peaks seem artificial, due to our strategy of imposing the bottom displacement and the fact that we do not have a proper energy closure. Studies in the literature that impose the landslide displacement using the same kind of simplified model (for example [18,19]) do not present such kind of peaks. However the dynamic of the landslide is significantly different.…”

“…From the description of the models §2.2, an important difference can be noticed by the fact that the simplified models ( 6) and ( 7) do not describe the dynamic of the landslide but only the dynamic of the water. Several strategies exist in the literature to model landslide using the same kind of simplified model [18,19]. However the difference of the dynamic of the landslide obtained by this strategy compared to the result of the multiphase Navier-Stokes model ( 2)-( 5) will make the comparison of the free surface unclear.…”

Asymptotic derivation and simulations of a non-local Exner model in large viscosity regime

“…This model gave satisfactory results when the density of the sediment layer is close to that of the water layer, but did not perform well for heavier sediment layers. Another improved description was proposed in [11], in which a shallow water-Exner system is derived from a stratified bi-fluid Navier-Stokes equations. It follows that the solid flux depends not only on the water velocity but also on the slope of the free surface.…”