A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

Abstract

Abstract.The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.Mathematics Subject Classification. 35Q30, 35Q35, 76D05.

A Multilayer Method for the Hydrostatic Navier-Stokes Equations: A Particular Weak Solution

“…The governing equations of the system are similar to the one exhibited and analyzed in [8], assuming hydrostatic pressure too and using therein an asymptotic development technique. However the source terms, namely the momentum transmission through the artificial interfaces, and the vertical velocities are handled in a quite different way.…”

“…Then, we impose a boundary condition until it reaches a stationary solution. For the shallow water equations it is a classic test, by imposing a constant inflow (see [4,8,10,13]). The multilayer approach allows us to introduce a variation of this test, by considering a constant in time inflow with a linear vertical profile.…”

“…The non-hydrostatic case will be issued in a forthcoming paper. The model we derive allows also mass and momentum transference between the neighboring layers as the one proposed in [8]. However, here the discontinuity of the velocity and the pressure at the interfaces are related to the normal flux jump conditions associated to mass and momentum conservation.…”

“…Multilayer Saint-Venant (or shallow water) models are commonly used techniques to study hydrodynamic flows with large friction, with significant water depth and/or with important wind effects, among others (see for example [5], [8], [17]). In these cases, the standard shallow water system is considered invalid since the horizontal velocity can hardly be approximated by a vertically constant velocity in the whole domain.…”

“…That model is also extended to 3D computations of free surface flows with friction and viscosity effects by Audusse et al in [6]. A different multilayer model with a hydrostatic framework is proposed by Audusse et al in [8]. Each layer is pre-set, described by its height and by a vertically constant horizontal velocity.…”