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

Abstract

In this work we present a mutilayer approach to the solution of non-stationnary 3D Navier-Stokes equations. We use piecewise smooth weak solutions. We approximate the velocity by a piecewise constant (in z) horizontal velocity and a linear (in z) vertical velocity in each layer, possibly discontinuous across layer interfaces. The multilayer approach is deduced by using the variational formulation and by considering a reduced family of test functions. The procedure naturally provides the mass and momentum interfaces conditions. The mass and momentum conservation across interfaces is formulated via normal flux jump conditions. The jump conditions associated to momentum conservation are formulated by means of an approximation of the vertical derivative of the velocity that appears in the stress tensor. We approximate the multilayer model for hydrostatic pressure, by using a PVM finite volume scheme and we present some numerical tests that show the main advantages of the model: it improves the approximation of the vertical velocity, provides good predictions for viscous effects and simulates re-circulations behind solid obstacles.

A fast finite volume solver for multi-layered shallow water flows with mass exchange

“…The main advantage of these models is the fact that the multi-layer shallow water model avoids the expensive three-dimensional Navier-Stokes equations and obtains stratified horizontal flow velocities as the pressure distribution is nearly hydrostatic. Applying the same idea to a single fluid lead to the development of a second class of multilayer models, see [7,3,6,14] among others. Both classes of multilayered models have some links but contain also huge differences since for the second class the introduction of layers along the vertical direction is no more linked to the nature of the flow but it is a way to obtain a more accurate description of the flow than the classical shallow water equations.…”

“…This later multi-layered model is adopted in the present paper and numerically solved. Recently, a rigorous derivation of multilayer shallow water equations has been presented in [14]. The essential difference between this class of models and the multilayer shallow water models developed in [7,6] and adopted in the present study lies on the derivation of the viscous terms in the models.…”

“…The essential difference between this class of models and the multilayer shallow water models developed in [7,6] and adopted in the present study lies on the derivation of the viscous terms in the models. The authors in [14] derived the multilayer model from the hydrostatic Navier-Stokes equations to obtain a fully justified viscosity tensor whereas, in [7] the multilayer model is derived from the the conventional Navier-Stokes equations using the shallow water assumption in the sense that some components of the viscosity tensor are very small and can be neglected. On the other hand, starting from the hydrostatic Euler equations multilayer model is derived in [6] without considering the shallow water assumption and ad-hoc vertical and horizontal viscous terms are added a posteriori.…”

“…On the other hand, starting from the hydrostatic Euler equations multilayer model is derived in [6] without considering the shallow water assumption and ad-hoc vertical and horizontal viscous terms are added a posteriori. We should point out that the viscous terms considered in the current model can be viewed as a simplified vertical viscosity used to impose local coupling between the layers while a fully detailed version of the viscous tensor in multilayer shallow water equations are given in [14]. Needless to mention that the emphasis in the present work is on the hyperbolic part of multilayer shallow water equations which is the same in all the models developed in [7,6,14].…”

“…We should point out that the viscous terms considered in the current model can be viewed as a simplified vertical viscosity used to impose local coupling between the layers while a fully detailed version of the viscous tensor in multilayer shallow water equations are given in [14]. Needless to mention that the emphasis in the present work is on the hyperbolic part of multilayer shallow water equations which is the same in all the models developed in [7,6,14]. The main objective of this study is to develop a fast finite volume solver for multi-layered shallow water flows with mass exchange.…”