The Navier-Stokes system with temperature and salinity for free surface flows Part I: Low-Mach approximation & layer-averaged formulation

Léa Boittin (1) , François Bouchut (2) , Marie-Odile Bristeau (1, 3) , Anne Mangeney (4, 1) , Jacques Sainte-Marie (1) , Fabien Souillé (1)

1 ANGE - Numerical Analysis, Geophysics and Ecology
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
3 Cerema - Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement
4 IPG Paris - Institut de Physique du Globe de Paris

We are interested in free surface flows where density variations coming e.g. from temperature or salinity differences play a significant role in the hydrody-namic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising from compressible fluid mechanics often requires a decoupling between these waves. Starting from the compressible Navier-Stokes system, we derive the so-called Navier-Stokes-Fourier system in an "incompressible" regime using the low-Mach scaling, hence filtering the acoustic waves, neglecting the density dependency on the fluid pressure but keeping its variations in terms of temperature and salinity. A slightly modified low-Mach asymptotics is proposed to obtain a model with thermo-mechanical compatibility. The case when the density depends only on the temperature is studied first. Then the variations of the fluid density with respect to temperature and salinity are considered, and it seems to be the first time that salinity dependency is considered in this low Mach limit. We give a layer-averaged formulation of the obtained models in an hydrostatic context, allowing to derive numerical schemes endowed with strong stability properties that are presented in a companion paper. Several stability properties of the layer-averaged Navier-Stokes-Fourier system are proved.

Mots clés: Compressible and incompressible fluids; Navier-Stokes equations; Layer-averaged formulation; Low-Mach approximation; Free sur- face flows; Variable density flows

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]; Modélisation et simulation

Léa Boittin, François Bouchut, Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, et al.. The Navier-Stokes system with temperature and salinity for free surface flows Part I: Low-Mach approximation & layer-averaged formulation. 2020. ⟨hal-02510711v1⟩