Low-Mach type approximation of the Navier–Stokes system with temperature and salinity for free surface flows

Abstract

We are interested in free surface flows where density variations coming, for example, from temperature or salinity differences, play a significant role in the hydrodynamic 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. The obtained models conserve the mass of the fluid but not the volume and satisfy the second principle of thermodynamics.

The Navier-Stokes system with temperature and salinity for free surface flows. Numerical scheme and validation

“…This model describes variable density flows with free surface, the density variations coming from differences in temperature and/or salinity. The model is presented in the companion paper (Boittin et al, 2020), in which a layer-averaged formulation is also given. The layer-averaged formulation suppresses the need for moving meshes (Decoene and Gerbeau, 2009), (Donea et al, 2004).…”

“…The paper is organized as follows. In section 2, the layer-averaged Navier-Stokes-Fourier system and the Euler with variable density introduced in (Boittin et al, 2020) are recalled. A numerical scheme for the layer-averaged Euler model with variable density is presented in Section 3, its properties are studied.…”

“…2 The layer-averaged models minder_models From now on, we just call the Euler system, the Euler system with variable density. We briefly recall the features of the multilayer models presented in (Boittin et al, 2020) and studied here. The multilayer Navier-Stokes-Fourier model is a layer-averaged version of the incompressible, hydrostatic Navier-Stokes-Fourier system…”

“…The system (1)-(3) was derived from the compressible Navier-Stokes system in (Boittin et al, 2020). More specifically, the derivation consisted in performing the incompressible limit.…”

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In this paper, we propose a numerical scheme for the layer-averaged Euler with variable density and the Navier-Stokes-Fourier systems presented in part I (Boittin et al, 2020). These systems model hydrostatic free surface flows with density variations.

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