Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications

Recently, A. Vasseur and C. Yu have proved the existence of global entropy-weak solutions to the compressible Navier-Stokes equations with viscosities $\nu(\varrho)=\mu\varrho$ and $\lambda(\varrho)=0$ and a pressure law under the form $p(\varrho)=a\varrho^\gamma$ with $a>0$ and $\gamma>1$ constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities initiated by E. Feiersl, B.J. Jin and A. Novotny. We present some mathematical results related to the weak-strong uniqueness, convergence to a dissipative solution of compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions of a viscous shallow water system to solutions of the inviscid shall-water system

Numerical scheme for multilayer shallow-water model in the low-Froude number regime

International audienceThe aim of this note is to present a multi-dimensional numerical scheme approximating the solutions of the multilayer shallow water model in the low Froude number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfy the dissipation of energy, which act for mathematical entropy, and the main physical properties required for simulations within oceanic flows. Résumé Schéma numérique pour lesmo eles de Saint-Venant multi-couchè a faible nombre de Froude. Le but de cette note est de présenter un schéma numérique multi-dimensionnel rapprochant les solutions dumo ele de Saint-Venant multi-couche en régime de faible nombre de Froude. La stratégie proposée est basée sur unmo ele régulariséò u la vitesse de transport est modifié par un gradient de pression dans le equations de la masse et de la quantité de mouvement. La solution numérique satisfait la dissipation denergie,jouantlerôledel'entropiedupointdevuemathématique,etlesprincipalespropriétésphysiquesnécessairesauxsimulationsdanslecadredeecoulementsocéanique

Centered-potential regularization for the advection upstream splitting method

International audienceThis paper is devoted to a centered IMEX scheme in a multidimensional framework for a wide class of multicomponent and isentropic flows. The proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations. The stability of the scheme is ensured by the dissipation of mechanic energy, which stands for a mathematical entropy, under an advective CFL condition. The main physical properties, such as positivity, conservation of the total momentum, and conservation of the steady state at rest, are satisfied. In addition, asymptotic preserving properties in the regimes (“incompressible” and “acoustic”) are analyzed. Finally, several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension

Roll-waves in bi-layer flows

In this paper, we derive consistent shallow water equations for bi-layer flows of Newtonian fluids flowing down a ramp. We carry out a complete spectral analysis of steady flows in the low frequency regime and show the occurence of hydrodynamic instabilities, so called roll-waves, when steady flows are unstable

Multi-Regime Shallow Free-Surface Flow Models for Quasi-Newtonian Fluids

International audienceThe mathematical modeling of thin free-surface laminar flows for quasi-Newtonian fluids (power-law rheology) is addressed with a particular attention to geophysical flows (e.g. ice or lava flows). Asymptotic thin-layer flow models (one-equation and two-equation models) consistent with various viscous regimes, corresponding to different basal boundary conditions (from adherence to pure slip), are derived. The challenge being to derive models consistent from slip to no-slip basal boundary condition, though at the price of balancing small friction by small mean slope. Starting from reference flows (the steady-state uniform ones) corresponding to different shear regimes, the exact expressions of all fields (\bsigma, \bu, p) are calculated formally by a perturbation expansion method.The calculations are such that all field expressions remain valid for any laminar viscous regimes. The calculations are presented either in a mean slope coordinatesystem with local variations of the topography or in the Prandtl coordinate system, hence valid in presence of any non flat basal topography.Formal error estimates proving the consistency of the derivations are stated. An unified one-equation model (lubrication type in the depth variable $h$) is derived at order $1$. Next, few unified two-equation models in variable $(q,h)$ (shallow water type) are stated and discussed.The classical first order models from the literature are recovered if considering the corresponding particular cases (generally, flat bottom with a particular regime and/or specific basal boundary condition). Two one-dimensional numerical examples illustrate the robustness of these new multi-regime formulations (the change of flow regimes being due either to a sharp change of the mean-slope topography or to a sharp change of basal boundary condition)

DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides

DassFlow is a computational software for free-surface flows includingvariational data assimilation (4D-VAR), sensitivity analysis, calibration features (adjoint method). The code version "shallow" solves shallow-water like models (Saint-Venant's type).The other version (ALE, not detailed in the present document) includes free-surface Stokes like models (low Reynolds, power-law rheology, ALE surface dynamics). All source files are written in Fortran 2003 / MPI. For more details and references, please consult DassFlow website.In the present manuscript, we describe: the equations, the compilation/execution instructions, the input / output files (user guide), the finite volume schemes, few validation test cases included in the archive, and the code structure (developer guide)

Matching of Asymptotic Expansions for a 2-D eigenvalue problem with two cavities linked by a narrow hole

One question of interest in an industrial conception of air planes motors is the study of the deviation of the acoustic resonance frequencies of a cavity which is linked to another one through a narrow hole. These frequencies have a direct impact on the stability of the combustion in one of these two cavities. In this work, we aim is analyzing the eigenvalue problem for the Laplace operator with Dirichlet boundary conditions. Using the Matched Asymptotic Expansions technique, we derive the asymptotic expansion of this eigenmodes. Then, these results are validated through error estimates. Finally, we show how we can design a numerical method to compute the eigenvalues of this problem. The results are compared with direct computations