For years, astrophysicists, plasma fusion, and fluid physicists have puzzled over Rayleigh-Taylor turbulent mixing layers. In particular, strong discrepancies in the growth rates have been observed between experiments and numerical simulations. Although two phenomenological mechanisms (mode-coupling and mode-competition) have brought some insight on these differences, convincing theoretical arguments are missing to explain the observed values. In this paper, we provide an analytical expression of the growth rate compatible with both mechanisms and is valid for a self-similar, low Atwood Rayleigh-Taylor turbulent mixing subjected to a constant or time-varying acceleration. The key step in this work is the presentation of foliated averages and foliated turbulent spectra highlighted in our three-dimensional numerical simulations. We show that the exact value of the Rayleigh-Taylor growth rate not only depends upon the acceleration history but is also bound to the power-law exponent of the foliated spectra at large scales.

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-In a recent paper [Poncet R., Peybernes M., Gasc T., De Vuyst F. (2016) Performance modeling of a compressible hydrodynamics solver on multicore CPUs, in "Parallel Computing: on the road to Exascale"], we have achieved the performance analysis of staggered Lagrange-remap schemes, a class of solvers widely used for hydrodynamics applications. This paper is devoted to the rethinking and redesign of the Lagrange-remap process for achieving better performance using today's computing architectures. As an unintended outcome, the analysis has lead us to the discovery of a new family of solvers -the so-called Lagrange-flux schemes -that appear to be promising for the CFD community.Résumé -Schémas Lagrange-flux : reformuler les schémas Lagrange-Projection d'ordre deux pour améliorer la performance HPC au niveau noeud de calcul -Dans un article récent [Poncet R., Peybernes M., Gasc T., De Vuyst F. (2016) Performance modeling of a compressible hydrodynamics solver on multicore CPUs, in "Parallel Computing: on the road to Exascale"], nous avons effectué l'analyse de la performance d'un schéma de type Lagrange+projection à variables décalées ; cette classe de solveurs est très utilisée pour les applications d'hydrodynamique. Dans cet article, on s'intéresse à la reformulation des solveurs Lagrange-projection afin d'améliorer leur performance globale sur architectures de calculs standards. De manière inattendue, l'analyse nous a conduit vers la découverte d'une nouvelle famille de solveurs -appelés schémas Lagrange-flux -qui apparaissent comme très prometteurs dans la communauté CFD.

We propose a free boundary shallow water model for which we give an existence theorem. The proof uses an original Lagrangian discrete scheme in order to build a sequence of approximate solutions. The properties of this scheme allow to treat the difficulties linked to the boundary motion. These approximate solutions verify some compactness results which allow us to pass to the limit in the discrete problem. Résumé Nous proposons un modèle de shallow water à frontière libre pour lequel nous donnons un théorème d'existence. La preuve utilise un schéma de discrétisation lagrangien original afin de construire une suite de solutions approchées. Les propriétés de ce schéma permettent de traiter les difficultés liées au mouvement de la frontière. Ces solutions approchées vérifient certaines estimations qui nous permettent de passer à la limite dans le problème discrétisé.