A low cost semi-implicit low-Mach relaxation scheme for the full Euler equations

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut, Chalons, Guisset (2019) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves

Numerical methods for all-speed flows in fluid-dynamics and non-linear elasticity

In this thesis we are concerned with the numerical simulation of compressible materials flows, including gases, liquids and elastic solids. These materials are described by a monolithic Eulerian model of conservation laws, closed by an hyperelastic state law that includes the different behaviours of the considered materials. A novel implicit relaxation scheme to solve compressible flows at all speeds is proposed, with Mach numbers ranging from very small to the order of unity. The scheme is general and has the same formulation for all the considered materials, since a direct dependence on the state law is avoided via the relaxation. It is based on a fully implicit time discretization, easily implemented thanks to the linearity of the transport operator in the relaxation system. The spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in different Mach regimes. The scheme is validated with one and two dimensional simulations of fluid flows and of deformations of compressible solids. We exploit the domain discretization through Cartesian grids, allowing for massively parallel computations (HPC) that drastically reduce the computational times on 2D test cases. Moreover, the scheme is adapted to the resolution on adaptive grids based on quadtrees, implementing adaptive mesh refinement techinques. The last part of the thesis is devoted to the numerical simulation of heterogeneous multi-material flows. A novel sharp interface method is proposed, with the derivation of implicit equilibrium conditions. The aim of the implicit framework is the solution of weakly compressible and low Mach flows, thus the proposed multi-material conditions are coupled with the implicit relaxation scheme that is solved in the bulk of the flow. Dans cette thèse on s’intéresse à la simulation numérique d’écoulements des matériaux compressibles, voir fluides et solides élastiques. Les matériaux considérés sont décrits avec un modèle monolithique eulérian, fermé avec une loi d’état hyperélastique qui considère les différents comportéments des matériaux. On propose un nouveau schéma de relaxation qui résout les écoulements compressibles dans des différents régimes, avec des nombres de Mach très petits jusqu’à l’ordre 1. Le schéma a une formulation générale qui est la même pour tous le matériaux considérés, parce que il ne dépend pas directement de la loi d’état. Il se base sur une discrétization complétement implicite, facile à implémenter grâce à la linearité de l’opérateur de transport du système de relaxation. La discrétization en éspace est donnée par la combinaison de flux upwind et centrés, pour retrouver la correcte viscosité numérique dans les différents régimes. L’utilisation de mailles cartésiennes pour les cas 2D s’adapte bien à une parallélisation massive, qui permet de réduire drastiquement le temps de calcul. De plus, le schéma a été adapté pour la résolution sur des mailles quadtree, pour implémenter l’adaptivité de la maille avec des critères entropiques. La dernière partie de la thèse concerne la simulation numérique d’écoulements multi-matériaux. On a proposé une nouvelle méthode d’interface “sharp”, en dérivant les conditions d’équilibre en implicite. L’objectif est la résolution d’interfaces physiques dans des régimes faiblement compressibles et avec un nombre de Mach faible, donc les conditions multi-matériaux sont couplées au schéma implicite de relaxation

Liquid vapor phase transitions : modeling, Riemann solvers and computation

The numerical approximation of liquid vapor flows within the compressible regime is a challenging task because complex physical effects at the phase interfaces govern the global flow behavior. We develop a sharp interface approach which treats the phase boundary like a shock wave discontinuity and takes capillarity effects into account. The approach relies on the solution of Riemann problems across the interface that separates the liquid and the vapor phase. The Riemann solution accounts for the relevant physics by enforcing appropriate jump conditions at the phase boundary. A wide variety of interface effects can be handled in a thermodynamically consistent way. This includes surface tension, as well as, mass and energy transfer by phase transition. Moreover, the local normal speed of the interface, which is needed to calculate the time evolution of the phase boundary, is given by the Riemann solution. The focus of this work is the development of isothermal and non-isothermal two-phase Riemann solvers for the sharp interface approach. To verify the solvers with respect to numerical and thermodynamic requirements, one-dimensional and radially symmetric problems are studied. Furthermore, the Riemann solvers and the sharp interface approach are successfully validated against shock tube experiments of real fluids (alkanes).Die numerische Approximation von Zweiphasenströmungen (flüssig/Dampf) in kompressiblen Medien ist eine Herausforderung, da komplexe physikalische Effekte an der Phasengrenze das globale Strömungsverhalten bestimmen. Wir entwickeln einen Sharp-Interface Ansatz, der Phasengrenzen als Schockwellen-Unstetigkeiten behandelt und Kapillareffekte berücksichtigt. Der Ansatz beruht auf der Lösung von Riemann-Problemen an der Grenzfläche zwischen Flüssigkeit und Dampf. Die Riemann-Lösung berücksichtigt relevante physikalische Effekte, indem Sprungbedingungen an der Phasengrenze vorgegeben werden. Dadurch kann eine Vielzahl an Grenzeffekten, wie Oberflächenspannung, Massen- und Energieaustausch durch Phasenübergänge, thermodynamisch konsistent gehandhabt werden. Darüber hinaus ist die lokale Geschwindigkeit in Normalenrichtung, die für die Berechnung der zeitlichen Entwicklung der Phasengrenze benötigt wird, durch die Riemann-Lösungen bestimmt. Der Schwerpunkt dieser Arbeit liegt auf der Entwicklung von isothermen und nicht-isothermen Zweiphasen-Riemannlösern für den Sharp-Interface Ansatz. Zur Verifizierung der Löser bezüglich numerischer und thermodynamischer Anforderungen werden eindimensionale und radial symmetrische Probleme untersucht. Darüber hinaus werden die Riemannlöser und der Sharp-Interface Ansatz erfolgreich durch den Vergleich mit Stoßrohr-Experimenten mit echten Fluiden (Alkane) validiert