71 research outputs found
Computing material fronts with a Lagrange-Projection approach
This paper reports investigations on the computation of material fronts in
multi-fluid models using a Lagrange-Projection approach. Various forms of the
Projection step are considered. Particular attention is paid to minimization of
conservation errors
Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes
This series of papers is devoted to the formulation and the approximation of
coupling problems for nonlinear hyperbolic equations. The coupling across an
interface in the physical space is formulated in term of an augmented system of
partial differential equations. In an earlier work, this strategy allowed us to
develop a regularization method based on a thick interface model in one space
variable. In the present paper, we significantly extend this framework and, in
addition, encompass equations in several space variables. This new formulation
includes the coupling of several distinct conservation laws and allows for a
possible covering in space. Our main contributions are, on one hand, the design
and analysis of a well-balanced finite volume method on general triangulations
and, on the other hand, a proof of convergence of this method toward entropy
solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a
single conservation law without coupling). The core of our analysis is, first,
the derivation of entropy inequalities as well as a discrete entropy
dissipation estimate and, second, a proof of convergence toward the entropy
solution of the coupling problem.Comment: 37 page
Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces
We investigate various analytical and numerical techniques for the coupling
of nonlinear hyperbolic systems and, in particular, we introduce here an
augmented formulation which allows for the modeling of the dynamics of
interfaces between fluid flows. The main technical difficulty to be overcome
lies in the possible resonance effect when wave speeds coincide and global
hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is
observed for the initial value problem which need to be supplemented with
further admissibility conditions. This first paper is devoted to investigating
these issues in the setting of self-similar vanishing viscosity approximations
to the Riemann problem for general hyperbolic systems. Following earlier works
by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the
Riemann problem under fairly general structural assumptions on the nonlinear
hyperbolic system and its regularization. Our main contribution consists of
nonlinear wave interaction estimates for solutions which apply to resonant wave
patterns.Comment: 28 page
A Positive and Entropy-Satisfying Finite Volume Scheme for the Baer-Nunziato Model
We present a relaxation scheme for approximating the entropy dissipating weak
solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is
straightforwardly obtained as an extension of the relaxation scheme designed in
[16] for the isentropic Baer-Nunziato model and consequently inherits its main
properties. To our knowledge, this is the only existing scheme for which the
approximated phase fractions, phase densities and phase internal energies are
proven to remain positive without any restrictive condition other than a
classical fully computable CFL condition. For ideal gas and stiffened gas
equations of state, real values of the phasic speeds of sound are also proven
to be maintained by the numerical scheme. It is also the only scheme for which
a discrete entropy inequality is proven, under a CFL condition derived from the
natural sub-characteristic condition associated with the relaxation
approximation. This last property, which ensures the non-linear stability of
the numerical method, is satisfied for any admissible equation of state. We
provide a numerical study for the convergence of the approximate solutions
towards some exact Riemann solutions. The numerical simulations show that the
relaxation scheme compares well with two of the most popular existing schemes
available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's
Godunov-type scheme [39] and Toro-Tokareva's HLLC scheme [42]. The relaxation
scheme also shows a higher precision and a lower computational cost (for
comparable accuracy) than a standard numerical scheme used in the nuclear
industry, namely Rusanov's scheme. Finally, we assess the good behavior of the
scheme when approximating vanishing phase solutions
Regularization and relaxation tools for interface coupling
We analyze a relaxation method for approximating the coupling of two Euler systems at a fixed interface and more generally for approximating fluid systems
La correction de Ristorcelli dans les modèles aux tensions de Reynolds compressibles
La motivation des travaux est l'étude
mathématique de l'interaction entre une onde de choc et une turbulence développée
lorsque cette interaction est modélisée par les équations de Navier-Stokes moyennées
avec tensions de Reynolds. L'analyse des profils de viscosité associés aux ondes planes
montre l'existence de phénomènes tangentiels non triviaux, susceptibles d'induire des
mécanismes d'instabilité violente dans les profils de compression d'amplitude
suffisamment grande. Ces mécanismes d'instabilité s'expriment dans les modèles aux
tensions de Reynolds les plus simples et sont intimement liés à la prédiction de taux de
production de turbulence anormalement élevés à la traversée d'un profil de compression.
L'étude éclaire l'influence stabilisante d'un terme de modélisation souvent ignoré, la
correction dite de Ristorcelli, dont l'existence est dictée par la nature compressible
du fluide. Nous montrons que choisir suffisamment grande la constante de temps associée
permet de restaurer la stabilité multidimensionnelle du choc plan (défini dans la limite
d'un nombre de Reynolds infini), sans contrevenir à la condition de production de
turbulence à la traversée de ce dernier. L'absence de formulation faible classique pour
décrire la limite de fluide parfait des équations complique l'analyse mathématique des
solutions chocs et de leur stabilité. Nous montrons que le formalisme des relations
cinétiques permet de proposer une extension très naturelle des conditions de stabilité
structurelle du choc droit dues à Majda puis des conditions de stabilité
multidimensionnelles introduites par D'aykov-Erpenbeck-Lopatinski dans le cadre
d'équations de fluide parfait purement conservatives
Automatic coupling and finite element discretization of the Navier-Stokes and heat equations
We consider the finite element discretization of the Navier-Stokes equations coupled with the heat equation where the viscosity depends on the temperature. We prove a posteriori error estimates which allow us to automatically determine the zone where the temperature-dependent viscosity must be inserted into the Navier-Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations
Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems
International audiencePhase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation at the location of phase boundaries. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel sub-iteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. Up to our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases
Two properties of two-velocity two-pressure models for two-phase flows
International audienceWe study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form
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