A rough scheme to couple free and porous media

International audienceThis paper is devoted to the computation of flows between free and porous media separated by a thin interface . The basic strategy relies on some ideas developed earlier by J.M. Greenberg and A.Y. Leroux on their work on well balanced schemes. This approach requires introducing a set of partial differential equations at the interface, in order to account for the sudden change of medium. The main features of the interface PDE are investigated. We afterwards propose to compute approximations of solutions with help of an approximate Godunov scheme. A linear interface Riemann solver is introduced, which aims at enforcing the continuity of the two (steady wave-) Riemann invariants. Numerical computations involving shock waves or rarefaction waves are examined and the agreement with the entropy inequality is tracked. Effects of the mesh refinement and the impact of the smoothing of the thin interface are also adressed in the paper

Un modèle hyperbolique diphasique bi-fluide en milieu poreux

International audienceAn hyperbolic two-fluid model in porous medium We introduce an hyperbolic two-fluid two-pressure model to compute unsteady two-phase flows in porous media. The closure laws comply with the entropy inequality, and a unique set of jump conditions holds within each field.On introduit dans cette note un modèle d'écoulement bifluide hyperbolique pour simuler les écoulements diphasiques en milieu poreux, en configuration instationnaire. Les lois de fermeture proposées sont consistantes avec l'inégalité d'entropie, et les relations de saut sont uniques champ par champ

A class of compressible multiphase flow models

International audienceWe propose in this note a class of entropy-consistent hyperbolic models for multi-phase barotropic flows. Relevant closure laws are derived and discussed

An hyperbolic three-phase flow model

International audienceWe introduce an hyperbolic entropy-consistant model to describe three-phase flows, which ensures that void fractions, mass fractions and pressures remain positive through single waves occuring in the one dimensional solution of the Riemann problem

A new approach for three-phase flows

International audienceWe present here a new model to describe three-field patterns or three-phase flows. The basic ideas rely on the counterpart of the two-fluid two-pressure model which has been introduced in the DDT framework, and more recently extended to water-vapour simulations. We show the system is hyperbolic without any constraining condition on the flow patterns. A detailed investigation of the structure of the Riemann problem is achieved. Regular solutions of the whole are in agreement with physical requirements on void fractions, densities and internal energies for a rather wide class of equations of state. Even more, this approach enables to perform computations of standard single pressure three-phase flow models, using relaxation techniques and coarse meshes. A few computational results confirm the stability of the whole approach

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

A fractional step method to compute a class of compressible gas–liquid flows

International audienceWe present in this paper some algorithms dedicated to the computation of numerical approximations of a class of two-fluid two-phase flow models. Governing equations for the statistical void fraction, partial mass, momentum, energy are presented first, and meaningful closure laws are given. Then we may give the main properties of the class of two-fluid models. The whole algorithm that relies on the fractional step method and complies with the entropy inequality is presented afterwards. Emphasis is given on the computation of pressure-velocity-temperature relaxation source terms. Conditions pertaining to the existence and uniqueness of discrete solutions of the relaxation step are given. While focusing on some one-dimensional test cases, the true rates of convergence may be obtained within the evolution step and the relaxation step. Eventually, some two-dimensional numerical simulations of a heated wall are shown and are briefly discussed. Some advantages and weaknesses of algorithms are also discussed

A compressible two-layer model for transient gas–liquid flows in pipes

International audienceThis work is dedicated to the modeling of gas-liquid flows in pipes. As a first step, a new two-layer model is proposed to deal with the stratified regime. The starting point is the isentropic Euler set of equations for each phase where the classical hydrostatic assumption is made for the liquid. The main difference with the models issued from the classical literature is that the liquid as well as the gas is assumed compressible. In that framework, an averaging process results in a five-equation system where the hydrostatic constraint has been used to define the interfacial pressure. Closure laws for the interfacial velocity and source terms such as mass and momentum transfer are provided following an entropy inequality. The resulting model is hyperbolic with non-conservative terms. Therefore, regarding the homogeneous part of the system, the definition and uniqueness of jump conditions is studied carefully and acquired. The nature of characteristic fields and the corresponding Riemann invariants are also detailed. Thus, one may build analytical solutions for the Riemann problem. In addition, positivity is obtained for heights and densities. The overall derivation deals with gas-liquid flows through rectangular channels, circular pipes with variable cross section and includes vapor-liquid flows

A two-fluid hyperbolic model in a porous medium

International audienceThe paper is devoted to the computation of two-phase flows in a porous medium when applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them corresponding to solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme, are shown to give wrong approximations in some cases involving sharp porous profiles. The third one, which is an extension of a scheme proposed by D. Kröner and M. D. Thanh (27) for the computation of single phase flows in varying cross section ducts , provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L 1 norm of the error

An interface condition to compute compressible flows in variable cross section ducts

International audienceWe propose an improved interface condition in order to account for head losses in pipe when some discontinuous cross sections occur