Approximate solutions of the Baer-Nunziato Model (original) (raw)
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Numerical approximation for a Baer–Nunziato model of two-phase flows
Applied Numerical Mathematics, 2011
We present a well-balanced numerical scheme for approximating the solution of the Baer-Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist-Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.
A splitting method for the isentropic Baer-Nunziato two-phase flow model
ESAIM: Proceedings, 2012
In the present work, we propose a fractional step method for computing approximate solutions of the isentropic Baer-Nunziato two-phase flow model. The scheme relies on an operator splitting method corresponding to a separate treatment of fast propagation phenomena due to the acoustic waves on the one hand and slow propagation phenomena due to the fluid motion on the other. The scheme is proved to preserve positive values of the statistical fractions and densities. We also provide two test-cases that assess the convergence of the method. Résumé. Nous proposons ici une méthodeà pas fractionnaires pour le calcul de solutions approchées pour la version isentropique du modèle diphasique de Baer-Nunziato. Le schéma s'appuie sur un splitting de l'opérateur temporel correspondantà la prise en compte différenciée des phénomènes de propagation rapide dus aux ondes acoustiques et des phénomènes de propagation lente dus aux ondes matérielles. On prouve que le schéma permet de préserver des valeurs positives pour les taux statistiques de présence des phases ainsi que pour les densités. Deux cas tests numériques permettent d'illustrer la convergence de la méthode.
Numerical Modeling of Two-Phase Flows Using the Two-Fluid Two-Pressure Approach
Mathematical Models and Methods in Applied Sciences, 2004
The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to ...
A Class of Two-fluid Two-phase Flow Models
42nd AIAA Fluid Dynamics Conference and Exhibit, 2012
We introduce a class of two-fluid models that complies with a few theoretical requirements that include : (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for nonviscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments.
The computation of compressible two-phase flows with the Baer-Nunziato model is addressed. Only the convective part of the model that exhibits non-conservative products is considered and the source terms of the model that represent the exchange between phases are neglected. Based on the solver proposed by Tokareva & Toro [42], a new HLLC-type Riemann solver is built. The key idea of this new solver lies in an approximation of the two-phase contact discontinuity of the model. Thus the Riemann invariants of the wave are approximated in the "subsonic" case. A major consequence of this approximation is that the resulting solver can deal with any Equation of State. It also allows to bypass the resolution of a non-linear equation based on those Riemann invariants. We assess the solver and compare it with others on 1D Riemann problems including grid convergence and efficiency studies. The ability of the proposed solver to deal with complex Equations Of State is also investigated....
Nuclear Engineering and Design, 2019
A new method is proposed to solve the two-phase two-fluid six-equation model. A Roe-type numerical flux is formulated based on a very structured Jacobian matrix. The Jacobian matrix with arbitrary equation of state is formulated and simplified with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the eigenvalue and eigenvector can be obtained analytically. An explicit Roetype numerical solver is constructed based on the analytical eigenvalue and eigenvector. A critical feature of the method is that the formulation of the solver does not depend on the form of the equation of state. The proposed method is applicable to realistic two-phase problems. It is applied to the BWR Full-size Fine-mesh Bundle Test (BFBT) benchmark. Considering simplified physical models, the solutions are in very good agreement with those from both existing codes and experiment data. The numerical solver using analytical eigenvalue and eigenvector is shown to be stable and robust.
Computer Methods in Applied Mechanics and Engineering, 2019
A novel Finite-Volume scheme for the numerical computations of compressible two-phase flows in pipelines is proposed for the fully non-equilibrium Baer-Nunziato model. The present FV approach is the extension of the method proposed in [1] in the context of the Euler equations to the Baer-Nunziato model. In addition, proper approximations of the non-conservative terms are proposed to consider jumps of volume fraction as well as jumps of cross-section in order to respect uniform pressure and velocity profiles preservation. In particular, focus is given to the numerical treatment of abrupt changes in area and to networks wherein several pipelines are connected at junctions. The proposed method makes it possible to avoid the use of an iterative procedure for the solution of the junction problem. The present approach can also deal with general Equations Of State. In addition, the fluid-structure interaction of compressible fluid flowing in flexible pipes is also considered. The proposed scheme is then assessed on a variety of shock-tubes and other transient flow problems and experiments demonstrating its capability to resolve such problems efficiently, accurately and robustly.
A Godunov-type method for the seven-equation model of compressible two-phase flow
Computers & Fluids, 2012
We are interested in the numerical approximation of the solutions of the compressible seven-equation two-phase flow model. We propose a numerical srategy based on the derivation of a simple, accurate and explicit approximate Riemann solver. The source terms associated with the external forces and the drag force are included in the definition of the Riemann problem, and thus receive an upwind treatment. The objective is to try to preserve, at the numerical level, the asymptotic property of the solutions of the model to behave like the solutions of a drift-flux model with an algebraic closure law when the source terms are stiff. Numerical simulations and comparisons with other strategies are proposed.
Modelling compressible dense and dilute two-phase flows
Physics of Fluids
Many two-phase flow situations, from engineering science to astrophysics, deal with transition from dense (high concentration of the condensed phase) to dilute concentration (low concentration of the same phase), covering the entire range of volume fractions. Some models are now well accepted at the two limits, but none is able to cover accurately the entire range, in particular regarding waves propagation. In the present work an alternative to the Baer and Nunziato (1986) (BN for short) model, initially designed for dense flows, is built. The corresponding model is hyperbolic and thermodynamically consistent. Contrarily to the BN model that involves 6 wave speeds, the new formulation involves 4 waves only, in agreement with the Marble (1963) model based on pressureless Euler equations for the dispersed phase, a well-accepted model for low particle volume concentrations. In the new model, the presence of pressure in the momentum equation of the particles and consideration of volume fractions in the two phases render the model valid for large particle concentrations. A symmetric version of the new model is derived as well for liquids containing gas bubbles. This model version involves 4 wave speeds as well, but with different wave's speeds. Last, the two sub-models with 4 waves are combined in a unique formulation, valid for the full range of volume fractions. It involves the same 6 wave's speeds as the BN model, but at a given point of space 4 waves only emerge, depending on the local volume fractions. The non-linear pressure waves propagate only in the phase with dominant volume fraction. The new model is tested numerically on various test problems ranging from separated phases in a shock tube to shock-particle cloud interaction. Its predictions are compared to BN and Marble models as well as against experimental data.
A relaxation method for two-phase flow models with hydrodynamic closure law
Numerische Mathematik, 2005
This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided. Classification (1991): 76T10, 76N15, 35L65, 65M06