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 Tokareva-Toro's HLLC scheme 44. 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.

[1]  Richard Saurel,et al.  Mathematical and numerical modeling of two-phase compressible flows with micro-inertia , 2002 .

[2]  Christophe Chalons,et al.  A robust numerical method for approximating solutions of a model of two-phase flows and its properties , 2012, Appl. Math. Comput..

[3]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[4]  Jean-Marc Hérard,et al.  A SPLITTING METHOD FOR THE ISENTROPIC BAER-NUNZIATO TWO-PHASE FLOW MODEL , 2012 .

[5]  Siegfried Müller,et al.  Modeling of multi-component flows with phase transition and application to collapsing bubbles , 2014 .

[6]  I. Suliciu.  On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure , 1990 .

[7]  Jean-Marc Hérard,et al.  A Robust Entropy-Satisfying Finite Volume Scheme for the Isentropic Baer-Nunziato Model , 2014 .

[8]  Sophie Dallet,et al.  A comparative study of numerical schemes for the Baer-Nunziato model , 2016 .

[9]  I. Suliciu.  Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations , 1992 .

[10]  David H. Sharp,et al.  RENORMALIZATION GROUP SOLUTION OF TWO-PHASE FLOW EQUATIONS FOR RAYLEIGH-TAYLOR MIXING , 1996 .

[11]  Yujie Liu,et al.  Contribution à la vérification et à la validation d'un modèle diphasique bifluide instationnaire , 2013 .

[12]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[13]  F. Archambeau,et al.  Code Saturne: A Finite Volume Code for the computation of turbulent incompressible flows - Industrial Applications , 2004 .

[14]  Rémi Abgrall,et al.  Discrete equations for physical and numerical compressible multiphase mixtures , 2003 .

[15]  C. Chalons,et al.  Relaxation and numerical approximation of a two-fluid two-pressure diphasic model , 2009 .

[16]  Steven F. Son,et al.  Two-phase modeling of DDT: Structure of the velocity-relaxation zone , 1997 .

[17]  Nguyen Thanh Nam,et al.  Numerical approximation for a Baer-Nunziato model of two-phase flows , 2011 .

[18]  Vincent Perrier,et al.  Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models , 2012, J. Comput. Phys..

[19]  Eleuterio F. Toro,et al.  HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow , 2010, J. Comput. Phys..

[20]  Frédéric Coquel,et al.  RELAXATION OF FLUID SYSTEMS , 2012 .

[21]  Eleuterio F. Toro,et al.  A flux splitting method for the Baer-Nunziato equations of compressible two-phase flow , 2016, J. Comput. Phys..

[22]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[23]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[24]  P. Galon,et al.  On the computation of the Baer-Nunziato model using ALE formulation with HLL- and HLLC-type solvers towards fluid-structure interactions , 2016, J. Comput. Phys..

[25]  B. Perthame,et al.  Some New Godunov and Relaxation Methods for Two-Phase Flow Problems , 2001 .

[26]  Jean-Marc Hérard,et al.  A three-phase flow model , 2007, Math. Comput. Model..

[27]  G. Warnecke,et al.  The Riemann problem for the Baer-Nunziato two-phase flow model , 2004 .

[28]  Pascal Galon,et al.  Comparison of two-fluid models on steam-water transients , 2016 .

[29]  Khaled Saleh,et al.  Analyse et Simulation Numérique par Relaxation d'Ecoulements Diphasiques Compressibles. Contribution au Traitement des Phases Evanescentes. , 2012 .

[30]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[31]  Tore Flåtten,et al.  Relaxation two-phase flow models and the subcharacteristic condition , 2011 .

[32]  T. Gallouët,et al.  Numerical modeling of two-phase flows using the two-fluid two-pressure approach , 2004 .

[33]  Jean-Marc Hérard,et al.  Closure laws for a two-fluid two-pressure model , 2002 .

[34]  Donald W. Schwendeman,et al.  The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow , 2006, J. Comput. Phys..

[35]  Jean-Marc Hérard,et al.  A fractional step method to compute a class of compressible gas–liquid flows , 2012 .

[36]  P. Raviart,et al.  A Godunov-type method for the seven-equation model of compressible two-phase flow , 2012 .

[37]  Samuel Kokh,et al.  Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows , 2011 .

[38]  M. Baer,et al.  Mathematical analysis of a two-phase continuum mixture theory , 1992 .

[39]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[40]  Pascal Richter,et al.  Closure conditions for non-equilibrium multi-component models , 2016 .

[41]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .