×

zbMATH — the first resource for mathematics

A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model. (English) Zbl 1378.76117
Summary: 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 [the first author et al., ESAIM, Math. Model. Numer. Anal. 48, No. 1, 165–206 (2014; Zbl 1286.76098)] 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 the Schwendeman-Wahle-Kapila’s Godunov-type scheme [D. W. Schwendeman et al., J. Comput. Phys. 212, No. 2, 490–526 (2006; Zbl 1161.76531)] and Tokareva-Toro’s HLLC scheme [S. A. Tokareva and E. F. Toro, ibid. 229, No. 10, 3573–3604 (2010; Zbl 1391.76440)]. 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.

MSC:
76T10 Liquid-gas two-phase flows, bubbly flows
76M12 Finite volume methods applied to problems in fluid mechanics
Software:
HLLE; Code_Saturne; HLLC
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atelier de vérification de schémas pour la simulation des modèles diphasiques, Chatou, 2015
[2] Abgrall, R.; Dallet, S., Large time-step numerical scheme for the seven-equation model of compressible two-phase flows, (Finite Volumes for Complex Applications VII, Parabolic and Hyperbolic Problems, (2014)), 749-757 · Zbl 1426.76326
[3] Abgrall, R.; Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures, J. Comput. Phys., 186, 2, 361-396, (2003) · Zbl 1072.76594
[4] Ambroso, A.; Chalons, C.; Coquel, F.; Galié, T., Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, M2AN Math. Model. Numer. Anal., 43, 6, 1063-1097, (2009) · Zbl 1422.76178
[5] Ambroso, A.; Chalons, C.; Raviart, P.-A., A Godunov-type method for the seven-equation model of compressible two-phase flow, Comput. Fluids, 54, 67-91, (2012) · Zbl 1291.76212
[6] Andrianov, N.; Warnecke, G., The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys., 195, 2, 434-464, (2004) · Zbl 1115.76414
[7] Archambeau, F.; Méchitoua, N.; Sakiz, M., Code saturne: a finite volume code for the computation of turbulent incompressible flows - industrial applications, Int. J. Finite Vol., 1, 1, (February 2004)
[8] T. Asmaa, Université Pierre et Marie Curie, PhD thesis, in preparation.
[9] Baer, M. R.; Nunziato, J. W., A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiph. Flow, 12, 6, 861-889, (1986) · Zbl 0609.76114
[10] Bouchut, F., Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, (2004), Birkhäuser Verlag Basel · Zbl 1086.65091
[11] Chalons, C.; Coquel, F.; Kokh, S.; Spillane, N., Large time-step numerical scheme for the seven-equation model of compressible two-phase flows, (Springer Proceedings in Mathematics, FVCA 6, 2011, vol. 4, (2011)), 225-233 · Zbl 1246.76071
[12] Coquel, F.; Gallouët, T.; Hérard, J.-M.; Seguin, N., Closure laws for a two-fluid two pressure model, C. R. Acad. Sci., I-334, 5, 927-932, (2002) · Zbl 0999.35057
[13] Coquel, F.; Godlewski, E.; Perthame, B.; In, A.; Rascle, P., Some new Godunov and relaxation methods for two-phase flow problems, (Godunov Methods, (2001), Kluwer/Plenum New York), 179-188, Oxford, 1999 · Zbl 1064.76545
[14] Coquel, F.; Godlewski, E.; Seguin, N., Relaxation of fluid systems, Math. Models Methods Appl. Sci., 22, 8, (2012) · Zbl 1248.35008
[15] Coquel, F.; Hérard, J.-M.; Saleh, K., A splitting method for the isentropic Baer-Nunziato two-phase flow model, ESAIM Proc., 38, 241-256, (2012) · Zbl 1329.76253
[16] Coquel, F.; Hérard, J.-M.; Saleh, K.; Seguin, N., A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model, Math. Model. Numer. Anal., 48, (2013)
[17] S. Dallet, A comparative study of numerical schemes for the Baer-Nunziato model, in preparation. · Zbl 1426.76326
[18] Daude, F.; Galon, P., On the computation of the Baer-Nunziato model using ALE formulation with HLL- and HLLC-type solvers towards fluid-structure interactions, J. Comput. Phys., 304, C, 189-230, (January 2016)
[19] Dumbser, M.; Hidalgo, A.; Castro, M.; Parés, C.; Toro, E. F., FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng., 199, 9-12, 625-647, (2010) · Zbl 1227.76043
[20] Embid, P.; Baer, M., Mathematical analysis of a two-phase continuum mixture theory, Contin. Mech. Thermodyn., 4, 4, 279-312, (1992) · Zbl 0760.76096
[21] Flåtten, T.; Lund, H., Relaxation two-phase flow models and the subcharacteristic condition, Math. Models Methods Appl. Sci., 21, 12, 2379-2407, (2011) · Zbl 1368.76070
[22] Franquet, E.; Perrier, V., Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models, J. Comput. Phys., 231, 11, 4096-4141, (2012) · Zbl 1426.76530
[23] Gallouët, T.; Hérard, J.-M.; Seguin, N., Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci., 14, 5, 663-700, (2004) · Zbl 1177.76428
[24] Gavrilyuk, S.; Saurel, R., Mathematical and numerical modeling of two-phase compressible flows with micro-inertia, J. Comput. Phys., 175, 1, 326-360, (2002) · Zbl 1039.76067
[25] Glimm, J.; Saltz, D.; Sharp, D. H., Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing, Phys. Lett. A, 222, 3, 171-176, (1996) · Zbl 0972.76512
[26] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, Appl. Math. Sci., vol. 118, (1996), Springer-Verlag New York · Zbl 1063.65080
[27] Han, E.; Hantke, M.; Müller, S., Modeling of multi-component flows with phase transition and application to collapsing bubbles, (2014), Institut für Geometrie une Praktische Mathematik Preprint No. 409
[28] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61, (1983) · Zbl 0565.65051
[29] Hérard, J.-M., A three-phase flow model, Math. Comput. Model., 45, 5-6, 732-755, (2007) · Zbl 1165.76382
[30] Hérard, J.-M.; Hurisse, O., A fractional step method to compute a class of compressible gas-liquid flows, Comput. Fluids, 55, 57-69, (2012) · Zbl 1291.76217
[31] Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Reactivity initiated accident (RIA)
[32] Jin, S.; Xin, Z. P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48, 3, 235-276, (1995) · Zbl 0826.65078
[33] Kapila, A. K.; Son, S. F.; Bdzil, J. B.; Menikoff, R.; Stewart, D. S., Two-phase modeling of DDT: structure of the velocity-relaxation zone, Phys. Fluids, 9, 12, 3885-3897, (1997)
[34] Liu, Y., Contribution à la vérification et à la validation d’un modèle diphasique bifluide instationnaire, (2013), Université Aix-Marseille, PhD thesis
[35] Lochon, H.; Daude, F.; Galon, P.; Hérard, J.-M., Comparison of two-fluid models on steam-water transients, Math. Model. Numer. Anal., (2016) · Zbl 1373.76136
[36] Müller, S.; Hantke, M.; Richter, P., Closure conditions for non-equilibrium multi-component models, Contin. Mech. Thermodyn., 28, 4, 1157-1189, (2016) · Zbl 1355.76069
[37] Saleh, K., Analyse et simulation numérique par relaxation d’ecoulements diphasiques compressibles. contribution au traitement des phases evanescentes, (2012), Université Pierre et Marie Curie Paris VI, PhD thesis
[38] Saurel, R.; Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150, 2, 425-467, (1999) · Zbl 0937.76053
[39] Schwendeman, D. W.; Wahle, C. W.; Kapila, A. K., The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow, J. Comput. Phys., 212, 2, 490-526, (2006) · Zbl 1161.76531
[40] Suliciu, I., On modelling phase transitions by means of rate-type constitutive equations. shock wave structure, Int. J. Eng. Sci., 28, 8, 829-841, (1990) · Zbl 0738.73007
[41] Suliciu, I., Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations, Int. J. Eng. Sci., 30, 4, 483-494, (1992) · Zbl 0752.73009
[42] Thanh, M. D.; Kröner, D.; Chalons, C., A robust numerical method for approximating solutions of a model of two-phase flows and its properties, Appl. Math. Comput., 219, 1, 320-344, (2012) · Zbl 1291.76325
[43] Thanh, M. D.; Kröner, D.; Nam, N. T., Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61, 5, 702-721, (2011) · Zbl 1429.76113
[44] Tokareva, S. A.; Toro, E. F., HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow, J. Comput. Phys., 229, 10, 3573-3604, (2010) · Zbl 1391.76440
[45] Tokareva, S. A.; Toro, E. F., A flux splitting method for the Baer-Nunziato equations of compressible two-phase flow, J. Comput. Phys., 323, 45-74, (2016)
[46] U.S. NRC: Glossary. Departure from Nucleate Boiling (DNB)
[47] U.S. NRC: Glossary. Loss of Coolant Accident (LOCA)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.