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Compressible Heterogeneous Two-Phase Flows

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Theory, Numerics and Applications of Hyperbolic Problems II (HYP 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 237))

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Abstract

The modeling and the numerical simulation of two-phase flows are investigated for several decades. When dealing with very heterogeneous problems, for instance a water flow with many bubbles, one has to make use of averaged models since the description of each phase and interface is out of reach. Whatever the average is, the resulting models often suffer from severe mathematical pathologies: lack of hyperbolicity, non-conservative products, non-preservation of admissible states... In 1986, Baer and Nunziato proposed an original model which possesses interesting features from the mathematical point of view. Our goal is to provide a (partial) state of art on this model and its derivatives, but also to list some open questions.

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References

  1. R. Abgrall, R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186(2), 361–396 (2003)

    Article  MathSciNet  Google Scholar 

  2. G. Allaire, G. Faccanoni, S. Kokh, Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium. \(M^2\) \(AN\) 46:1029–1054 (2012)

    Google Scholar 

  3. A. Ambroso, C. Chalons, F. Coquel, T. Galié, E. Godlewski, P.-A. Raviart, N. Seguin, The drift-flux asymptotic limit of barotropic two-phase two-pressure models. Commun. Math. Sci. 6(2), 521–529 (2008)

    Article  MathSciNet  Google Scholar 

  4. B. Andreianov, N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete Contin. Dyn. Syst. 32(6), 1939–1964 (2012)

    Article  MathSciNet  Google Scholar 

  5. M.R. Baer, J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12, 861–889 (1986)

    Article  Google Scholar 

  6. F. Bouchut, A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyperb. Diff. Equ. 1(1), 149–170 (2004)

    Article  MathSciNet  Google Scholar 

  7. D. Bresch, M. Hillairet, Note on the derivation of multi-component flow systems. Proc. Am. Math. Soc. 143(8), 3429–3443 (2015)

    Article  MathSciNet  Google Scholar 

  8. G.Q. Chen, C.D. Levermore, T.P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47(6), 787–830 (1994)

    Article  MathSciNet  Google Scholar 

  9. A. Chinnayya, E. Daniel, R. Saurel, Modelling detonation waves in heterogeneous energetic materials. J. Comput. Phys. 196(2), 490–538 (2004)

    Article  MathSciNet  Google Scholar 

  10. F. Coquel, T. Gallouët, J.-M. Hérard, N. Seguin, Closure laws for a two-fluid two-pressure model. C. R. Math. Acad. Sci. Paris 334(10), 927–932 (2002)

    Article  MathSciNet  Google Scholar 

  11. F. Coquel, J.-M. Hérard, K. Saleh, A positive and entropy-satisfying finite volume scheme for the Baer–Nunziato model. J. Comput. Phys. (to appear)

    Google Scholar 

  12. F. Coquel, J.-M. Hérard, K. Saleh, N. Seguin, A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model. M2AN Math. Model. Numer. Anal. 48, 165–206 (2013)

    Article  MathSciNet  Google Scholar 

  13. F. Coquel, J.-M. Hérard, K. Saleh, N. Seguin, Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. 12(3), 593–600 (2014)

    Article  MathSciNet  Google Scholar 

  14. D.A. Drew, S. Passman, Theory of Multicomponent Fluids (Springer, New York, 1998)

    MATH  Google Scholar 

  15. P. Embid, M. Baer, Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4(4), 279–312 (1992)

    Article  MathSciNet  Google Scholar 

  16. S. Evje, T. Flåtten, On the wave structure of two-phase flow models. SIAM J. Appl. Math. 67(2), 487–511 (2006/07)

    Article  MathSciNet  Google Scholar 

  17. T. Gallouët, J.-M. Hérard, N. Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Models Methods Appl. Sci. 14(5), 663–700 (2004)

    Article  MathSciNet  Google Scholar 

  18. S. Gavrilyuk, R. Saurel, Mathematical and numerical modelling of two phase compressible flows with micro-inertia. J. Comp. Phys. 175(1), 326–360 (2002)

    Article  Google Scholar 

  19. P. Goatin, P.G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(6), 881–902 (2004)

    Article  MathSciNet  Google Scholar 

  20. H. Guillard, F. Duval, A Darcy law for the drift velocity in a two-phase flow model. J. Comput. Phys. 224(1), 288–313 (2007)

    Article  MathSciNet  Google Scholar 

  21. P. Helluy, N. Seguin, Relaxation models of phase transition flows. M2AN Math. Model. Numer. Anal. 40(2), 331–352 (2006)

    Article  MathSciNet  Google Scholar 

  22. J.-M. Hérard, A three-phase flow model. Math. Comput. Model. 45(5–6), 732–755 (2007)

    Article  MathSciNet  Google Scholar 

  23. E. Isaacson, B. Temple, Convergence of the \(2\times 2\) Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55(3), 625–640 (1995)

    Article  MathSciNet  Google Scholar 

  24. M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flows (Collection de la Direction des Etudes et Recherches d’Électicité de France, 1975)

    Google Scholar 

  25. M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow (Springer, New York, 2006). With a foreword by Lefteri H. Tsoukalas

    Book  Google Scholar 

  26. F. James, H. Mathis, Modeling phase transition and metastable phases, in Finite Volumes for Complex Applications. VII. Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics and Statistics, vol. 78 (Springer, Cham, 2014), pp. 865–872

    MATH  Google Scholar 

  27. A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13(10), 3002–3024 (2001)

    Article  Google Scholar 

  28. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975)

    Article  MathSciNet  Google Scholar 

  29. P.J.M. Ferrer, T. Flåtten, S.T. Munkejord, On the effect of temperature and velocity relaxation in two-phase flow models. ESAIM Math. Model. Numer. Anal. 46(2), 411–442 (2012)

    Article  MathSciNet  Google Scholar 

  30. C. Mascia, R. Natalini, On relaxation hyperbolic systems violating the Shizuta-Kawashima condition. Arch. Ration. Mech. Anal. 195(3), 729–762 (2010)

    Article  MathSciNet  Google Scholar 

  31. H. Mathis, C. Cancès, E. Godlewski, N. Seguin, Dynamic model adaptation for multiscale simulation of hyperbolic systems with relaxation. J. Sci. Comput. 63(3), 820–861 (2015)

    Article  MathSciNet  Google Scholar 

  32. A. Morin, T. Flåtten, A two-fluid four-equation model with instantaneous thermodynamical equilibrium. ESAIM Math. Model. Numer. Anal. 50(4), 1167–1192 (2016)

    Article  MathSciNet  Google Scholar 

  33. R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis of Systems of Conservation Laws (Aachen, 1997). Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 99 (Chapman & Hall/CRC, Boca Raton, FL, 1999), pp. 128–198

    Google Scholar 

  34. Y.-J. Peng, V. Wasiolek, Uniform global existence and parabolic limit for partially dissipative hyperbolic systems. J. Diff. Eq. 260(9), 7059–7092 (2016)

    Article  MathSciNet  Google Scholar 

  35. V.H. Ransom, D.L. Hicks, Hyperbolic two-pressure models for two-phase flow. J. Comp. Phys. 53(1), 124–151 (1984)

    Article  MathSciNet  Google Scholar 

  36. V.H. Ransom, D.L. Hicks, Hyperbolic two-pressure models for two-phase flow revisited. J. Comp. Phys. 75(2), 498–504 (1988)

    Article  Google Scholar 

  37. R. Saurel, R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150(2), 425–467 (1999)

    Article  MathSciNet  Google Scholar 

  38. D.W. Schwendeman, C.W. Wahle, A.K. Kapila, 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)

    Article  MathSciNet  Google Scholar 

  39. N. Seguin, J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Mod. Meth. Appl. Sci. (\(M^3\) \(AS\)) 13(2), 221–250 (2003)

    Article  MathSciNet  Google Scholar 

  40. Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)

    Article  MathSciNet  Google Scholar 

  41. H.B. Stewart, B. Wendroff, Two-phase flow: models and methods. J. Comput. Phys. 56(3), 363–409 (1984)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

During his position in UPMC-Paris 6, the author has been supported by the LRC Manon (Modélisation et approximation numérique orientées pour l’énergie nucléaire—CEA/DM2S-LJLL).

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Authors and Affiliations

  1. IRMAR, Université de Rennes 1, 35042, Rennes Cédex, France

    Nicolas Seguin

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  1. Nicolas Seguin
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Correspondence to Nicolas Seguin .

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Editors and Affiliations

  1. Department of Mathematics, Würzburg University, Würzburg, Germany

    Christian Klingenberg

  2. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Michael Westdickenberg

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Seguin, N. (2018). Compressible Heterogeneous Two-Phase Flows. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_43

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