DOI QR코드

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WELL-BALANCED ROE-TYPE NUMERICAL SCHEME FOR A MODEL OF TWO-PHASE COMPRESSIBLE FLOWS

  • 투고 : 2013.04.23
  • 발행 : 2014.01.01

초록

We present a multi-stage Roe-type numerical scheme for a model of two-phase flows arisen from the modeling of deflagration-to-detonation transition in granular materials. The first stage in the construction of the scheme computes the volume fraction at every time step. The second stage deals with the nonconservative terms in the governing equations which produces states on both side of the contact wave at each node. In the third stage, a Roe matrix for the two-phase is used to apply on the states obtained from the second stage. This scheme is shown to capture stationary waves and preserves the positivity of the volume fractions. Finally, we present numerical tests which all indicate that the proposed scheme can give very good approximations to the exact solution.

키워드

참고문헌

  1. A. Ambroso, C. Chalons, F. Coquel, and T. Galie, Relaxation and numerical approximation of a two-fluid two-pressure diphasic model, Math. Model. Numer. Anal. 43 (2009), no. 6, 1063-1097. https://doi.org/10.1051/m2an/2009038
  2. N. Andrianov and G.Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys. 195 (2004), no. 2, 434-464. https://doi.org/10.1016/j.jcp.2003.10.006
  3. E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004), no. 6, 2050-2065. https://doi.org/10.1137/S1064827503431090
  4. M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the de agration-to-detonation transition (ddt) in reactive granular materials, Int. J. Multiphase Flow 12 (1986), no. 6, 861-889. https://doi.org/10.1016/0301-9322(86)90033-9
  5. R. Botchorishvili, B. Perthame, and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Compu. 72 (2003), no. 241, 131-157.
  6. R. Botchorishvili and O. Pironneau, Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws, J. Comput. Phys. 187 (2003), no. 2, 391-427. https://doi.org/10.1016/S0021-9991(03)00086-X
  7. J. B. Bzil, R. Menikoff, S. F. Son, A. K. Kapila, and D. S. Steward, Two-phase modelling of a de agration-to-detonation transition in granular materials: A critical examination of modelling issues, Phys. Fluids 11 (1999), no. 2, 378-402. https://doi.org/10.1063/1.869887
  8. A. Chinnayya, A.-Y. LeRoux, and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon, Int. J. Finite Vol. 1 (2004), no. 1, 33 pp.
  9. G. Dal Maso, P. G. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483-548.
  10. P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory, Contin. Mech. Thermodyn. 4 (1992), no. 4, 279-312. https://doi.org/10.1007/BF01129333
  11. T. Gallouet, J.-M. Herard, and N. Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci. 14 (2004), no. 5, 663-700. https://doi.org/10.1142/S0218202504003404
  12. P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004), no. 6881-902.
  13. J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1-16. https://doi.org/10.1137/0733001
  14. J. M. Greenberg, A. Y. Leroux, R. Baraille, and A. Noussair, Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal. 34 (1997), no. 5, 1980-2007. https://doi.org/10.1137/S0036142995286751
  15. S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math. 22 (2004), no. 2, 230-249.
  16. S. Karni and G. Hernandez-Duenas, A hybrid algorithm for the Baer-Nunziato model using the Riemann invariants, J. Sci. Comput. 45 (2010), no. 1-3, 382-403. https://doi.org/10.1007/s10915-009-9332-y
  17. B. L. Keyfitz, R. Sander, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 4, 541-563. https://doi.org/10.3934/dcdsb.2003.3.541
  18. D. Kroner, P. G. LeFloch, and M. D. Thanh, The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section, Math. Model. Numer. Anal. 42 (2008), no. 3, 425-442. https://doi.org/10.1051/m2an:2008011
  19. D. Kroner and M. D. Thanh, Numerical solutions to compressible flows in a nozzle with variable cross-section, SIAM J. Numer. Anal. 43 (2005), no. 2, 796-824. https://doi.org/10.1137/040607460
  20. M.-H. Lallemand and R. Saurel, Pressure relaxation procedures for multiphase compressible flows, INRIA Report (2000), No. 4038.
  21. P. G. LeFloch and M. D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. Math. Sci. 1 (2003), no. 4, 763-797. https://doi.org/10.4310/CMS.2003.v1.n4.a6
  22. P. G. LeFloch and M. D. Thanh, The Riemann problem for the shallow water equations with discontinuous topography, Commun. Math. Sci. 5 (2007), no. 4, 865-885. https://doi.org/10.4310/CMS.2007.v5.n4.a7
  23. P. G. LeFloch and M. D. Thanh, A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J. Comput. Phys. 230 (2011), no. 20, 7631-7660. https://doi.org/10.1016/j.jcp.2011.06.017
  24. S. T. Munkejord, Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation, Computers & Fluids 36 (2007), no. 6, 1061-1080. https://doi.org/10.1016/j.compfluid.2007.01.001
  25. R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425-467. https://doi.org/10.1006/jcph.1999.6187
  26. D. W. Schwendeman, C. W. Wahle, and A. K. Kapila, The Riemann problem and a high-resolution godunov method for a model of compressible two-phase flow, J. Comput. Phys. 212 (2006), no. 2, 490-526. https://doi.org/10.1016/j.jcp.2005.07.012
  27. M. D. Thanh, A phase decomposition approach and the Riemann problem for a model of two-phase flows, preprint.
  28. M. D. Thanh, The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math. 69 (2009), no. 6, 1501-1519. https://doi.org/10.1137/080724095
  29. M. D. Thanh, Exact solutions of a two-fluid model of two-phase compressible flows with gravity, Nonlinear Anal. Real World Appl. 13 (2012), no. 2, 987-998. https://doi.org/10.1016/j.nonrwa.2011.09.009
  30. M. D. Thanh, On a two-fluid model of two-phase compressible flows and its numerical approximation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 1, 195-211. https://doi.org/10.1016/j.cnsns.2011.05.010
  31. M. D. Thanh and A. Izani Md. Ismail, A well-balanced scheme for a one-pressure model of two-phase flows, Phys. Scr. 79 (2009), no. 6, 065401, 7pp. https://doi.org/10.1088/0031-8949/79/06/065401
  32. M. D. Thanh, Md. Fazlul Karim, and A. Izani Md. Ismail, Well-balanced scheme for shallow water equations with arbitrary topography, Int. J. Dyn. Syst. Differ. Equ. 1 (2008), no. 3, 196-204.
  33. M. D. Thanh, D. Kroner, and C. Chalons, A robust numerical method for approximating solutions of a model of two-phase flows and its properties, Appl. Math. Comput. 219 (2012), no. 1, 320-344. https://doi.org/10.1016/j.amc.2012.06.022
  34. M. D. Thanh, D. Kroner, and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math. 61 (2011), no. 5, 702-721. https://doi.org/10.1016/j.apnum.2011.01.004
  35. F. M. White, Fluid Mechanics, 7th ed. McGraw-Hill, 2010.