본문 바로가기
KERD
논문KCI 등재

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

Mai Duc Thanh
대한수학회 (The Korean Mathematical Society)|2014.01|대한수학회지 (Journal of the KMS)|제51권 제1호|pp. 163-187

0

조회

0

다운로드

1

인용

0

참고문헌

키워드

two-phase flowsource termnonconservativeshock wavebalance lawnu- merical approximationwell-balanced schemeRoe-type schemerarefaction wavecontact discontinuity.
주제분류수학

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 con- struction 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 ap- ply 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.

Abstract

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 con- struction 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 ap- ply 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.

참고문헌 (35)

  1. A. Ambroso, "Relaxation and numerical approximation of a two-fluid two-pressure diphasic model", Math. Model. Numer. Anal, 43(6), pp.1063~1097 (2009)
  2. N. Andrianov, "The Riemann problem for the Baer-Nunziato two-phase flow model", J. Comput. Phys, 195(2), pp.434~464 (2004)
  3. E. Audusse, "A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows", SIAM J. Sci. Comput, 25(6), pp.2050~2065 (2004)
  4. M. R. Baer, "A two-phase mixture theory for the de agration-to-detonation transition (ddt) in reactive granular materials", Int. J. Multiphase Flow, 12(6), pp.861~889 (1986)
  5. R. Botchorishvili, "Equilibrium schemes for scalar conservation laws with stiff sources", Math. Compu, 72(241), pp.131~157 (2003)
  6. R. Botchorishvili, "Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws", J. Comput. Phys, 187(2), pp.391~427 (2003)
  7. J. B. Bzil, "Two-phase modelling of a de agration-to-detonation transition in granular materials: A critical examination of modelling issues", Phys. Fluids, 11(2), pp.378~402 (1999)
  8. A. Chinnayya, "A well-balanced numerical scheme for the approximation of the shallow water equations with topography: the resonance phenomenon", Int. J. Finite Vol, 1(1), pp.33~ (2004)
  9. G. Dal Maso, "Definition and weak stability of nonconservative products", J. Math. Pures Appl. (9), 74(6), pp.483~548 (1995)
  10. P. Embid, "Mathematical analysis of a two-phase continuum mixture theory", Contin. Mech. Thermodyn, 4(4), pp.279~312 (1992)
  11. T. Gallouet, "Numerical modeling of two-phase flows using the two-fluid two-pressure approach", Math. Models Methods Appl. Sci, 14(5), pp.663~700 (2004)
  12. P. Goatin, "The Riemann problem for a class of resonant hyperbolic systems of balance laws", Ann. Inst. H. Poincare Anal. Non Lineaire, 21(6881-902.) (2004)
  13. J. M. Greenberg, "A well-balanced scheme for the numerical processing of source terms in hyperbolic equations", SIAM J. Numer. Anal, 33(1), pp.1~16 (1996)
  14. J. M. Greenberg, "R. Baraille, and A. Noussair, Analysis and approximation of conservation laws with source terms", SIAM J. Numer. Anal, 34(5), pp.1980~2007 (1997)
  15. S. Jin, "An efficient method for computing hyperbolic systems with geometrical source terms having concentrations", J. Comput. Math, 22(2), pp.230~249 (2004)
  16. S. Karni, "A hybrid algorithm for the Baer-Nunziato model using the Riemann invariants", J. Sci. Comput, 45(1-3), pp.382~403 (2010)
  17. B. L. Keyfitz, "Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow", Discrete Contin. Dyn. Syst. Ser. B, 3(4), pp.541~563 (2003)
  18. D. Kroner, "The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section", Math. Model. Numer. Anal, 42(3), pp.425~442 (2008)
  19. D. Kroner, "Numerical solutions to compressible flows in a nozzle with variable cross-section", SIAM J. Numer. Anal, 43(2), pp.796~824 (2005)
  20. M.-H. Lallemand, "Pressure relaxation procedures for multiphase compressible flows", INRIA Report, (4038) (2000)
  21. P. G. LeFloch, "The Riemann problem for fluid flows in a nozzle with discontinuous cross-section", Commun. Math. Sci, 1(4), pp.763~797 (2003)
  22. P. G. LeFloch, "The Riemann problem for the shallow water equations with discontinuous topography", Commun. Math. Sci, 5(4), pp.865~885 (2007)
  23. P. G. LeFloch, "A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime", J. Comput. Phys, 230(20), pp.7631~7660 (2011)
  24. S. T. Munkejord, "Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation", Computers & Fluids, 36(6), pp.1061~1080 (2007)
  25. R. Saurel, "A multiphase Godunov method for compressible multifluid and multiphase flows", J. Comput. Phys, 150(2), pp.425~467 (1999)
  26. D. W. Schwendeman, "The Riemann problem and a high-resolution godunov method for a model of compressible two-phase flow", J. Comput. Phys, 212(2), pp.490~526 (2006)
  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(6), pp.1501~1519 (2009)
  29. M. D. Thanh, "Exact solutions of a two-fluid model of two-phase compressible flows with gravity", Nonlinear Anal. Real World Appl, 13(2), pp.987~998 (2012)
  30. M. D. Thanh, "On a two-fluid model of two-phase compressible flows and its numerical approximation", Commun. Nonlinear Sci. Numer. Simul, 17(1), pp.195~211 (2012)
  31. M. D. Thanh, "A well-balanced scheme for a one-pressure model of two-phase flows", Phys. Scr, 79(6), pp.7~ (2009)
  32. M. D. Thanh, "Well-balanced scheme for shallow water equations with arbitrary topography", Int. J. Dyn. Syst. Differ. Equ, 1(3), pp.196~204 (2008)
  33. M. D. Thanh, "A robust numerical method for approximating solutions of a model of two-phase flows and its properties", Appl. Math. Comput, 219(1), pp.320~344 (2012)
  34. M. D. Thanh, "Numerical approximation for a Baer-Nunziato model of two-phase flows", Appl. Numer. Math, 61(5), pp.702~721 (2011)
  35. F. M. White, "Fluid Mechanics" (2010), McGraw-Hill

관련 논문AI추천

A HIGH-RESOLUTION VAN LEER-TYPE SCHEME FOR A MODEL OF FLUID FLOWS IN A NOZZLE WITH VARIABLE CROSS-SECTIONDao Huy Cuong, Mai Duc Thanh대한수학회 · 대한수학회지2017 · 제54권 제1호Splitting technique and Godunov-type schemes for 2D shallow water equations with variable topographyDao Huy Cuong, Mai Duc Thanh대한수학회 · 대한수학회보2024 · 제61권 제4호Adaptive mesh refinement for weighted essentially non-oscillatory schemes윤대기, 황운재, Hongjoong Kim대한수학회 · 대한수학회보2008 · 제45권 제4호MATHEMATICAL ANALYSIS OF A MULTIFLUID INTERPENETRATION MIX MODEL진현성대한수학회 · 대한수학회보2012 · 제49권 제2호A Numerical scheme with a mesh on characteristics for the Cauchy problem for one-dimensional hyperbolic conservation laws황운재, Daeki Yoon, Hongjoong Kim대한수학회 · 대한수학회논문집2009 · 제24권 제3호Stability of two-phase flow models진현성대한수학회 · 대한수학회논문집2007 · 제22권 제4호

이 간행물의 다른 논문 (11건)

Conditional central limit theorems for a sequence of conditional independent random variablesLan Lei, Li-Ran Wei, De-Mei Yuanpp. 1-15ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C¹ VECTOR FIELDSLe Huy Tien, Xiao Wen, 이건희pp. 17-53TIGHT TOUGHNESS CONDITION FOR FRACTIONAL (g,f,n)-CRITICAL GRAPHSJuxiang Zhou, Li Liang, Wei Gao, Tianwei Xupp. 55-65A PRIORI L² ERROR ANALYSIS FOR AN EXPANDED MIXED FINITE ELEMENT METHOD FOR QUASILINEAR PSEUDO-PARABOLIC EQUATIONS엄미례, 신준용, 이현영pp. 67-86A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULESEhsan Momtahan, Mohammad Baziar, Saeed Safaeeyanpp. 87-98
원문 보기