Conservative formulation for compressible multiphase flows
Authors:
Evgeniy Romenski, Alexander A. Belozerov and Ilya M. Peshkov
Journal:
Quart. Appl. Math. 74 (2016), 113-136
MSC (2010):
Primary 35L65, 76T99
DOI:
https://doi.org/10.1090/qam/1409
Published electronically:
December 3, 2015
MathSciNet review:
3472522
Full-text PDF Free Access
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Additional Information
Abstract: Derivation of governing equations for multiphase flow on the base of thermodynamically compatible systems theory is presented. The mixture is considered as a continuum in which the multiphase character of the flow is taken into account. The resulting governing equations of the formulated model belong to the class of hyperbolic systems of conservation laws. In order to examine the reliability of the model, the one-dimensional Riemann problem for the four-phase flow is studied numerically with the use of the MUSCL-Hancock method in conjunction with the GFORCE flux.
References
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- G. La Spina, M. de’ Michieli Vitturi, and E. Romenski, A compressible single-temperature conservative two-phase model with phase transitions, Internat. J. Numer. Methods Fluids 76 (2014), no. 5, 282–311. MR 3257934, DOI 10.1002/fld.3934
- O. Le Métayer, J. Massoni, and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers, J. Comput. Phys. 205 (2005), no. 2, 567–610. MR 2134994, DOI 10.1016/j.jcp.2004.11.021
- E. Romenski, Hyperbolic systems of conservation laws for compressible multiphase flows based on thermodynamically compatible systems theory, Numerical Analysis and Applied Mathematics ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics, vol. 1479, AIP Publishing, 2012, pp. 62–65.
- Evgeniy Romenski, Dimitris Drikakis, and Eleuterio Toro, Conservative models and numerical methods for compressible two-phase flow, J. Sci. Comput. 42 (2010), no. 1, 68–95. MR 2576365, DOI 10.1007/s10915-009-9316-y
- E. Romenski, A. D. Resnyansky, and E. F. Toro, Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures, Quart. Appl. Math. 65 (2007), no. 2, 259–279. MR 2330558, DOI 10.1090/S0033-569X-07-01051-2
- E. Romenski and E. F. Toro, Compressible two-phase flows: two-pressure models and numerical methods, Comput. Fluid Dyn. J. 13 (2004), 403–416.
- E. I. Romensky, Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, Godunov methods (Oxford, 1999) Kluwer/Plenum, New York, 2001, pp. 745–761. MR 1963637
- Richard Saurel and Rémi Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425–467. MR 1684902, DOI 10.1006/jcph.1999.6187
- H. Staedtke, G. Franchello, B. Worth, U. Graf, P. Romstedt, A. Kumbaro, J. García-Cascales, H. Paillere, H. Deconinck, M. Ricchiuto et al., Advanced three-dimensional two-phase flow simulation tools for application to reactor safety (astar), Nuclear Engineering and Design 235 (2005), no. 2, 379–400.
- H. Bruce Stewart and Burton Wendroff, Two-phase flow: models and methods, J. Comput. Phys. 56 (1984), no. 3, 363–409. MR 768670, DOI 10.1016/0021-9991(84)90103-7
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd ed., Springer-Verlag, Berlin, 1999. A practical introduction. MR 1717819, DOI 10.1007/978-3-662-03915-1
- E. F. Toro and V. A. Titarev, MUSTA fluxes for systems of conservation laws, J. Comput. Phys. 216 (2006), no. 2, 403–429. MR 2235378, DOI 10.1016/j.jcp.2005.12.012
- D. Zeidan, On a further work of two-phase mixture conservation laws, Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, vol. 1389, AIP Publishing, 2011, pp. 163–166.
- Ali Zein, Maren Hantke, and Gerald Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comput. Phys. 229 (2010), no. 8, 2964–2998. MR 2595804, DOI 10.1016/j.jcp.2009.12.026
References
- M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials, International Journal of Multiphase Flow 12 (1986), no. 6, 861–889.
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR 2169977 (2006d:35159)
- K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392. MR 0062932 (16,44c)
- Sergei K. Godunov and Evgenii I. Romenskii, Elements of continuum mechanics and conservation laws, Kluwer Academic/Plenum Publishers, New York, 2003. Translated from the 1998 Russian edition by Tamara Rozhkovskaya. MR 1999156 (2005e:74004), DOI 10.1007/978-1-4757-5117-8
- Jean-Marc Hérard, A three-phase flow model, Math. Comput. Modelling 45 (2007), no. 5-6, 732–755. MR 2287317 (2007j:76156), DOI 10.1016/j.mcm.2006.07.018
- M. Ishii, Thermo-fluid dynamic theory of two-phase flow, NASA STI/Recon Technical Report A 75 (1975), 29657.
- Giuseppe La Spina and Mattia de’ Michieli Vitturi, High-resolution finite volume central schemes for a compressible two-phase model, SIAM J. Sci. Comput. 34 (2012), no. 6, B861–B880. MR 3029835, DOI 10.1137/12087089X
- G. La Spina, M. de’ Michieli Vitturi, and E. Romenski, A compressible single-temperature conservative two-phase model with phase transitions, Internat. J. Numer. Methods Fluids 76 (2014), no. 5, 282–311. MR 3257934
- O. Le Métayer, J. Massoni, and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers, J. Comput. Phys. 205 (2005), no. 2, 567–610. MR 2134994 (2005m:76136), DOI 10.1016/j.jcp.2004.11.021
- E. Romenski, Hyperbolic systems of conservation laws for compressible multiphase flows based on thermodynamically compatible systems theory, Numerical Analysis and Applied Mathematics ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics, vol. 1479, AIP Publishing, 2012, pp. 62–65.
- Evgeniy Romenski, Dimitris Drikakis, and Eleuterio Toro, Conservative models and numerical methods for compressible two-phase flow, J. Sci. Comput. 42 (2010), no. 1, 68–95. MR 2576365 (2010m:65198), DOI 10.1007/s10915-009-9316-y
- E. Romenski, A. D. Resnyansky, and E. F. Toro, Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures, Quart. Appl. Math. 65 (2007), no. 2, 259–279. MR 2330558 (2009f:35216)
- E. Romenski and E. F. Toro, Compressible two-phase flows: two-pressure models and numerical methods, Comput. Fluid Dyn. J. 13 (2004), 403–416.
- E. I. Romensky, Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, Godunov methods (Oxford, 1999) Kluwer/Plenum, New York, 2001, pp. 745–761. MR 1963637 (2004a:74001)
- Richard Saurel and Rémi Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), no. 2, 425–467. MR 1684902 (99m:76097), DOI 10.1006/jcph.1999.6187
- H. Staedtke, G. Franchello, B. Worth, U. Graf, P. Romstedt, A. Kumbaro, J. García-Cascales, H. Paillere, H. Deconinck, M. Ricchiuto et al., Advanced three-dimensional two-phase flow simulation tools for application to reactor safety (astar), Nuclear Engineering and Design 235 (2005), no. 2, 379–400.
- H. Bruce Stewart and Burton Wendroff, Two-phase flow: models and methods, J. Comput. Phys. 56 (1984), no. 3, 363–409. MR 768670 (86a:76040), DOI 10.1016/0021-9991(84)90103-7
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, A practical introduction, 2nd ed., Springer-Verlag, Berlin, 1999. MR 1717819 (2000f:76091), DOI 10.1007/978-3-662-03915-1
- E. F. Toro and V. A. Titarev, MUSTA fluxes for systems of conservation laws, J. Comput. Phys. 216 (2006), no. 2, 403–429. MR 2235378, DOI 10.1016/j.jcp.2005.12.012
- D. Zeidan, On a further work of two-phase mixture conservation laws, Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, vol. 1389, AIP Publishing, 2011, pp. 163–166.
- Ali Zein, Maren Hantke, and Gerald Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comput. Phys. 229 (2010), no. 8, 2964–2998. MR 2595804 (2011c:80007), DOI 10.1016/j.jcp.2009.12.026
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Additional Information
Evgeniy Romenski
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russian Federation
Email:
evrom@math.nsc.ru
Alexander A. Belozerov
Affiliation:
Novosibirsk State University, Novosibirsk, 630090, Russian Federation
Email:
belozerov314@gmail.com
Ilya M. Peshkov
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russian Federation
Address at time of publication:
Aix-Marseille Université, CNRS, IUSTI UMR 7343, Marseille, France
Email:
peshkov@math.nsc.ru
Keywords:
Hyperbolic system of conservation laws,
multiphase compressible flow,
four phase flow,
finite-volume method,
Riemann problem
Received by editor(s):
May 15, 2014
Received by editor(s) in revised form:
July 3, 2014
Published electronically:
December 3, 2015
Additional Notes:
The financial support from the Russian Foundation for Basic Research (grants 13-05-00076 and 13-05-12051), Presidium of Russian Academy of Sciences (Programme of Fundamental Research No. 15, project 121), and the Siberian Branch of Russian Academy of Sciences (Integration Projects No. 127 and No. 30) is greatly acknowledged.
The financial support from the Labex MEC (ANR-10-LABX-0092) and A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Ave nir” French government program managed by the French National Research Agency (ANR) is acknowledged
Article copyright:
© Copyright 2015
Brown University
