The Riemann problem for a two-phase mixture hyperbolic system with phase function and multi-component equation of state
Authors:
Maren Hantke, Christoph Matern, Gerald Warnecke and Hazem Yaghi
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35Q15, 76T30, 76T10
DOI:
https://doi.org/10.1090/qam/1664
Published electronically:
June 26, 2023
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Additional Information
Abstract: In this paper a hyperbolic system of partial differential equations for two-phase mixture flows with $N$ components is studied. It is derived from a more complicated model involving diffusion and exchange terms. Important features of the model are the assumption of isothermal flow, the use of a phase field function to distinguish the phases and a mixture equation of state involving the phase field function as well as an affine relation between partial densities and partial pressures in the liquid phase. This complicates the analysis. A complete solution of the Riemann initial value problem is given. Some interesting examples are suggested as benchmarks for numerical schemes.
References
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- Ch. Chalons, P. Engel, and Ch. Rohde, A conservative and convergent scheme for undercompressive shock waves, SIAM J. Numer. Anal. 52 (2014), no. 1, 554–579. MR 3168613, DOI 10.1137/120897821
- Christophe Chalons, Christian Rohde, and Maria Wiebe, A finite volume method for undercompressive shock waves in two space dimensions, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 5, 1987–2015. MR 3731557, DOI 10.1051/m2an/2017027
- Dieter Bothe and Wolfgang Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech. 226 (2015), no. 6, 1757–1805. MR 3347503, DOI 10.1007/s00707-014-1275-1
- Wolfgang Dreyer, Jan Giesselmann, and Christiane Kraus, A compressible mixture model with phase transition, Phys. D 273/274 (2014), 1–13. MR 3173090, DOI 10.1016/j.physd.2014.01.006
- Stefan Fechter, Claus-Dieter Munz, Christian Rohde, and Christoph Zeiler, A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys. 336 (2017), 347–374. MR 3622620, DOI 10.1016/j.jcp.2017.02.001
- Maren Hantke, Wolfgang Dreyer, and Gerald Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71 (2013), no. 3, 509–540. MR 3112826, DOI 10.1090/S0033-569X-2013-01290-X
- Maren Hantke, Christoph Matern, Gerald Warnecke, and Hazem Yaghi, A new method to discretize a model for isothermal flow with a multi-component equation of state, J. Comput. Appl. Math. 422 (2023), Paper No. 114876, 10. MR 4500707, DOI 10.1016/j.cam.2022.114876
- Maren Hantke and Siegfried Müller, Analysis and simulation of a new multi-component two-phase flow model with phase transitions and chemical reactions, Quart. Appl. Math. 76 (2018), no. 2, 253–287. MR 3769896, DOI 10.1090/qam/1498
- Jean-Marc Hérard, A three-phase flow model, Math. Comput. Modelling 45 (2007), no. 5-6, 732–755. MR 2287317, DOI 10.1016/j.mcm.2006.07.018
- S. Müller, M. Hantke, and P. Richter, Closure conditions for non-equilibrium multi-component models, Contin. Mech. Thermodyn. 28 (2016), no. 4, 1157–1189. MR 3513192, DOI 10.1007/s00161-015-0468-8
- 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
- Veronika Schleper, A HLL-type Riemann solver for two-phase flow with surface forces and phase transitions, Appl. Numer. Math. 108 (2016), 256–270. MR 3528315, DOI 10.1016/j.apnum.2015.12.010
- Ferdinand Thein, Evgeniy Romenski, and Michael Dumbser, Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase flows, J. Sci. Comput. 93 (2022), no. 3, Paper No. 83, 60. MR 4507140, DOI 10.1007/s10915-022-02028-x
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 3rd ed., Springer-Verlag, Berlin, 2009. A practical introduction. MR 2731357, DOI 10.1007/b79761
- H. Yaghi, Analytical and numerical studies of Riemann problems for a multiphase mixture model, Ph.D. Thesis, Otto-von-Guericke University Magdeburg, 2023.
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.
- Ch. Chalons, P. Engel, and Ch. Rohde, A conservative and convergent scheme for undercompressive shock waves, SIAM J. Numer. Anal. 52 (2014), no. 1, 554–579. MR 3168613, DOI 10.1137/120897821
- Christophe Chalons, Christian Rohde, and Maria Wiebe, A finite volume method for undercompressive shock waves in two space dimensions, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 5, 1987–2015. MR 3731557, DOI 10.1051/m2an/2017027
- Dieter Bothe and Wolfgang Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech. 226 (2015), no. 6, 1757–1805. MR 3347503, DOI 10.1007/s00707-014-1275-1
- Wolfgang Dreyer, Jan Giesselmann, and Christiane Kraus, A compressible mixture model with phase transition, Phys. D 273/274 (2014), 1–13. MR 3173090, DOI 10.1016/j.physd.2014.01.006
- Stefan Fechter, Claus-Dieter Munz, Christian Rohde, and Christoph Zeiler, A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys. 336 (2017), 347–374. MR 3622620, DOI 10.1016/j.jcp.2017.02.001
- Maren Hantke, Wolfgang Dreyer, and Gerald Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71 (2013), no. 3, 509–540. MR 3112826, DOI 10.1090/S0033-569X-2013-01290-X
- Maren Hantke, Christoph Matern, Gerald Warnecke, and Hazem Yaghi, A new method to discretize a model for isothermal flow with a multi-component equation of state, J. Comput. Appl. Math. 422 (2023), Paper No. 114876, 10. MR 4500707, DOI 10.1016/j.cam.2022.114876
- Maren Hantke and Siegfried Müller, Analysis and simulation of a new multi-component two-phase flow model with phase transitions and chemical reactions, Quart. Appl. Math. 76 (2018), no. 2, 253–287. MR 3769896, DOI 10.1090/qam/1498
- Jean-Marc Hérard, A three-phase flow model, Math. Comput. Modelling 45 (2007), no. 5-6, 732–755. MR 2287317, DOI 10.1016/j.mcm.2006.07.018
- S. Müller, M. Hantke, and P. Richter, Closure conditions for non-equilibrium multi-component models, Contin. Mech. Thermodyn. 28 (2016), no. 4, 1157–1189. MR 3513192, DOI 10.1007/s00161-015-0468-8
- 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
- Veronika Schleper, A HLL-type Riemann solver for two-phase flow with surface forces and phase transitions, Appl. Numer. Math. 108 (2016), 256–270. MR 3528315, DOI 10.1016/j.apnum.2015.12.010
- Ferdinand Thein, Evgeniy Romenski, and Michael Dumbser, Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase flows, J. Sci. Comput. 93 (2022), no. 3, Paper No. 83, 60. MR 4507140, DOI 10.1007/s10915-022-02028-x
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 3rd ed., Springer-Verlag, Berlin, 2009. A practical introduction. MR 2731357, DOI 10.1007/b79761
- H. Yaghi, Analytical and numerical studies of Riemann problems for a multiphase mixture model, Ph.D. Thesis, Otto-von-Guericke University Magdeburg, 2023.
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Additional Information
Maren Hantke
Affiliation:
Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle (Saale), Germany
MR Author ID:
822591
Email:
maren.hantke@mathematik.uni-halle.de
Christoph Matern
Affiliation:
Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D–40225 Düsseldorf, Germany
Email:
christoph.matern@hhu.de
Gerald Warnecke
Affiliation:
Institute of Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D–39016 Magdeburg, Germany
MR Author ID:
261694
Email:
gerald.warnecke@ovgu.de
Hazem Yaghi
Affiliation:
Institute of Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D–39016 Magdeburg, Germany
MR Author ID:
1531416
Email:
hazem.yaghi@ovgu.de
Received by editor(s):
September 30, 2022
Received by editor(s) in revised form:
February 2, 2023
Published electronically:
June 26, 2023
Article copyright:
© Copyright 2023
Brown University
