Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition
Authors:
Maren Hantke, Wolfgang Dreyer and Gerald Warnecke
Journal:
Quart. Appl. Math. 71 (2013), 509-540
MSC (2010):
Primary 80A22, 76T15, 35L65
DOI:
https://doi.org/10.1090/S0033-569X-2013-01290-X
Published electronically:
May 17, 2013
MathSciNet review:
3112826
Full-text PDF Free Access
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Abstract: We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition.
References
- Rohan Abeyaratne and James K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal. 114 (1991), no. 2, 119–154. MR 1094433, DOI https://doi.org/10.1007/BF00375400
- M. Baer and J. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows, 12 (1986), pp. 861-889.
- Sylvie Benzoni-Gavage, Raphaël Danchin, Stéphane Descombes, and Didier Jamet, Stability issues in the Euler-Korteweg model, Control methods in PDE-dynamical systems, Contemp. Math., vol. 426, Amer. Math. Soc., Providence, RI, 2007, pp. 103–127. MR 2311523, DOI https://doi.org/10.1090/conm/426/08186
- M. Bond and H. Struchtrup, Mean evaporation and condensation coefficients based on energy dependent condensation probability, Phys. Rev. E, 70 (2004) 061605.
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377
- W. Döring, Die Überhitzungsgrenze und Zerreißfestigkeit von Flüssigkeiten, Z. physikal. Chem., Abt. B, Bd. 86 (1937), Heft 5/6.
- W. Dreyer, On jump conditions at phase boundaries for ordered and disordered phases, WIAS Preprint, 869 (2003). [http://www.wias-berlin.de/main/publications/wias-publ/]
- W. Dreyer, F. Duderstadt, M. Hantke, and G. Warnecke, On phase change of a vapor bubble in liquid water, WIAS Preprint, 1424 (2009). [http://www.wias-berlin.de/main/publications/wias-publ/]
- W. Dreyer, J. Giesselmann, C. Kraus, and C. Rohde, Asymptotic Analysis for Korteweg models, WIAS Preprint, 1545 (2010). [http://www.wias-berlin.de/main/publications/wias-publ/]
- U. Grigull, S. Straub, and P. Schiebener, Steam Tables in SI-Units, Wasserdampftafeln, Springer-Verlag, Berlin, 1990.
- Dietmar Kröner, Numerical schemes for conservation laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1997. MR 1437144
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
- Randall J. LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. MR 1077828
- Ralph Menikoff and Bradley J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Modern Phys. 61 (1989), no. 1, 75–130. MR 977944, DOI https://doi.org/10.1103/RevModPhys.61.75
- C. Merkle, Dynamical Phase Transitions in Compressible Media, Doctoral Thesis, Univ. Freiburg, 2006.
- Ingo Müller and Wolfgang H. Müller, Fundamentals of thermodynamics and applications, Springer-Verlag, Berlin, 2009. With historical annotations and many citations from Avogadro to Zermelo. MR 2721954
- I. Müller, Thermodynamics, Pitman, London, 1985.
- Siegfried Müller and Alexander Voß, The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves, SIAM J. Sci. Comput. 28 (2006), no. 2, 651–681. MR 2231725, DOI https://doi.org/10.1137/040619909
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd ed., Springer-Verlag, Berlin, 1999. A practical introduction. MR 1717819
- A. Voss, Exact Riemann Solution for the Euler Equations with Nonconvex and Nonsmooth Equation of State, Doctoral Thesis, RWTH Aachen 2005.
- W. Wagner and H.-J. Kretzschmar, International Steam Tables, Springer-Verlag, Berlin, Heidelberg, 2008.
- A. Zein, Numerical methods for multiphase mixture conservation laws with phase transition, Doctoral Thesis, Univ. Magdeburg, 2010.
- 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 https://doi.org/10.1016/j.jcp.2009.12.026
References
- R. Abeyaratne and J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Arch. Rational Mech. Anal., 114 (1991), pp. 119-154. MR 1094433 (92a:73006)
- M. Baer and J. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows, 12 (1986), pp. 861-889.
- S. Benzoni-Gavage, R. Danchin, S. Descombes, and D. Jamet, Stability issues in the Euler-Korteweg model, Contemp. Math. 426 (2007), pp. 103-127. MR 2311523 (2008k:35368)
- M. Bond and H. Struchtrup, Mean evaporation and condensation coefficients based on energy dependent condensation probability, Phys. Rev. E, 70 (2004) 061605.
- C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, Heidelberg, 2010. MR 2574377 (2011i:35150)
- W. Döring, Die Überhitzungsgrenze und Zerreißfestigkeit von Flüssigkeiten, Z. physikal. Chem., Abt. B, Bd. 86 (1937), Heft 5/6.
- W. Dreyer, On jump conditions at phase boundaries for ordered and disordered phases, WIAS Preprint, 869 (2003). [http://www.wias-berlin.de/main/publications/wias-publ/]
- W. Dreyer, F. Duderstadt, M. Hantke, and G. Warnecke, On phase change of a vapor bubble in liquid water, WIAS Preprint, 1424 (2009). [http://www.wias-berlin.de/main/publications/wias-publ/]
- W. Dreyer, J. Giesselmann, C. Kraus, and C. Rohde, Asymptotic Analysis for Korteweg models, WIAS Preprint, 1545 (2010). [http://www.wias-berlin.de/main/publications/wias-publ/]
- U. Grigull, S. Straub, and P. Schiebener, Steam Tables in SI-Units, Wasserdampftafeln, Springer-Verlag, Berlin, 1990.
- D. Kröner, Numerical schemes for conservation laws, John Wiley and Sons, Chichester, 1997. MR 1437144 (98b:65003)
- P. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Society for Industrial and Applied Mathematics, Philadelphia, 1990. MR 0350216 (50:2709)
- R. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel - Boston - Berlin, 1990. MR 1077828 (91j:65142)
- R. Menikoff and B. J. Plohr, The Riemann problem for fluid flow of real materials, Rev. of Mod. Phys., 61 (1989), pp. 75-104. MR 977944 (90a:35142)
- C. Merkle, Dynamical Phase Transitions in Compressible Media, Doctoral Thesis, Univ. Freiburg, 2006.
- I. Müller and W. Müller, Fundamentals of Thermodynamics and Applications, Springer-Verlag, Berlin, 2009. MR 2721954 (2012b:80001)
- I. Müller, Thermodynamics, Pitman, London, 1985.
- S. Müller and A. Voss, The Riemann Problem for the Euler equations with nonconvex and nonsmooth equation of state: Construction of wave curves, SIAM J. Sci. Comput., 28 (2006), pp. 651-681. MR 2231725 (2007c:35106)
- J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. MR 1301779 (95g:35002)
- E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, Heidelberg, 1999. MR 1717819 (2000f:76091)
- A. Voss, Exact Riemann Solution for the Euler Equations with Nonconvex and Nonsmooth Equation of State, Doctoral Thesis, RWTH Aachen 2005.
- W. Wagner and H.-J. Kretzschmar, International Steam Tables, Springer-Verlag, Berlin, Heidelberg, 2008.
- A. Zein, Numerical methods for multiphase mixture conservation laws with phase transition, Doctoral Thesis, Univ. Magdeburg, 2010.
- A. Zein, M. Hantke, and G. Warnecke, Modeling phase transitions for compressible two-phase flows applied to metastable liquids, J. Comput. Phys., 229 (2010), pp. 2964-2998. MR 2595804 (2011c:80007)
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Additional Information
Maren Hantke
Affiliation:
Institute for Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D-39016 Magdeburg, Germany
MR Author ID:
822591
Email:
maren.hantke@ovgu.de
Wolfgang Dreyer
Affiliation:
Weierstraß-Institut für angewandte Analysis and Stochastik (WIAS), Mohrenstraße 39, D-10117 Berlin, Germany
Email:
dreyer@wias-berlin.de
Gerald Warnecke
Affiliation:
Institute for Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D-39016 Magdeburg, Germany
MR Author ID:
261694
Email:
gerald.warnecke@ovgu.de
Keywords:
Conservation laws,
phase transitions,
nonclassical shocks,
two-phase flow model,
exact Riemann solver,
sharp interface model,
thermodynamics
Received by editor(s):
August 19, 2011
Published electronically:
May 17, 2013
Additional Notes:
This work was supported by the DFG grant Wa 633/17 within the DFGCNRS research group FOR 563/1 and by the DFG grant DR 401/4-1. The authors thank the DFG for this funding
Article copyright:
© Copyright 2013
Brown University
The copyright for this article reverts to public domain 28 years after publication.
