Analysis and simulation of a new multi-component two-phase flow model with phase transitions and chemical reactions
Authors:
Maren Hantke and Siegfried Müller
Journal:
Quart. Appl. Math. 76 (2018), 253-287
MSC (2010):
Primary 76T30, 76T10, 74A15, 80A32, 82C26
DOI:
https://doi.org/10.1090/qam/1498
Published electronically:
January 18, 2018
MathSciNet review:
3769896
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Additional Information
Abstract: A class-II-model for multi-component mixtures recently introduced in D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757–1805, is investigated for simple mixtures. Bothe and Dreyer were aiming at deriving physically admissible closure conditions. Here the focus is on mathematical properties of this model. In particular, hyperbolicity of the inviscid flux Jacobian is verified for non-resonance states. Although the eigenvalues cannot be determined explicitly but have to be computed numerically an eigenvector basis is constructed depending on the eigenvalues. This basis is helpful to apply standard numerical solvers for the discretization of the model. This is verified by numerical computations for two- and three-component mixtures with and without phase transition and chemical reactions.
References
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- Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI https://doi.org/10.1006/jcph.1998.5892
- Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
- Peter Deuflhard and Andreas Hohmann, Numerische Mathematik, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 1991 (German). Eine algorithmisch orientierte Einführung. [An algorithmically oriented introduction]. MR 1197354
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- Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261
- Wolfgang Dreyer and Frank Duderstadt, Towards the thermodynamic modeling of nucleation and growth of liquid droplets in single crystals, Free boundary problems (Trento, 2002) Internat. Ser. Numer. Math., vol. 147, Birkhäuser, Basel, 2004, pp. 113–130. MR 2044568
- Wolfgang Dreyer, Frank Duderstadt, Maren Hantke, and Gerald Warnecke, Bubbles in liquids with phase transition. Part 1. On phase change of a single vapor bubble in liquid water, Contin. Mech. Thermodyn. 24 (2012), no. 4-6, 461–483. MR 2992847, DOI https://doi.org/10.1007/s00161-011-0225-6
- W. Dreyer, C. Guhlke, and R. Müller, A new perspective on the electron transfer: Recovering the Butler–Volmer equation in non-equilibrium thermodynamics, WIAS preprint 2204.
- Nils Gerhard, Francesca Iacono, Georg May, Siegfried Müller, and Roland Schäfer, A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible flows, J. Sci. Comput. 62 (2015), no. 1, 25–52. MR 3295028, DOI https://doi.org/10.1007/s10915-014-9846-9
- Nils Gerhard and Siegfried Müller, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case, Comput. Appl. Math. 35 (2016), no. 2, 321–349. MR 3514813, DOI https://doi.org/10.1007/s40314-014-0134-y
- Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987
- Nune Hovhannisyan, Siegfried Müller, and Roland Schäfer, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws, Math. Comp. 83 (2014), no. 285, 113–151. MR 3120584, DOI https://doi.org/10.1090/S0025-5718-2013-02732-9
- Kolumban Hutter and Klaus Jöhnk, Continuum methods of physical modeling, Springer-Verlag, Berlin, 2004. Continuum mechanics, dimensional analysis, turbulence. MR 2060165
- A. Kapila, R. Menikoff, J. Bdzil, S. Son, and D. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluid 13 (2001), 3002–3024.
- L. D. Landau and E. M. Lifshitz, Statistical physics, Course of Theoretical Physics, Vol. 5, Pergamon Press Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1958. Translated from the Russian by E. Peierls and R. F. Peierls. MR 0136378
- 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
- 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 https://doi.org/10.1007/s00161-015-0468-8
- M. J. Pilling and P. W. Seakins, Reaction Kinetics. Oxford Science Publications, 1999.
- A. Zein, Numerical methods for multiphase mixture conservation laws with phase transition, Ph.D. thesis, Otto-von-Guericke University, 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
- Rémi Abgrall and Smadar Karni, A comment on the computation of non-conservative products, J. Comput. Phys. 229 (2010), no. 8, 2759–2763. MR 2595792
- P. Atkins and J. de Paula, Physical Chemistry, OUP Oxford, 2014.
- M. R. Baer and J. W. 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.
- Dieter Bothe and Wolfgang Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech. 226 (2015), no. 6, 1757–1805. MR 3347503
- Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652
- Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
- Peter Deuflhard and Andreas Hohmann, Numerische Mathematik, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, 1991 (German). Eine algorithmisch orientierte Einführung. [An algorithmically oriented introduction]. MR 1197354
- D. Drew, Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech., 15 (1983), pp. 261–291.
- Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261
- Wolfgang Dreyer and Frank Duderstadt, Towards the thermodynamic modeling of nucleation and growth of liquid droplets in single crystals, Free boundary problems (Trento, 2002) Internat. Ser. Numer. Math., vol. 147, Birkhäuser, Basel, 2004, pp. 113–130. MR 2044568
- Wolfgang Dreyer, Frank Duderstadt, Maren Hantke, and Gerald Warnecke, Bubbles in liquids with phase transition. Part 1. On phase change of a single vapor bubble in liquid water, Contin. Mech. Thermodyn. 24 (2012), no. 4-6, 461–483. MR 2992847
- W. Dreyer, C. Guhlke, and R. Müller, A new perspective on the electron transfer: Recovering the Butler–Volmer equation in non-equilibrium thermodynamics, WIAS preprint 2204.
- Nils Gerhard, Francesca Iacono, Georg May, Siegfried Müller, and Roland Schäfer, A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible flows, J. Sci. Comput. 62 (2015), no. 1, 25–52. MR 3295028
- Nils Gerhard and Siegfried Müller, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case, Comput. Appl. Math. 35 (2016), no. 2, 321–349. MR 3514813
- Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987
- Nune Hovhannisyan, Siegfried Müller, and Roland Schäfer, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws, Math. Comp. 83 (2014), no. 285, 113–151. MR 3120584
- Kolumban Hutter and Klaus Jöhnk, Continuum methods of physical modeling, Springer-Verlag, Berlin, 2004. Continuum mechanics, dimensional analysis, turbulence. MR 2060165
- A. Kapila, R. Menikoff, J. Bdzil, S. Son, and D. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluid 13 (2001), 3002–3024.
- L. D. Landau and E. M. Lifshitz, Statistical physics, Course of Theoretical Physics. Vol. 5. Translated from the Russian by E. Peierls and R. F. Peierls, Pergamon Press Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1958. MR 0136378
- 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
- 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
- M. J. Pilling and P. W. Seakins, Reaction Kinetics. Oxford Science Publications, 1999.
- A. Zein, Numerical methods for multiphase mixture conservation laws with phase transition, Ph.D. thesis, Otto-von-Guericke University, 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
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Additional Information
Maren Hantke
Affiliation:
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, PSF 4120, D-39016 Magdeburg, Germany
MR Author ID:
822591
Email:
maren.hantke@ovgu.de
Siegfried Müller
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Email:
mueller@igpm.rwth-aachen.de
Received by editor(s):
May 23, 2017
Published electronically:
January 18, 2018
Additional Notes:
The authors are very grateful to the Mathematisches Forschungsinstitut Oberwolfach (MFO) for its hospitality and giving us the opportunity to spend four weeks at the MFO within the Research in Pairs programme.
Article copyright:
© Copyright 2018
Brown University