Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations
Authors:
Maren Hantke and Ferdinand Thein
Journal:
Quart. Appl. Math. 73 (2015), 575-591
MSC (2010):
Primary 35Q31, 82B26, 82C26, 80A22, 76B10
DOI:
https://doi.org/10.1090/qam/1393
Published electronically:
June 16, 2015
MathSciNet review:
3400760
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Phase transitions are in the focus of the modeling of multiphase flows. A large number of models are available to describe such processes. We consider several different two phase models that are based on the Euler equations of compressible fluid flows and that take into account phase transitions between a liquid phase and its vapor. Especially we consider the flow of liquid water and water vapor. We give a mathematical proof that all these models are not able to describe the process of condensation by compression. This behavior is in agreement with observations in experiments that simulate adiabatic flows and shows that the Euler equations give a fairly good description of the process. The mathematical proof is valid for the official standard IAPWS-IF97 for water and for any other good equation of state. Also the opposite case of expanding the liquid phase will be discussed.
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.R. Baer, J.W. Nunziato, A two-phase mixture theory of the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows 12 (1986), 861–889.
- T. Barberon, P. Helluy, Finite volume simulations of cavitating flows, Computers & Fluids 34 (2005), 832–858.
- 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
- M. Dumbser, U. Iben, C.D. Munz, Efficient implementation of high order unstructured WENO schemes for cavitating flows, Computers & Fluids 86 (2013), 141–168.
- 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 https://doi.org/10.1090/S0033-569X-2013-01290-X
- U. Iben, Modelling of cavitation, Systems Analysis Modelling Simulation, 42:9 (2002), 1283–1307.
- U. Iben, F. Wrona, C.D. Munz, M. Beck, Cavitation in hydraulic tools based on thermodynamic properties of liquid and gas, Journal of Fluids Engineering - Transactions of the ASME 124 (2002), 1011–1017.
- U. Iben, Entwicklung und Untersuchung von Kavitationsmodellen im Zusammenhang mit transienten Leitungsströmungen, Fortschritt-Berichte VDI, Reihe 7, Strömngstechnik, VDI Verlag, 2004.
- 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
- O. Le Metayer, J. Massoni, R. Saurel, Elaborating equations of state of a liquid and its vapor for two phase flow models (in French), Int. J. Thermal Sci. 43(3) (2004), 265–276.
- Randall J. LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. MR 1077828
- 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.
- F. Petitpas, J. Massoni, R. Saurel, E. Lapebie, L. Munier, Diffuse interface models for high speed cavitating underwater systems, Int. J. Multiphase Flows 35 (2009), 747–759.
- 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 https://doi.org/10.1006/jcph.1999.6187
- Richard Saurel, Fabien Petitpas, and Remi Abgrall, Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607 (2008), 313–350. MR 2436919, DOI https://doi.org/10.1017/S0022112008002061
- 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
- W. Wagner and H.-J. Kretzschmar, International steam tables, Springer-Verlag, Berlin - Heidelberg, 2008.
- W. Wagner and A. Kruse, Properties of Water and Steam - The Industrial Standard IAPWS-IF97 / Zustandsgrößen von Wasser und Wasserdampf - Der Industriestandard IAPWS-IF97, Springer-Verlag, Berlin, 1998.
- W. Wagner et al., The IAPWS Industrial Formulation 1997 for the thermodynamic properties of water and steam, Journal of Engineering for Gas, Turbines and Power 122 (2000), 150–182.
- 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
- 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 (92a:73006), DOI https://doi.org/10.1007/BF00375400
- M.R. Baer, J.W. Nunziato, A two-phase mixture theory of the deflagration-to-detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows 12 (1986), 861–889.
- T. Barberon, P. Helluy, Finite volume simulations of cavitating flows, Computers & Fluids 34 (2005), 832–858.
- 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 (2011i:35150)
- M. Dumbser, U. Iben, C.D. Munz, Efficient implementation of high order unstructured WENO schemes for cavitating flows, Computers & Fluids 86 (2013), 141–168.
- 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 https://doi.org/10.1090/S0033-569X-2013-01290-X
- U. Iben, Modelling of cavitation, Systems Analysis Modelling Simulation, 42:9 (2002), 1283–1307.
- U. Iben, F. Wrona, C.D. Munz, M. Beck, Cavitation in hydraulic tools based on thermodynamic properties of liquid and gas, Journal of Fluids Engineering - Transactions of the ASME 124 (2002), 1011–1017.
- U. Iben, Entwicklung und Untersuchung von Kavitationsmodellen im Zusammenhang mit transienten Leitungsströmungen, Fortschritt-Berichte VDI, Reihe 7, Strömngstechnik, VDI Verlag, 2004.
- Dietmar Kröner, Numerical schemes for conservation laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons Ltd., Chichester, 1997. MR 1437144 (98b:65003)
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216 (50 \#2709)
- O. Le Metayer, J. Massoni, R. Saurel, Elaborating equations of state of a liquid and its vapor for two phase flow models (in French), Int. J. Thermal Sci. 43(3) (2004), 265–276.
- Randall J. LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. MR 1077828 (91j:65142)
- C. Merkle, Dynamical phase transitions in compressible media, Doctoral Thesis, Univ. Freiburg, 2006.
- Ingo Müller and Wolfgang H. Müller, Fundamentals of thermodynamics and applications, with historical annotations and many citations from Avogadro to Zermelo. Springer-Verlag, Berlin, 2009. MR 2721954 (2012b:80001)
- I. Müller, Thermodynamics, Pitman, London, 1985.
- F. Petitpas, J. Massoni, R. Saurel, E. Lapebie, L. Munier, Diffuse interface models for high speed cavitating underwater systems, Int. J. Multiphase Flows 35 (2009), 747–759.
- 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 https://doi.org/10.1006/jcph.1999.6187
- Richard Saurel, Fabien Petitpas, and Remi Abgrall, Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607 (2008), 313–350. MR 2436919 (2009i:80011), DOI https://doi.org/10.1017/S0022112008002061
- 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 (95g:35002)
- Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, A practical introduction, 2nd ed., Springer-Verlag, Berlin, 1999. MR 1717819 (2000f:76091)
- W. Wagner and H.-J. Kretzschmar, International steam tables, Springer-Verlag, Berlin - Heidelberg, 2008.
- W. Wagner and A. Kruse, Properties of Water and Steam - The Industrial Standard IAPWS-IF97 / Zustandsgrößen von Wasser und Wasserdampf - Der Industriestandard IAPWS-IF97, Springer-Verlag, Berlin, 1998.
- W. Wagner et al., The IAPWS Industrial Formulation 1997 for the thermodynamic properties of water and steam, Journal of Engineering for Gas, Turbines and Power 122 (2000), 150–182.
- 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 https://doi.org/10.1016/j.jcp.2009.12.026
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35Q31,
82B26,
82C26,
80A22,
76B10
Retrieve articles in all journals
with MSC (2010):
35Q31,
82B26,
82C26,
80A22,
76B10
Additional Information
Maren Hantke
Affiliation:
Institute for Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D–39016 Magdeburg, Germany
Address at time of publication:
University of Leipzig, Mathematical Institute, PF 100920, D-04009 Leipzig, Germany
MR Author ID:
822591
Email:
maren.hantke@ovgu.de
Ferdinand Thein
Affiliation:
Institute for Analysis and Numerics, Otto-von-Guericke-University Magdeburg, PSF 4120, D–39016 Magdeburg, Germany
Email:
ferdinand.thein@ovgu.de
Received by editor(s):
December 10, 2013
Published electronically:
June 16, 2015
Additional Notes:
The second author was supported by Landesgraduiertenstipendium Sachsen-Anhalt and is supported by the DFG grant HA 6471/2-1. The authors thankfully acknowledge the support.
Article copyright:
© Copyright 2015
Brown University
