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A nonisothermal thermodynamical model of liquid-vapor interaction with metastability
1. | Université de Nantes & CNRS UMR 6629, Laboratoire de Mathematiques Jean Leray, 2, rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 3, France |
2. | Université d'Orléans, Université de Tours & CNRS UMR 7013, Institut Denis Poisson, Rue de Chartres, BP 6759, F-45067 Orléans Cedex 2, France |
The paper concerns the construction of a compressible liquid-vapor relaxation model which is able to capture the metastable states of the non isothermal van der Waals model as well as saturation states. Starting from the Gibbs formalism, we propose a dynamical system which complies with the second law of thermodynamics. Numerical simulations illustrate the expected behaviour of metastable states: an initial metastable condition submitted to a certain perturbation may stay in the metastable state or reaches a saturation state. The dynamical system is then coupled to the dynamics of the compressible fluid using an Euler set of equations supplemented by convection equations on the fractions of volume, mass and energy of one of the phases.
References:
[1] |
M. R. Baer and J. W. Nunziato,
A two phase mixture theory for the deflagration to detonation (ddt) transition in reactive granular materials, Int. J. Multiphase Flow, 12 (1986), 861-889.
doi: 10.1016/0301-9322(86)90033-9. |
[2] |
D. W. Ball, Physical Chemistry, Cengage Learning, 2002. Google Scholar |
[3] |
T. Barberon and P. Helluy,
Finite volume simulation of cavitating flows, Computers and Fluids, 34 (2005), 832-858.
doi: 10.1016/j.compfluid.2004.06.004. |
[4] |
J. Bartak,
A study of the rapid depressurization of hot water and the dynamics of vapour bubble generation in superheated water, Int. J. Multiph. Flow, 16 (1990), 789-98.
doi: 10.1016/0301-9322(90)90004-3. |
[5] |
H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2$^{nd}$ edition, Wiley and Sons, 1985.
doi: 10.1119/1.19071. |
[6] |
F. Caro, Modélisation et simulation numérique des transitions de phase liquide vapeur, PhD thesis, Ecole Polytechnique X, 2004. Google Scholar |
[7] |
M. De Lorenzo, Modelling and numerical simulation of metastable two-phase flows, PhD thesis, Université Paris-Saclay, 2018. Google Scholar |
[8] |
M. De Lorenzo, P. Lafon, M. Di Matteo, M. Pelanti, J.-M. Seynhaeve and Y. Bartosiewicz,
Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations, Int. J. Multiph. Flow, 95 (2017), 199-219.
doi: 10.1016/j.ijmultiphaseflow.2017.06.001. |
[9] |
M. De Lorenzo, P. Lafon and M. Pelanti,
A hyperbolic phase-transition model with non-instantaneous EoS-independent relaxation procedures, J. Comput. Phys., 379 (2019), 279-308.
doi: 10.1016/j.jcp.2018.12.002. |
[10] |
G. Faccanoni, S. Kokh and G. Allaire,
Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium, ESAIM Math. Model. Numer. Anal., 46 (2012), 1029-1054.
doi: 10.1051/m2an/2011069. |
[11] |
S. Fechter, C.-D. Munz, C. Rohde and C. Zeiler,
A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys., 336 (2017), 347-374.
doi: 10.1016/j.jcp.2017.02.001. |
[12] |
T. Gallouët, J.-M. Hérard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS, Internat. J. Numer. Methods Fluids, 39 (2002), 1073–1138.
doi: 10.1002/fld.346. |
[13] |
S. Gavrilyuk, The structure of pressure relaxation terms: The one-velocity case, Technical Report, EDF, (2014), H-I83-2014-0276-EN. Google Scholar |
[14] |
H. Ghazi, Modélisation d'écoulements compressibles avec transition de phase et prise en compte des états métastables, PhD thesis, 2018. Google Scholar |
[15] |
H. Ghazi, F. James and H. Mathis,
Vapour-liquid phase transition and metastability, ESAIM: Proceedings and Surveys, 66 (2019), 22-41.
doi: 10.1051/proc/201966002. |
[16] |
J. W. Gibbs, The Collected Works of J. Willard Gibbs, vol I: Thermodynamics, Yale University Press, 1948.
doi: 10.2307/3609900. |
[17] |
E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118 Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0713-9. |
[18] |
P. Helluy, O. Hurisse and E. Le Coupanec, Verification of a two-phase flow code based on an homogeneous model, Int. J. Finite Vol., 24 (2015). |
[19] |
P. Helluy and H. Mathis,
Pressure laws and fast Legendre transform, Math. Models Methods Appl. Sci., 21 (2011), 745-775.
doi: 10.1142/S0218202511005209. |
[20] |
J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis, in Grundlehren Text Editions, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[21] |
O. Hurisse, Application of an homogeneous model to simulate the heating of two-phase flows, Int. J. Finite Vol., 11 (2014), 37 pp. |
[22] |
O. Hurisse, Numerical simulations of steady and unsteady two-phase flows using a homogeneous model, Comput. & Fluids, 152 (2017), 88–103, 2017.
doi: 10.1016/j.compfluid.2017.04.007. |
[23] |
O. Hurisse and L. Quibel, A homogeneous model for compressible three-phase flows involving heat and mass transfer, ESAIM: Proceedings and Surveys, (2019), 84 – 108.
doi: 10.1051/proc/201966005. |
[24] |
F. James and H. Mathis,
A relaxation model for liquid-vapor phase change with metastability, Commun. Math. Sci., 14 (2016), 2179-2214.
doi: 10.4310/CMS.2016.v14.n8.a4. |
[25] |
A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son and D. S. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluids, 13 (2001), 3002-3024. Google Scholar |
[26] |
L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5. Course of Theoretical Physics, Pergamon Press, 1969. |
[27] |
B. J. Lee, E. F. Toro, C. E. Castro and N. Nikiforakis,
Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246 (2013), 165-183.
doi: 10.1016/j.jcp.2013.03.046. |
[28] |
R. G. Mortimer, Physical Chemistry, Academic Press Elsevier, 2008. Google Scholar |
[29] |
D. Y. Peng and D. B. Robinson,
A new two-constant equation of state, Industrial and Engineering Chemistry: Fundamentals, 15 (1976), 59-64.
doi: 10.1021/i160057a011. |
[30] |
R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. |
[31] |
R. Saurel, F. Petitpas and R. Abgrall,
Modelling phase transition in metastable liquids: Application to cavitating and flashing flows, J. Fluid Mech., 607 (2008), 313-350.
doi: 10.1017/S0022112008002061. |
[32] |
N. Shamsundarl and J. H. Lienhard,
Equations of state and spinodal lines–A review, Nuclear Engineering and Design, 141 (1993), 269-287.
doi: 10.1016/0029-5493(93)90106-J. |
[33] |
G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chemical Engineering Science, 27 (1972), 1197-1203. Google Scholar |
[34] |
E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2009.
doi: 10.1007/b79761. |
[35] |
A. Zein, M. Hantke and G. Warnecke,
Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comp. Phys., 229 (2010), 1964-2998.
doi: 10.1016/j.jcp.2009.12.026. |
show all references
References:
[1] |
M. R. Baer and J. W. Nunziato,
A two phase mixture theory for the deflagration to detonation (ddt) transition in reactive granular materials, Int. J. Multiphase Flow, 12 (1986), 861-889.
doi: 10.1016/0301-9322(86)90033-9. |
[2] |
D. W. Ball, Physical Chemistry, Cengage Learning, 2002. Google Scholar |
[3] |
T. Barberon and P. Helluy,
Finite volume simulation of cavitating flows, Computers and Fluids, 34 (2005), 832-858.
doi: 10.1016/j.compfluid.2004.06.004. |
[4] |
J. Bartak,
A study of the rapid depressurization of hot water and the dynamics of vapour bubble generation in superheated water, Int. J. Multiph. Flow, 16 (1990), 789-98.
doi: 10.1016/0301-9322(90)90004-3. |
[5] |
H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2$^{nd}$ edition, Wiley and Sons, 1985.
doi: 10.1119/1.19071. |
[6] |
F. Caro, Modélisation et simulation numérique des transitions de phase liquide vapeur, PhD thesis, Ecole Polytechnique X, 2004. Google Scholar |
[7] |
M. De Lorenzo, Modelling and numerical simulation of metastable two-phase flows, PhD thesis, Université Paris-Saclay, 2018. Google Scholar |
[8] |
M. De Lorenzo, P. Lafon, M. Di Matteo, M. Pelanti, J.-M. Seynhaeve and Y. Bartosiewicz,
Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations, Int. J. Multiph. Flow, 95 (2017), 199-219.
doi: 10.1016/j.ijmultiphaseflow.2017.06.001. |
[9] |
M. De Lorenzo, P. Lafon and M. Pelanti,
A hyperbolic phase-transition model with non-instantaneous EoS-independent relaxation procedures, J. Comput. Phys., 379 (2019), 279-308.
doi: 10.1016/j.jcp.2018.12.002. |
[10] |
G. Faccanoni, S. Kokh and G. Allaire,
Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium, ESAIM Math. Model. Numer. Anal., 46 (2012), 1029-1054.
doi: 10.1051/m2an/2011069. |
[11] |
S. Fechter, C.-D. Munz, C. Rohde and C. Zeiler,
A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys., 336 (2017), 347-374.
doi: 10.1016/j.jcp.2017.02.001. |
[12] |
T. Gallouët, J.-M. Hérard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS, Internat. J. Numer. Methods Fluids, 39 (2002), 1073–1138.
doi: 10.1002/fld.346. |
[13] |
S. Gavrilyuk, The structure of pressure relaxation terms: The one-velocity case, Technical Report, EDF, (2014), H-I83-2014-0276-EN. Google Scholar |
[14] |
H. Ghazi, Modélisation d'écoulements compressibles avec transition de phase et prise en compte des états métastables, PhD thesis, 2018. Google Scholar |
[15] |
H. Ghazi, F. James and H. Mathis,
Vapour-liquid phase transition and metastability, ESAIM: Proceedings and Surveys, 66 (2019), 22-41.
doi: 10.1051/proc/201966002. |
[16] |
J. W. Gibbs, The Collected Works of J. Willard Gibbs, vol I: Thermodynamics, Yale University Press, 1948.
doi: 10.2307/3609900. |
[17] |
E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118 Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-0713-9. |
[18] |
P. Helluy, O. Hurisse and E. Le Coupanec, Verification of a two-phase flow code based on an homogeneous model, Int. J. Finite Vol., 24 (2015). |
[19] |
P. Helluy and H. Mathis,
Pressure laws and fast Legendre transform, Math. Models Methods Appl. Sci., 21 (2011), 745-775.
doi: 10.1142/S0218202511005209. |
[20] |
J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis, in Grundlehren Text Editions, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-56468-0. |
[21] |
O. Hurisse, Application of an homogeneous model to simulate the heating of two-phase flows, Int. J. Finite Vol., 11 (2014), 37 pp. |
[22] |
O. Hurisse, Numerical simulations of steady and unsteady two-phase flows using a homogeneous model, Comput. & Fluids, 152 (2017), 88–103, 2017.
doi: 10.1016/j.compfluid.2017.04.007. |
[23] |
O. Hurisse and L. Quibel, A homogeneous model for compressible three-phase flows involving heat and mass transfer, ESAIM: Proceedings and Surveys, (2019), 84 – 108.
doi: 10.1051/proc/201966005. |
[24] |
F. James and H. Mathis,
A relaxation model for liquid-vapor phase change with metastability, Commun. Math. Sci., 14 (2016), 2179-2214.
doi: 10.4310/CMS.2016.v14.n8.a4. |
[25] |
A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son and D. S. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluids, 13 (2001), 3002-3024. Google Scholar |
[26] |
L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5. Course of Theoretical Physics, Pergamon Press, 1969. |
[27] |
B. J. Lee, E. F. Toro, C. E. Castro and N. Nikiforakis,
Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246 (2013), 165-183.
doi: 10.1016/j.jcp.2013.03.046. |
[28] |
R. G. Mortimer, Physical Chemistry, Academic Press Elsevier, 2008. Google Scholar |
[29] |
D. Y. Peng and D. B. Robinson,
A new two-constant equation of state, Industrial and Engineering Chemistry: Fundamentals, 15 (1976), 59-64.
doi: 10.1021/i160057a011. |
[30] |
R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. |
[31] |
R. Saurel, F. Petitpas and R. Abgrall,
Modelling phase transition in metastable liquids: Application to cavitating and flashing flows, J. Fluid Mech., 607 (2008), 313-350.
doi: 10.1017/S0022112008002061. |
[32] |
N. Shamsundarl and J. H. Lienhard,
Equations of state and spinodal lines–A review, Nuclear Engineering and Design, 141 (1993), 269-287.
doi: 10.1016/0029-5493(93)90106-J. |
[33] |
G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chemical Engineering Science, 27 (1972), 1197-1203. Google Scholar |
[34] |
E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2009.
doi: 10.1007/b79761. |
[35] |
A. Zein, M. Hantke and G. Warnecke,
Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comp. Phys., 229 (2010), 1964-2998.
doi: 10.1016/j.jcp.2009.12.026. |



















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