The paper concerns the modelling of a compressible mixture of a liquid, its vapor and a gas. The gas and the vapor are miscible while the liquid is immiscible with the gaseous phases. This assumption leads to non symmetric constraints on the void fractions. We derive a three-phase three-pressure model endowed with an entropic structure. We show that interfacial pressures are uniquely defined and propose entropy-consistent closure laws for the source terms. Naturally one exhibits that the mechanical relaxation complies with Dalton’s law on the phasic pressures. Then the hyperbolicity and the eigenstructure of the homogeneous model are investigated and we prove that it admits a symmetric form leading to a local existence result. We also derive a barotropic variant which possesses similar properties.
Accepté le :
DOI : 10.1051/m2an/2019028
Mots-clés : Multi-component compressible flows, entropy, relaxation, phase transition, miscibility constraint, closure conditions, hyperbolicity
@article{M2AN_2019__53_4_1373_0,
author = {H\'erard, J.-M. and Mathis, H.},
title = {A three-phase flow model with two miscible phases},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1373--1389},
publisher = {EDP-Sciences},
volume = {53},
number = {4},
year = {2019},
doi = {10.1051/m2an/2019028},
zbl = {1423.76479},
mrnumber = {3980063},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019028/}
}
TY - JOUR AU - Hérard, J.-M. AU - Mathis, H. TI - A three-phase flow model with two miscible phases JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 1373 EP - 1389 VL - 53 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019028/ DO - 10.1051/m2an/2019028 LA - en ID - M2AN_2019__53_4_1373_0 ER -
%0 Journal Article %A Hérard, J.-M. %A Mathis, H. %T A three-phase flow model with two miscible phases %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 1373-1389 %V 53 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019028/ %R 10.1051/m2an/2019028 %G en %F M2AN_2019__53_4_1373_0
Hérard, J.-M.; Mathis, H. A three-phase flow model with two miscible phases. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 4, pp. 1373-1389. doi : 10.1051/m2an/2019028. http://www.numdam.org/articles/10.1051/m2an/2019028/
[1] and , 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 | Zbl
[2] and , Finite volume simulation of cavitating flows. Comput. Fluids 34 (2005) 832–858. | DOI | Zbl
[3] and , Continuum thermodynamics of chemically reacting fluid mixtures. Acta Mech. 226 (2015) 1757–1805. | DOI | MR | Zbl
[4] , , , , and , A renormalization group scaling analysis for compressible two-phase flow. Phys. Fluids A 5 (1993) 2929–2937. | DOI | MR | Zbl
[5] , , and , A class of two-fluid two-phase flow models. In: 42nd AIAA Fluid Dynamics Conference and Exhibit, Fluid Dynamics and Co-located Conferences Available at: (2012). | DOI
[6] , , and , Two properties of two-velocity two-pressure models for two-phase flows. Commun. Math. Sci. 12 (2014) 593–600. | DOI | MR | Zbl
[7] , Relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909–936. | DOI | Numdam | MR | Zbl
[8] , Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 15 (1983) 261–291. | DOI | Zbl
[9] and , Theory of multicomponent fluids. In: Vol. 135 of Applied Mathematical Sciences. Springer-Verlag, New York (1999). | MR | Zbl
[10] and , Mathematical analysis of a two-phase continuum mixture theory. Contin. Mech. Thermodyn. 4 (1992) 279–312. | DOI | MR | Zbl
[11] and , Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21 (2011) 2379–2407. | DOI | MR | Zbl
[12] , and , Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Models Methods Appl. Sci. 14 (2004) 663–700. | DOI | MR | Zbl
[13] , The structure of pressure relaxation terms: the one-velocity case Technical report, EDF, H-I83-2014-0276-EN (2014).
[14] and , Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175 (2002) 326–360. | DOI | MR | Zbl
[15] , and , Waves in continuous media. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser/Springer, Cham (2017). | DOI | MR
[16] U. S. NRC: Glossary. Reactivity initiated accident (ria). https://www.irsn.fr/FR/connaissances/Installations_nucleaires/Les-centrales-nucleaires/criteres_surete_ria_aprp/Pages/1-accident-reactivite-RIA.aspx#.XAkgQq17SE8.
[17] , Modelling and numerical simulation of two-phase flows using the two-fluid two-pressure approach. Theses, Université de Provence – Aix-Marseille I (2007).
[18] , and , Efficient and robust relaxation procedures for multi-component mixtures including phase transition. J. Comput. Phys. 338 (2017) 217–239. | DOI | MR | Zbl
[19] and , Analysis and simulation of a new multi-component two-phase flow model with phase transitions and chemical reactions. Quart. Appl. Math. 76 (2018) 253–287. | DOI | MR | Zbl
[20] and , Closure conditions for a one temperature non-equilibrium multi-component model of Baer-Nunziato type. To appear in: ESAIM Procs. (2019). | MR
[21] , A three-phase flow model. Math. Comput. Model. 45 (2007) 732–755. | DOI | MR | Zbl
[22] , Un modèle hyperbolique diphasique bi-fluide en milieu poreux. C.R. Méc. 742 (2008) 623–683.
[23] , A class of compressible multiphase flow models. C.R. Math. Acad. Sci. Paris 354 (2016) 954–959. | DOI | MR | Zbl
[24] and , A homogeneous model for compressible three-phase flows involving heat and mass transfer. To appear in: ESAIM Procs. (2019). | MR
[25] and , Continuum Methods of Physical Modeling. Continuum Mechanics, Dimensional Analysis, Turbulenc. Springer-Verlag, Berlin (2004). | DOI | MR | Zbl
[26] , Contribution to the simulation of low-velocity compressible two-phase flows with high pressure jumps using homogeneous and two-fluid approaches. Theses, Aix-Marseille Université (2018).
[27] , , , and , Two-phase modelling of ddt in granular materials: reduced equations. Phys. Fluids 13 (2001) 3002–3024. | DOI | Zbl
[28] , The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58 (1975) 181–205. | DOI | MR | Zbl
[29] , Modelling and simulation of steam-water transients using the two-fluid approach. Theses, Aix Marseille Université (2016).
[30] , , and , Computation of fast depressurization of water using a two-fluid model: revisiting Bilicki modelling of mass transfer. Comput. Fluids 156 (2017) 162–174. | DOI | MR | Zbl
[31] , A thermodynamically consistent model of a liquid-vapor fluid with a gas. ESAIM: M2AN 53 (2019) 63–84. | DOI | Numdam | MR | Zbl
[32] , and , Closure conditions for non-equilibrium multi-component models. Contin. Mech. Thermodyn. 28 (2016) 1157–1189. | DOI | MR | Zbl
[33] , and , Numerical simulation of a single bubble by compressible two-phase fluids. Int. J. Numer. Methods Fluids 62 (2010) 591–631. | DOI | MR | Zbl
[34] , and , A numerical model for three-phase liquid-vapor-gas flows with relaxation processes. Hyp2016 | Zbl
[35] , A relaxation scheme for a hyperbolic multiphase flow model. Part I: Barotropic eos. To appear in: ESAIM: M2AN (2019). | Numdam | MR | Zbl
Cité par Sources :
