A two-fluid four-equation model with instantaneous thermodynamical equilibrium

Morin, Alexandre; Flåtten, Tore

doi:10.1051/m2an/2015074

Morin, Alexandre ^1,² ; Flåtten, Tore ²

¹ Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Kolbjørn Hejes vei 1B, NO-7491 Trondheim, Norway.
² SINTEF Materials and Chemistry, P. O. Box 4760 Sluppen, NO-7465 Trondheim, Norway.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1167-1192.

Résumé

We consider an equilibrium version of a common two-fluid model for pipe flow, containing one mixture mass equation and one mixture energy equation. This model can be derived from a five-equation model with instantaneous thermal equilibrium, to which additional phase relaxation terms are added. An original contribution of this paper is a quasilinear formulation of the instantaneous phase relaxation limit. From this, the mixture sound speed intrinsic to the model can be extracted. This allows us to directly prove some subcharacteristic conditions with respect to a previously established model hierarchy of different relaxation processes. These subcharacteristic conditions reveal the fundamental insight of this paper; in the hierarchy, thermodynamic versus velocity relaxation both reduce the mixture sound velocity with a factor that is independent of whether the other type of relaxation has been performed.

Reçu le : 2015-03-28
Accepté le : 2015-09-18

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/m2an/2015074

Classification : 76T10, 35L65
Mots clés : Two-phase flows, relaxation, two-fluid model, subcharacteristic condition

Affiliations des auteurs :

Morin, Alexandre ^{1, 2} ; Flåtten, Tore ²

@article{M2AN_2016__50_4_1167_0,
     author = {Morin, Alexandre and Fl\r{a}tten, Tore},
     title = {A two-fluid four-equation model with instantaneous thermodynamical equilibrium},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1167--1192},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {4},
     year = {2016},
     doi = {10.1051/m2an/2015074},
     mrnumber = {3521716},
     zbl = {1346.76196},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015074/}
}

TY  - JOUR
AU  - Morin, Alexandre
AU  - Flåtten, Tore
TI  - A two-fluid four-equation model with instantaneous thermodynamical equilibrium
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 1167
EP  - 1192
VL  - 50
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015074/
DO  - 10.1051/m2an/2015074
LA  - en
ID  - M2AN_2016__50_4_1167_0
ER  -

%0 Journal Article
%A Morin, Alexandre
%A Flåtten, Tore
%T A two-fluid four-equation model with instantaneous thermodynamical equilibrium
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 1167-1192
%V 50
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015074/
%R 10.1051/m2an/2015074
%G en
%F M2AN_2016__50_4_1167_0

Morin, Alexandre; Flåtten, Tore. A two-fluid four-equation model with instantaneous thermodynamical equilibrium. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1167-1192. doi : 10.1051/m2an/2015074. http://www.numdam.org/articles/10.1051/m2an/2015074/

Bibliographie
Cité par

G. Allaire, S. Clerc and S. Kokh, A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577–616. | DOI | MR | Zbl

G. Allaire, G. Faccanoni and S. Kokh, A strictly hyperbolic equilibrium phase transition model. C. R. Acad. Sci. Paris Ser. I 344 (2007) 135–140. | DOI | MR | Zbl

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 Flow 12 (1986) 861–889. | DOI | Zbl

K.H. Bendiksen, D. Malnes, R. Moe and S. Nuland, The dynamic two-fluid model OLGA: Theory and application. SPE Prod. Eng. 6 (1991) 171–180. | DOI

M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec and Y. Penel, Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law. ESAIM: M2AN 48 (2014) 1639–1679. | DOI | Numdam | MR | Zbl

D. Bestion, The physical closure laws in the CATHARE code. Nucl. Eng. Des. 124 (1990) 229–245. | DOI

F. Bouchut, A reduced stability condition for nonlinear relaxation to conservation laws. J. Hyperbol. Differ. Eq. 1 (2004) 149–170. | DOI | MR | Zbl

C.-H. Chang and M.-S. Liou, A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM $^{+}$ -up scheme. J. Comput. Phys. 225 (2007) 850–873. | MR | Zbl

G.-Q. Chen, C.D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47 (1994) 787–830. | DOI | MR | Zbl

F. Coquel, K. El Amine and E. Godlewski, A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272–288. | DOI | MR | Zbl

F. Coquel, T. Gallouët, J.M. Hérard and N. Seguin, Closure laws for a two-fluid two-pressure model. C. R. Acad. Sci. Paris Ser. I 334 (2002) 927–932. | DOI | MR | Zbl

J. Cortes, A. Debussche and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems. J. Comput. Phys. 147 (1998) 463–484. | DOI | MR | Zbl

S. Evje and T. Flåtten, Hybrid Flux-Splitting Schemes for a Common Two-Fluid Model. J. Comput. Phys. 192 (2003) 175–210. | DOI | Zbl

G. Faccanoni, S. Kokh and G. Allaire, Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium. ESAIM: M2AN 46 (2012) 1029–1054. | DOI | Numdam | MR | Zbl

T. Flåtten and H. Lund, Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21 (2011) 2379-2407. | DOI | MR | Zbl

T. Flåtten and A. Morin, On Interface Transfer Terms in Two-Fluid Models. Int. J. Multiphase Flow 45 (2012) 24–29. | DOI

T. Flåtten, A. Morin and S.T. Munkejord, Wave propagation in multicomponent flow models. SIAM J. Appl. Math. 70 (2010) 2861–2882. | DOI | MR | Zbl

T. Gallouët, J.-M. Hérard and N. Seguin, 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

D. Gidaspow, Modeling of two-phase flow. Round table discussion (RT-1-2). Proc. of 5th Int. Heat Transfer Conf. VII (1974) 163. | DOI

M. Hammer and A. Morin, A method for simulating two-phase pipe flow with real equations of state. Comput. Fluids 100 (2014) 45–58. | DOI | MR | Zbl

P. Helluy and H. Mathis, Pressure laws and fast Legendre transform. Math. Models Methods Appl. Sci. 21 (2011) 745–775. | DOI | MR | Zbl

P. Helluy and N. Seguin, Relaxation models of phase transition flows. ESAIM: M2AN 40 (2006) 331–352. | DOI | Numdam | MR | Zbl

K.H. Karlsen, C. Klingenberg and N.H. Risebro, A relaxation system for conservation laws with a discontinuous coefficient. Math. Comput. 73 (2004) 1235–1259. | DOI | MR | Zbl

A. Kumbaro and M. Ndjinga, Influence of interfacial pressure term on the hyperbolicity of a general multifluid model. J. Comput. Multiphase Flows 3 (2011) 177–195. | DOI | MR

G. Linga, A hierarchy of non-equilibrium two-phase flow models. Submitted (2015).

T.-P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108 (1987) 153–175. | DOI | MR | Zbl

H. Lund, A hierarchy of relaxation models for two-phase flow. SIAM J. Appl. Math. 72 (2012) 1713–1741. | DOI | MR | Zbl

H. Lund, Relaxation models for two-phase with applications to CO2 transport. Ph.D. thesis, Norwegian University of Science and Technology (2013).

P.J. Martínez Ferrer, T. Flåtten and S.T. Munkejord, On the effect of temperature and velocity relaxation in two-phase flow models. ESAIM: M2AN 46 (2011) 411–442. | DOI | Numdam | MR | Zbl

A. Morin, T. Flåtten and S.T. Munkejord, A Roe scheme for a compressible six-equation two-fluid model. Int. J. Numer. Meth. Fluids 72 (2012) 478–504. | DOI | MR | Zbl

S.T. Munkejord and M. Hammer, Depressurization of CO2-rich mixtures in pipes: Two-phase flow modelling and comparison with experiments. Int. J. Greenh. Gas Control 37 (2015) 398–411. | DOI

S.T. Munkejord, S. Evje and T. Flåtten, A Musta scheme for a nonconservative two-fluid model. SIAM J. Sci. Comput. 31 (2009) 2587–2622. | DOI | MR | Zbl

A. Murrone and H. Guillard, A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005) 664–698. | DOI | MR | Zbl

R. Natalini, Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws. Monogr. Surv. Pure Appl. Math. Chapman & Hall/CRC (1999) 128–198. | MR | Zbl

H. Paillère, C. Corre and J.R. García Cascales, On the extension of the AUSM $^{+}$ scheme to compressible two-fluid models. Comput. Fluids 32 (2003) 891–916. | DOI | MR | Zbl

L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. | MR | Zbl

M. Pelanti and K.-M. Shyue, A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259 (2014) 331–357. | DOI | MR | Zbl

R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. | DOI | MR | Zbl

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 | MR | Zbl

A.L. Schor, M.S. Kazimi and N.E. Todreas, Advances in two-phase flow modeling for LMFBR applications. Nucl. Eng. Des. 82 (1984) 127–155. | DOI

U. Setzmann and W. Wagner, A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 20 (1991) 1061–1151. | DOI

S. Solem, P. Aursand and T. Flåtten, Wave dynamics of linear hyperbolic relaxation systems. J. Hyperbol. Differ. Eq. 12 (2015) 655–670. | DOI | MR | Zbl

R. Span and W. Wagner, A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 25 (1996) 1509–1596. | DOI

H.B. Stewart and B. Wendroff, Two-phase flow: Models and methods. J. Comput. Phys. 56 (1984) 363–409. | DOI | MR | Zbl

J.H. Stuhmiller, The influence of interfacial pressure forces on the character of two-phase flow model equations. Int. J. Multiphase Flow 3 (1977) 551–560. | DOI | Zbl

I. Toumi, An Upwind Numerical Method for Two-Fluid Two-Phase Flow Models. Nucl. Sci. Eng. 123 (1996) 147–168. | DOI

I. Toumi and A. Kumbaro, An Approximate Linearized Riemann Solver for a Two-Fluid Model. J. Comput. Phys. 124 (1996) 286–300. | DOI | MR | Zbl

Q.H. Tran, M. Baudin and F. Coquel, A relaxation method via the Born-Infeld system. Math. Models Methods Appl. Sci. 19 (2009) 1203–1240. | DOI | MR | Zbl

WAHA3 Code Manual. JSI Report IJS-DP-8841, Jožef Stefan Insitute, Ljubljana, Slovenia (2004).

Cité par Sources :