# On the effect of temperature and velocity relaxation in two-phase flow models

Pedro José Martínez Ferrer; Tore Flåtten; Svend Tollak Munkejord

- Volume: 46, Issue: 2, page 411-442
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMartínez Ferrer, Pedro José, Flåtten, Tore, and Munkejord, Svend Tollak. "On the effect of temperature and velocity relaxation in two-phase flow models." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 411-442. <http://eudml.org/doc/273143>.

@article{MartínezFerrer2012,

abstract = {We study a two-phase pipe flow model with relaxation terms in the momentum and energy equations, driving the model towards dynamic and thermal equilibrium. These equilibrium states are characterized by the velocities and temperatures being equal in each phase. For each of these relaxation processes, we consider the limits of zero and infinite relaxation times. By expanding on previously established results, we derive a formulation of the mixture sound velocity for the thermally relaxed model. This allows us to directly prove a subcharacteristic condition; each level of equilibrium assumption imposed reduces the propagation velocity of pressure waves. Furthermore, we show that each relaxation procedure reduces the mixture sound velocity with a factor that is independent of whether the other relaxation procedure has already been performed. Numerical simulations indicate that thermal relaxation in the two-fluid model has negligible impact on mass transport dynamics. However, the velocity difference of sonic propagation in the thermally relaxed and unrelaxed two-fluid models may significantly affect practical simulations.},

author = {Martínez Ferrer, Pedro José, Flåtten, Tore, Munkejord, Svend Tollak},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {two-fluid model; relaxation system; subcharacteristic condition},

language = {eng},

number = {2},

pages = {411-442},

publisher = {EDP-Sciences},

title = {On the effect of temperature and velocity relaxation in two-phase flow models},

url = {http://eudml.org/doc/273143},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Martínez Ferrer, Pedro José

AU - Flåtten, Tore

AU - Munkejord, Svend Tollak

TI - On the effect of temperature and velocity relaxation in two-phase flow models

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 2

SP - 411

EP - 442

AB - We study a two-phase pipe flow model with relaxation terms in the momentum and energy equations, driving the model towards dynamic and thermal equilibrium. These equilibrium states are characterized by the velocities and temperatures being equal in each phase. For each of these relaxation processes, we consider the limits of zero and infinite relaxation times. By expanding on previously established results, we derive a formulation of the mixture sound velocity for the thermally relaxed model. This allows us to directly prove a subcharacteristic condition; each level of equilibrium assumption imposed reduces the propagation velocity of pressure waves. Furthermore, we show that each relaxation procedure reduces the mixture sound velocity with a factor that is independent of whether the other relaxation procedure has already been performed. Numerical simulations indicate that thermal relaxation in the two-fluid model has negligible impact on mass transport dynamics. However, the velocity difference of sonic propagation in the thermally relaxed and unrelaxed two-fluid models may significantly affect practical simulations.

LA - eng

KW - two-fluid model; relaxation system; subcharacteristic condition

UR - http://eudml.org/doc/273143

ER -

## References

top- [1] R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys.186 (2003) 361–396. Zbl1072.76594MR1973195
- [2] 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 Flow12 (1986) 861–889. Zbl0609.76114
- [3] M. Baudin, F. Coquel and Q.H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline. SIAM J. Sci. Comput.27 (2005) 914–936. Zbl1130.76384MR2199914
- [4] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q.H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law. Numer. Math.99 (2005) 411–440. Zbl1204.76025MR2117734
- [5] 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.
- [6] D. Bestion, The physical closure laws in the CATHARE code. Nucl. Eng. Des.124 (1990) 229–245.
- [7] 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. Zbl1192.76030MR2346701
- [8] 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. Zbl0806.35112MR1280989
- [9] P. Cinnella, Roe-type schemes for dense gas flow computations. Comput. Fluids35 (2006) 1264–1281. Zbl1177.76216
- [10] F. Coquel, K. El Amine, E. Godlewski, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys.136 (1997) 272–288. Zbl0893.76052MR1474408
- [11] F. Coquel, Q.L. Nguyen, M. Postel and Q.H. Tran, Entropy-satisfying relaxation method with large time-steps for Euler IBVPs. Math. Comput.79 (2010) 1493–1533. Zbl05776275MR2630001
- [12] S. Evje and K.K. Fjelde, Relaxation schemes for the calculation of two-phase flow in pipes. Math. Comput. Modelling36 (2002) 535–567. Zbl1129.76345MR1928608
- [13] S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys.192 (2003) 175–210. Zbl1032.76696
- [14] S. Evje and T. Flåtten, Hybrid central-upwind schemes for numerical resolution of two-phase flows. ESAIM: M2AN 39 (2005) 253–273. Zbl1130.76057MR2143949
- [15] S. Evje and T. Flåtten, On the wave structure of two-phase flow models. SIAM J. Appl. Math.67 (2007) 487–511. Zbl1191.76100MR2285874
- [16] T. Flåtten and S.T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model. ESAIM: M2AN 40 (2006) 735–764. Zbl1123.76038MR2274776
- [17] T. Flåtten, A. Morin and S.T. Munkejord, Wave propagation in multicomponent flow models. SIAM J. Appl. Math.70 (2010) 2861–2882. Zbl05876566MR2735107
- [18] T. Flåtten, A. Morin and S.T. Munkejord, On solutions to equilibrium problems for systems of stiffened gases. SIAM J. Appl. Math.71 (2011) 41–67. Zbl05894938MR2765648
- [19] H. Guillard and F. Duval, A Darcy law for the drift velocity in a two-phase flow model. J. Comput. Phys.224 (2007) 288–313. Zbl1119.76067MR2322272
- [20] S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math.48 (1995) 235–276. Zbl0826.65078MR1322811
- [21] K.H. Karlsen, C. Klingenberg and N.H. Risebro, A relaxation scheme for conservation laws with a discontinuous coefficient. Math. Comput.73 (2004) 1235–1259. Zbl1078.65076MR2047086
- [22] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK (2002). Zbl1010.65040MR1925043
- [23] T.-P. Liu, Hyperbolic conservation laws with relaxation. Commun. Math. Phys.108 (1987) 153–175. Zbl0633.35049MR872145
- [24] P.J. Martínez Ferrer, Numerical and mathematical analysis of a five-equation model for two-phase flow. Master's thesis, SINTEF Energy Research, Trondheim, Norway (2010). Available from http://www.sintef.no/Projectweb/CO2-Dynamics/Publications/.
- [25] J.M. Masella, Q.H. Tran, D. Ferre and C. Pauchon, Transient simulation of two-phase flows in pipes. Int. J. Multiphase Flow24 (1998) 739–755. Zbl1121.76459
- [26] S.T. Munkejord, Partially-reflecting boundary conditions for transient two-phase flow. Commun. Numer. Meth. Eng. 22 (2007) 781–795. Zbl1115.76053MR2244956
- [27] 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. Zbl05770802MR2520291
- [28] A. Murrone and H. Guillard, A five equation reduced model for compressible two phase flow problems. J. Comput. Phys.202 (2005) 664–698. Zbl1061.76083MR2145395
- [29] R. Natalini, Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws, in Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 99. Chapman & Hall/CRC, Boca Raton, FL (1999) 128–198. Zbl0940.35127MR1679940
- [30] H. Paillère, C. Corre and J.R. Carcía Gascales, On the extension of the AUSM+ scheme to compressible two-fluid models. Comput. Fluids32 (2003) 891–916. Zbl1040.76044MR1966060
- [31] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput.25 (2005) 129–155. Zbl1203.65111MR2231946
- [32] V.H. Ransom, Faucet Flow, in Numerical Benchmark Tests, Multiph. Sci. Technol. 3, edited by G.F. Hewitt, J.M. Delhaye and N. Zuber. Hemisphere-Springer, Washington, USA (1987) 465–467.
- [33] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys.43 (1981) 357–372. Zbl0474.65066MR640362
- [34] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys.150 (1999) 425–467. Zbl0937.76053MR1684902
- [35] 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. Zbl1147.76060MR2436919
- [36] H.B. Stewart and B. Wendroff, Two-phase flow: models and methods. J. Comput. Phys.56 (1984) 363–409. Zbl0596.76103MR768670
- [37] J.H. Stuhmiller, The influence of interfacial pressure forces on the character of two-phase flow model equations. Int. J. Multiphase Flow3 (1977) 551–560. Zbl0368.76085
- [38] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys.136 (1997) 503–521. Zbl0918.76050
- [39] I. Toumi, A weak formulation of Roe's approximate Riemann solver. J. Comput. Phys.102 (1992) 360–373. Zbl0783.65068MR1187694
- [40] I. Toumi, An upwind numerical method for two-fluid two-phase flow models. Nucl. Sci. Eng.123 (1996) 147–168.
- [41] Q.H. Tran, M. Baudin and F. Coquel, A relaxation method via the Born-Infeld system. Math. Mod. Methods Appl. Sci.19 (2009) 1203–1240. Zbl1182.35162MR2555469
- [42] J.A. Trapp and R.A. Riemke, A nearly-implicit hydrodynamic numerical scheme for two-phase flows. J. Comput. Phys.66 (1986) 62–82. Zbl0622.76110MR865704
- [43] B. van Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14 (1974) 361–370. Zbl0276.65055
- [44] B. van Leer, Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys. 23 (1977) 276–299. Zbl0339.76056
- [45] WAHA3 Code Manual, JSI Report IJS-DP-8841. Jožef Stefan Institute, Ljubljana, Slovenia (2004).
- [46] A. Zein, M. Hantke and G. Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys.229 (2010) 2964–2998. Zbl1307.76079MR2595804
- [47] N. Zuber and J.A. Findlay, Average volumetric concentration in two-phase flow systems. J. Heat Transfer87 (1965) 453–468.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.