The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 4, p. 735-764

We construct a Roe-type numerical scheme for approximating the solutions of a drift-flux two-phase flow model. The model incorporates a set of highly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible. Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure laws is derived. In particular, a fully analytical Roe matrix is obtained for the special case of the Zuber-Findlay law describing bubbly flows. First and second-order accurate versions of the scheme are demonstrated by numerical examples.

DOI : https://doi.org/10.1051/m2an:2006032
Classification:  35L65,  76M12,  76T10
Keywords: two-phase flow, drift-flux model, Riemann solver, Roe scheme
@article{M2AN_2006__40_4_735_0,
author = {Fl\aa tten, Tore and Munkejord, Svend Tollak},
title = {The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {4},
year = {2006},
pages = {735-764},
doi = {10.1051/m2an:2006032},
zbl = {1123.76038},
mrnumber = {2274776},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_4_735_0}
}

Flåtten, Tore; Munkejord, Svend Tollak. The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 4, pp. 735-764. doi : 10.1051/m2an:2006032. http://www.numdam.org/item/M2AN_2006__40_4_735_0/

[1] R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361-396. | Zbl 1072.76594

[2] 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. | Zbl pre02146841

[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. | Zbl 1130.76384

[4] K.H. Bendiksen, An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphas. Flow 10 (1984) 467-483.

[5] S. Benzoni-Gavage, Analyse numérique des modèles hydrodynamiques d'écoulements diphasiques instationnaires dans les réseaux de production pétrolière. Thèse ENS Lyon, France (1991).

[6] 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. | Zbl 0917.76047

[7] S. Evje and K.K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175 (2002) 674-201. | Zbl pre01763718

[8] S. Evje and K.K. Fjelde, On a rough AUSM scheme for a one-dimensional two-phase model. Comput. Fluids 32 (2003) 1497-1530. | Zbl 1128.76337

[9] S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192 (2003) 175-210. | Zbl 1032.76696

[10] I. Faille and E. Heintzé, A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids 28 (1999) 213-241. | Zbl 0964.76050

[11] K.K. Fjelde and K.H. Karlsen, High-resolution hybrid primitive-conservative upwind schemes for the drift-flux model. Comput. Fluids 31 (2002) 335-367. | Zbl 1059.76044

[12] F. França and R.T. Lahey, Jr., The use of drift-flux techniques for the analysis of horizontal two-phase flows. Int. J. Multiphas. Flow 18 (1992) 787-801. | Zbl pre05349986

[13] A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. | Zbl 0565.65050

[14] S. Jin and Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pur. Appl. Math. 48 (1995) 235-276. | Zbl 0826.65078

[15] S. Karni, E. Kirr, A. Kurganov and G. Petrova, Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477-493. | Numdam | Zbl 1079.76045

[16] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK (2002). | MR 1925043 | Zbl 1010.65040

[17] J.M. Masella, Q.H. Tran, D. Ferre and C. Pauchon, Transient simulation of two-phase flows in pipes. Int. J. Multiphas. Flow 24 (1998) 739-755. | Zbl 1121.76459

[18] S.T. Munkejord, S. Evje and T. Flåtten, The multi-stage centred-scheme approach applied to a drift-flux two-phase flow model. Int. J. Numer. Meth. Fl. 52 (2006) 679-705. | Zbl 1113.76057

[19] A. Murrone and H. Guillard, A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005) 664-698. | Zbl 1061.76083

[20] S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21 (1984) 217-235. | Zbl 0592.65069

[21] V.H. Ransom and D.L. Hicks, Hyperbolic two-pressure models for two-phase flow. J. Comput. Phys. 53 (1984) 124-151. | Zbl 0537.76070

[22] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | Zbl 0474.65066

[23] J.E. Romate, An approximate Riemann solver for a two-phase flow model with numerically given slip relation. Comput. Fluids 27 (1998) 455-477. | Zbl 0968.76052

[24] L. Sainsaulieu, Finite volume approximation of two-phase fluid flow based on an approximate Roe-type Riemann solver. J. Comput. Phys. 121 (1995) 1-28. | Zbl 0834.76070

[25] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425-467. | Zbl 0937.76053

[26] H.B. Stewart and B. Wendroff, Review article; Two-phase flow: models and methods. J. Comput. Phys. 56 (1984) 363-409. | Zbl 0596.76103

[27] V.A. Titarev and E.F. Toro, MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. Int. J. Numer. Meth. Fl. 49 (2005) 117-147. | Zbl 1073.65553

[28] E.F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer-Verlag, Berlin (1999). | MR 1717819 | Zbl 0801.76062

[29] I. Toumi, An upwind numerical method for two-fluid two-phase flow models. Nucl. Sci. Eng. 123 (1996) 147-168.

[30] I. Toumi and D. Caruge, An implicit second-order numerical method for three-dimensional two-phase flow calculations. Nucl. Sci. Eng. 130 (1998) 213-225.

[31] I. Toumi and A. Kumbaro, An approximate linearized Riemann solver for a two-fluid model. J. Comput. Phys. 124 (1996) 286-300. | Zbl 0847.76056

[32] B. Van Leer, Towards the ultimate conservative difference scheme IV. New approach to numerical convection. J. Comput. Phys. 23 (1977) 276-299. | Zbl 0339.76056

[33] N. Zuber and J.A. Findlay, Average volumetric concentration in two-phase flow systems. J. Heat Transfer 87 (1965) 453-468.