Home

The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model

Abstract

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.

References

Hide All
[1] Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361396.
[2] Baudin, M., Berthon, C., Coquel, F., Masson, R. and Tran, Q.H., A relaxation method for two-phase flow models with hydrodynamic closure law. Numer. Math. 99 (2005) 411440.
[3] Baudin, M., Coquel, F. and Tran, Q.H., A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline. SIAM J. Sci. Comput. 27 (2005) 914936.
[4] Bendiksen, K.H., An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphas. Flow 10 (1984) 467483.
[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.
[7] Evje, S. and Fjelde, K.K., Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175 (2002) 674201.
[8] Evje, S. and Fjelde, K.K., On a rough AUSM scheme for a one-dimensional two-phase model. Comput. Fluids 32 (2003) 14971530.
[9] Evje, S. and Flåtten, T., Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192 (2003) 175210.
[10] Faille, I. and Heintzé, E., A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids 28 (1999) 213241.
[11] Fjelde, K.K. and Karlsen, K.H., High-resolution hybrid primitive-conservative upwind schemes for the drift-flux model. Comput. Fluids 31 (2002) 335367.
[12] França, F. and Lahey, R.T., The, Jr. use of drift-flux techniques for the analysis of horizontal two-phase flows. Int. J. Multiphas. Flow 18 (1992) 787801.
[13] Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357393.
[14] Jin, S. and Xin, Z.P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pur. Appl. Math. 48 (1995) 235276.
[15] Karni, S., Kirr, E., Kurganov, A. and Petrova, G., Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477493.
[16] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK (2002).
[17] Masella, J.M., Tran, Q.H., Ferre, D. and Pauchon, C., Transient simulation of two-phase flows in pipes. Int. J. Multiphas. Flow 24 (1998) 739755.
[18] Munkejord, S.T., Evje, S. and Flåtten, T., The multi-stage centred-scheme approach applied to a drift-flux two-phase flow model. Int. J. Numer. Meth. Fl. 52 (2006) 679705.
[19] Murrone, A. and Guillard, H., A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005) 664698.
[20] Osher, S., Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21 (1984) 217235.
[21] Ransom, V.H. and Hicks, D.L., Hyperbolic two-pressure models for two-phase flow. J. Comput. Phys. 53 (1984) 124151.
[22] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357372.
[23] Romate, J.E., An approximate Riemann solver for a two-phase flow model with numerically given slip relation. Comput. Fluids 27 (1998) 455477.
[24] Sainsaulieu, L., Finite volume approximation of two-phase fluid flow based on an approximate Roe-type Riemann solver. J. Comput. Phys. 121 (1995) 128.
[25] Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425467.
[26] H.B. Stewart and B. Wendroff, Review article; Two-phase flow: models and methods. J. Comput. Phys. 56 (1984) 363–409.
[27] Titarev, V.A. and Toro, E.F., MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. Int. J. Numer. Meth. Fl. 49 (2005) 117147.
[28] E.F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer-Verlag, Berlin (1999).
[29] Toumi, I., An upwind numerical method for two-fluid two-phase flow models. Nucl. Sci. Eng. 123 (1996) 147168.
[30] Toumi, I. and Caruge, D., An implicit second-order numerical method for three-dimensional two-phase flow calculations. Nucl. Sci. Eng. 130 (1998) 213225.
[31] Toumi, I. and Kumbaro, A., An approximate linearized Riemann solver for a two-fluid model. J. Comput. Phys. 124 (1996) 286300.
[32] van Leer, B., Towards the ultimate conservative difference scheme IV. New approach to numerical convection. J. Comput. Phys. 23 (1977) 276299.
[33] Zuber, N. and Findlay, J.A., Average volumetric concentration in two-phase flow systems. J. Heat Transfer 87 (1965) 453468.

The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model

Metrics

Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *