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

Published online by Cambridge University Press:  15 November 2006

Tore Flåtten  and
Svend Tollak Munkejord
Tore Flåtten
Affiliation:
The International Research Institute of Stavanger, Prof. Olav Hanssensvei 15, Stavanger, Norway. Tore.Flaatten@irisresearch.no Current address: Centre of Mathematics for Applications, 1053 Blindern, 0316 Oslo, Norway.
Svend Tollak Munkejord
Affiliation:
Department of Energy and Process Engineering, Norwegian University of Science and Technology, Kolbjørn Hejes veg 1A, 7491 Trondheim, Norway. stm@pvv.ntnu.no
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Abstract

We construct a Roe-type numerical scheme for approximating the solutionsof a drift-flux two-phase flow model. The model incorporates a set ofhighly 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 bynumerical examples.

Keywords

Two-phase flowdrift-flux modelRiemann solverRoe scheme.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 4 , July 2006 , pp. 735 - 764
Copyright
© EDP Sciences, SMAI, 2006

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References

Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361–396. CrossRef
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) 411–440. CrossRef
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) 914–936. CrossRef
Bendiksen, K.H., An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphas. Flow 10 (1984) 467–483. CrossRef
S. Benzoni-Gavage, Analyse numérique des modèles hydrodynamiques d'écoulements diphasiques instationnaires dans les réseaux de production pétrolière. Thèse ENS Lyon, France (1991).
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.
Evje, S. and Fjelde, K.K., Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175 (2002) 674–201. CrossRef
Evje, S. and Fjelde, K.K., On a rough AUSM scheme for a one-dimensional two-phase model. Comput. Fluids 32 (2003) 1497–1530. CrossRef
Evje, S. and Flåtten, T., Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192 (2003) 175–210. CrossRef
Faille, I. and Heintzé, E., A rough finite volume scheme for modeling two-phase flow in a pipeline. Comput. Fluids 28 (1999) 213–241. CrossRef
Fjelde, K.K. and Karlsen, K.H., High-resolution hybrid primitive-conservative upwind schemes for the drift-flux model. Comput. Fluids 31 (2002) 335–367. CrossRef
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) 787–801. CrossRef
Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. CrossRef
Jin, S. and Xin, Z.P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pur. Appl. Math. 48 (1995) 235–276. CrossRef
Karni, S., Kirr, E., Kurganov, A. and Petrova, G., Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477–493. CrossRef
R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK (2002).
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) 739–755. CrossRef
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) 679–705. CrossRef
Murrone, A. and Guillard, H., A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202 (2005) 664–698. CrossRef
Osher, S., Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21 (1984) 217–235. CrossRef
Ransom, V.H. and Hicks, D.L., Hyperbolic two-pressure models for two-phase flow. J. Comput. Phys. 53 (1984) 124–151. CrossRef
Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357–372. CrossRef
Romate, J.E., An approximate Riemann solver for a two-phase flow model with numerically given slip relation. Comput. Fluids 27 (1998) 455–477. CrossRef
Sainsaulieu, L., Finite volume approximation of two-phase fluid flow based on an approximate Roe-type Riemann solver. J. Comput. Phys. 121 (1995) 1–28. CrossRef
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 (1999) 425–467. CrossRef
H.B. Stewart and B. Wendroff, Review article; Two-phase flow: models and methods. J. Comput. Phys. 56 (1984) 363–409.
Titarev, V.A. and Toro, E.F., MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. Int. J. Numer. Meth. Fl. 49 (2005) 117–147. CrossRef
E.F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer-Verlag, Berlin (1999).
Toumi, I., An upwind numerical method for two-fluid two-phase flow models. Nucl. Sci. Eng. 123 (1996) 147–168. CrossRef
Toumi, I. and Caruge, D., An implicit second-order numerical method for three-dimensional two-phase flow calculations. Nucl. Sci. Eng. 130 (1998) 213–225. CrossRef
Toumi, I. and Kumbaro, A., An approximate linearized Riemann solver for a two-fluid model. J. Comput. Phys. 124 (1996) 286–300. CrossRef
van Leer, B., Towards the ultimate conservative difference scheme IV. New approach to numerical convection. J. Comput. Phys. 23 (1977) 276–299. CrossRef
Zuber, N. and Findlay, J.A., Average volumetric concentration in two-phase flow systems. J. Heat Transfer 87 (1965) 453–468. CrossRef