Hybrid central-upwind schemes for numerical resolution of two-phase flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, p. 253-273

In this paper we present a methodology for constructing accurate and efficient hybrid central-upwind (HCU) type schemes for the numerical resolution of a two-fluid model commonly used by the nuclear and petroleum industry. Particularly, we propose a method which does not make use of any information about the eigenstructure of the jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolution equation which describes how pressure evolves in time. By applying a quasi-staggered Lax-Friedrichs type discretization for this pressure equation together with a Modified Lax-Friedrich type discretization of the convective terms, we obtain a central type scheme which allows to cope with the nonlinearity (nonlinear pressure waves) of the two-fluid model in a robust manner. Then, in order to obtain an accurate resolution of mass fronts, we employ a modification of the convective mass fluxes by hybridizing the central type mass flux components with upwind type components. This hybridization is based on a splitting of the mass fluxes into components corresponding to the pressure and volume fraction variables, recovering an accurate resolution of a contact discontinuity. In the numerical simulations, the resulting HCU scheme gives results comparable to an approximate Riemann solver while being superior in efficiency. Furthermore, the HCU scheme yields better robustness than other popular Riemann-free upwind schemes.

DOI : https://doi.org/10.1051/m2an:2005011
Classification:  35L65,  65M12,  76N10,  76T10
Keywords: two-phase flow, two-fluid model, hyperbolic system of conservation laws, central discretization, upwind discretization, pressure evolution equation, hybrid scheme
@article{M2AN_2005__39_2_253_0,
author = {Evje, Steinar and Fl\aa tten, Tore},
title = {Hybrid central-upwind schemes for numerical resolution of two-phase flows},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
pages = {253-273},
doi = {10.1051/m2an:2005011},
zbl = {1130.76057},
mrnumber = {2143949},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_2_253_0}
}

Evje, Steinar; Flåtten, Tore. Hybrid central-upwind schemes for numerical resolution of two-phase flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 253-273. doi : 10.1051/m2an:2005011. http://www.numdam.org/item/M2AN_2005__39_2_253_0/

[1] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations. J. Comput. Phys. 125 (1996) 150-160. | Zbl 0847.76060

[2] F. Barre et al., The CATHARE code strategy and assessment. Nucl. Eng. Des. 124 (1990) 257-284.

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

[4] 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. | Zbl 0893.76052

[5] 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

[6] S. Evje and K.K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175 (2002) 674-701. | Zbl pre01763718

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

[8] 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

[9] S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model. SIAM J. Sci. Comput., accepted. | MR 2142581 | Zbl pre02207113

[10] T. Flåtten, Hybrid flux-splitting schemes for numerical resolution of two-phase flows. Dr.ing.-thesis, Norwegian University of Science and Technology (2003) 114.

[11] M. Larsen, E. Hustvedt, P. Hedne and T. Straume, PeTra: A novel computer code for simulation of slug flow, in SPE Annual Technical Conference and Exhibition, SPE 38841 (October 1997) 1-12.

[12] M.-S. Liou, A sequel to AUSM: AUSM(+). J. Comput. Phys. 129 (1996) 364-382. | Zbl 0870.76049

[13] Y.Y. Niu, Simple conservative flux splitting for multi-component flow calculations. Num. Heat Trans. 38 (2000) 203-222.

[14] Y.Y. Niu, Advection upwinding splitting method to solve a compressible two-fluid model. Internat. J. Numer. Methods Fluids 36 (2001) 351-371. | Zbl 1044.76041

[15] H. Paillère, C. Corre and J.R.G. Cascales, On the extension of the AUSM+ scheme to compressible two-fluid models. Comput. Fluids 32 (2003) 891-916. | Zbl 1040.76044

[16] V.H. Ransom, Numerical bencmark tests. Multiphase Sci. Tech. 3 (1987) 465-473.

[17] V.H. Ransom et al., RELAP5/MOD3 Code Manual, NUREG/CR-5535, Idaho National Engineering Laboratory (1995).

[18] 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

[19] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp. 168 (1984) 369-381. | Zbl 0587.65058

[20] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys. 136 (1997) 503-521. | Zbl 0918.76050

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

[22] 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

[23] J.A. Trapp and R.A. Riemke, A nearly-implicit hydrodynamic numerical scheme for two-phase flows. J. Comput. Phys. 66 (1986) 62-82. | Zbl 0622.76110

[24] Y. Wada and M.-S. Liou, An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM J. Sci. Comput. 18 (1997) 633-657. | Zbl 0879.76064