# Hybrid central-upwind schemes for numerical resolution of two-phase flows

- Volume: 39, Issue: 2, page 253-273
- ISSN: 0764-583X

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topEvje, Steinar, and Flåtten, Tore. "Hybrid central-upwind schemes for numerical resolution of two-phase flows." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.2 (2005): 253-273. <http://eudml.org/doc/244887>.

@article{Evje2005,

abstract = {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.},

author = {Evje, Steinar, Flåtten, Tore},

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

keywords = {two-phase flow; two-fluid model; hyperbolic system of conservation laws; central discretization; upwind discretization; pressure evolution equation; hybrid scheme},

language = {eng},

number = {2},

pages = {253-273},

publisher = {EDP-Sciences},

title = {Hybrid central-upwind schemes for numerical resolution of two-phase flows},

url = {http://eudml.org/doc/244887},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Evje, Steinar

AU - Flåtten, Tore

TI - Hybrid central-upwind schemes for numerical resolution of two-phase flows

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

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 2

SP - 253

EP - 273

AB - 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.

LA - eng

KW - two-phase flow; two-fluid model; hyperbolic system of conservation laws; central discretization; upwind discretization; pressure evolution equation; hybrid scheme

UR - http://eudml.org/doc/244887

ER -

## References

top- [1] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations. J. Comput. Phys. 125 (1996) 150–160. Zbl0847.76060
- [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. Zbl0893.76052
- [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. Zbl0917.76047
- [6] S. Evje and K.K. Fjelde, Hybrid flux-splitting schemes for a two-phase flow model. J. Comput. Phys. 175 (2002) 674–701. Zbl1197.76132
- [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. Zbl1128.76337
- [8] S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192 (2003) 175–210. Zbl1032.76696
- [9] S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model. SIAM J. Sci. Comput., accepted. Zbl1149.76673MR2142581
- [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. Zbl0870.76049
- [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. Zbl1044.76041
- [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. Zbl1040.76044
- [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. Zbl0937.76053
- [19] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp. 168 (1984) 369–381. Zbl0587.65058
- [20] I. Tiselj and S. Petelin, Modelling of two-phase flow with second-order accurate scheme. J. Comput. Phys. 136 (1997) 503–521. Zbl0918.76050
- [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. Zbl0847.76056
- [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. Zbl0622.76110
- [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. Zbl0879.76064

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