Compressible two-phase flows by central and upwind schemes
Karni, Smadar ; Kirr, Eduard ; Kurganov, Alexander ; Petrova, Guergana
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 477-493 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004024
Classification:  35L65,  65M06,  76N15,  76T99 
@article{M2AN_2004__38_3_477_0,
     author = {Karni, Smadar and Kirr, Eduard and Kurganov, Alexander and Petrova, Guergana},
     title = {Compressible two-phase flows by central and upwind schemes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {477-493},
     doi = {10.1051/m2an:2004024},
     mrnumber = {2075756},
     zbl = {1079.76045},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_3_477_0}
}

Karni, Smadar; Kirr, Eduard; Kurganov, Alexander; Petrova, Guergana. Compressible two-phase flows by central and upwind schemes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 477-493. doi : 10.1051/m2an:2004024. http://gdmltest.u-ga.fr/item/M2AN_2004__38_3_477_0/

[1] R. Abgrall and S. Karni, Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. | Zbl 1033.76029

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

[3] 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

[4] D.A. Drew, Mathematical modelling of tow-phase flow. Ann. Rev. Fluid Mech. 15 (1983) 261-291. | Zbl 0569.76104

[5] B. Einfeldt, C.-D. Munz, P.L. Roe and B. Sjogreen, On Godunov-type methods near low densities. J. Comput. Phys. 92 (1991) 273-295. | Zbl 0709.76102

[6] A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes. I. SIAM J. Numer. Anal. 24 (1987) 279-309. | Zbl 0627.65102

[7] S. Karni, Multi-component flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112 (1994) 31-43. | Zbl 0811.76044

[8] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. ESAIM: M2AN 36 (2002) 397-425. | Numdam | Zbl 1137.65398

[9] A. Kurganov, S. Noelle and G. Petrova, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23 (2001) 707-740. | Zbl 0998.65091

[10] A. Kurganov and G. Petrova, Central schemes and contact discontinuities. ESAIM: M2AN 34 (2000) 1259-1275. | Numdam | Zbl 0972.65055

[11] B. Van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136. | Zbl 0939.76063

[12] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | Zbl 0697.65068

[13] V.H. Ransom, Numerical benchmark tests, G.F. Hewitt, J.M. Delhay and N. Zuber Eds., Hemisphere, Washington, DC Multiphase Science and Technology 3 (1987).

[14] P.-A. Raviart and L. Sainsaulieu, Nonconservative hyperbolic systems and two-phase flows, International Conference on Differential Equations (Barcelona, 1991) World Sci. Publishing, River Edge, NJ 1, 2 (1993) 225-233. | Zbl 0938.35575

[15] P.-A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Models Methods Appl. Sci. 5 (1995) 297-333. | Zbl 0837.76089

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

[17] P.L. Roe, Fluctuations and signals - A framework for numerical evolution problems, in Numerical Methods for Fluid Dynamics, K.W. Morton and M.J. Baines Eds., Academic Press (1982) 219-257. | Zbl 0569.76072

[18] P.L. Roe and J. Pike, Efficient construction and utilisation of approximate Riemann solutions, in Computing methods in applied sciences and engineering, VI (Versailles, 1983) North-Holland, Amsterdam (1984) 499-518. | Zbl 0558.76001

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

[20] 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

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

[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