This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problem between its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, and tends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction with one's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension, oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.
@article{M2AN_2008__42_6_991_0,
author = {Chertock, Alina and Karni, Smadar and Kurganov, Alexander},
title = {Interface tracking method for compressible multifluids},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {991-1019},
doi = {10.1051/m2an:2008036},
mrnumber = {2473317},
zbl = {pre05370464},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2008__42_6_991_0}
}
Chertock, Alina; Karni, Smadar; Kurganov, Alexander. Interface tracking method for compressible multifluids. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 991-1019. doi : 10.1051/m2an:2008036. http://gdmltest.u-ga.fr/item/M2AN_2008__42_6_991_0/
[1] , Generalization of the Roe scheme for the computation of mixture of perfect gases. Rech. Aérosp. 6 (1988) 31-43. | Zbl 0662.76097
[2] , How to prevent pressure oscillations in multicomponent flows: A quasi conservative approach. J. Comp. Phys. 125 (1996) 150-160. | MR 1381808 | Zbl 0847.76060
[3] and , Ghost-fluids for the poor: a single fluid algorithm for multifluids, in Hyperbolic problems: theory, numerics, applications, Vols. I, II (Magdeburg, 2000), Birkhäuser, Basel, Internat. Ser. Numer. Math. 140 (2001) 1-10. | MR 1882900
[4] and , Computations of compressible multifluids. J. Comp. Phys. 169 (2001) 594-623. | MR 1836526 | Zbl 1033.76029
[5] and , Discrete equations for physical and numerical compressible multiphase flow mixtures. J. Comp. Phys. 186 (2003) 361-396. | MR 1973195 | Zbl 1072.76594
[6] , and , Efficient numerical approximation of compressible multi-material flow for unstructured meshes. Comput. Fluids 4 (2003) 571-605. | MR 1966640 | Zbl 1084.76543
[7] , , , and , Front tracking for gas dynamics. J. Comp. Phys. 62 (1986) 83-110. | MR 825892 | Zbl 0577.76068
[8] and , Conservative locally moving mesh method for multifluid flows. Proceedings of the Fourth International Symposium on Finite Volumes for Complex Applications, Marrakech (2005) 273-284. | MR 1500025
[9] , , , and , A numerical method using upwind schemes for the resolution of two-phase flows. J. Comp. Phys. 136 (1997) 272-288. | MR 1474408 | Zbl 0893.76052
[10] , An interface tracking method for hyperbolic systems of conservation laws. Appl. Numer. Math. 10 (1992) 447-472. | MR 1192634 | Zbl 0766.65067
[11] , , and , A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comp. Phys. 152 (1999) 457-492. | MR 1699710 | Zbl 0957.76052
[12] , , , , and , Three-dimensional front tracking. SIAM J. Sci. Comput. 19 (1998) 703-727. | MR 1616658 | Zbl 0912.65075
[13] , , and , Conservative front tracking and level set algorithms. Proc. Natl. Acad. Sci. USA 98 (2001) 14198-14201. | MR 1867514 | Zbl 1005.65091
[14] , , and , Conservative front tracking with improved accuracy. SIAM J. Numer. Anal. 41 (2003) 1926-1947. | MR 2035012 | Zbl 1053.35093
[15] and , Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075
[16] and , The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case. Numer. Math. 97 (2004) 81-130. | MR 2045460 | Zbl 1063.65080
[17] , , , The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II. The case of systems. ESAIM: M2AN 39 (2005) 649-692. | Numdam | MR 2165674 | Zbl 1095.65084
[18] , and , High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89-112. | MR 1854647 | Zbl 0967.65098
[19] and , Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181 (1987) 313-336.
[20] and , Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235-269. | MR 707200 | Zbl 0565.65049
[21] and , Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal. 24 (1987) 279-309. | MR 881365 | Zbl 0627.65102
[22] , , and , Some results on uniformly high order accurate essentially non-oscillatory schemes. Appl. Numer. Math. 2 (1986) 347-377. | MR 863993 | Zbl 0627.65101
[23] , and , Correction of conservative Euler solvers for gas mixtures. J. Comp. Phys. 132 (1997) 91-107. | MR 1440335 | Zbl 0879.76059
[24] , Multicomponent flow calculations by a consistent primitive algorithm. J. Comp. Phys. 112 (1994) 31-43. | MR 1277497 | Zbl 0811.76044
[25] , Compressible bubbles with surface tension, in Sixteenth International Conference on Numerical Methods in Fluid Dynamics (Arcachon, 1998), Springer, Berlin, Lecture Notes in Physics 515 (1998) 506-511. | MR 1730427
[26] , , and , Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN 38 (2004) 477-493. | Numdam | MR 2075756 | Zbl 1079.76045
[27] , Numerical Schemes for Conservation Laws. Wiley, Chichester (1997). | MR 1437144 | Zbl 0872.76001
[28] and , On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2 (2007) 141-163. | MR 2305919 | Zbl pre05371717
[29] and , New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comp. Phys. 160 (2000) 241-282. | MR 1756766 | Zbl 0987.65085
[30] , and , Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (2001) 707-740. | MR 1860961 | Zbl 0998.65091
[31] , How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comp. Phys. 95 (1991) 59-84. | MR 1112315 | Zbl 0725.76090
[32] , Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040
[33] and , On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24 (2003) 1157-1174. | MR 1976211 | Zbl 1038.65078
[34] , and , Computing interface motion in compressible gas dynamics. J. Comp. Phys. 100 (1992) 209-228. | MR 1167743 | Zbl 0758.76044
[35] and , Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87 (1990) 408-463. | MR 1047564 | Zbl 0697.65068
[36] and , On the dynamics of a shock-bubble interaction. J. Fluid Mech. 318 (1996) 129-163. | Zbl 0877.76046
[37] , Fluctuations and signals - a framework for numerical evolution problems, in Numerical Methods for Fluid Dynamics, Academic Press, New York (1982) 219-257. | Zbl 0569.76072
[38] and , A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comp. Phys. 150 (1999) 425-467. | MR 1684902 | Zbl 0937.76053
[39] , An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comp. Phys. 142 (1998) 208-242. | MR 1618088 | Zbl 0934.76062
[40] , A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comp. Phys. 156 (1999) 43-88. | MR 1727633 | Zbl 0957.76039
[41] , High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995-1011. | MR 760628 | Zbl 0565.65048
[42] , Improved shock-capturing methods for multicomponent and reacting flows. J. Comp. Phys. 128 (1996) 237-253. | MR 1412056 | Zbl 0860.76060
[43] , Riemann solvers and numerical methods for fluid dynamics. A practical introduction. Second edition, Springer-Verlag, Berlin (1999). | MR 1717819 | Zbl 0801.76062
[44] , , , , , , , and , A front-tracking method for the computations of multiphase flow. J. Comp. Phys. 169 (2001) 708-759. | Zbl 1047.76574
[45] , Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comp. Phys. 32 (1979) 101-136. | MR 1703646 | Zbl 0939.76063
[46] and , Five-equation model for compressible two-fluid flow. Report MAS-E0414, CWI, Amsterdam (2004). Available at http://ftp.cwi.nl/CWIreports/MAS/MAS-E0414.pdf
[47] , , , and , A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows. J. Comp. Phys. 195 (2004) 528-559. | MR 2046109 | Zbl 1115.76377
[48] , Underwater explosion test cases. IHTR 2069 (1998).