Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-20T03:39:58.728Z Has data issue: false hasContentIssue false
  • English
  • Français

Interface tracking method for compressible multifluids

Published online by Cambridge University Press:  25 September 2008

Alina Chertock ,
Smadar Karni  and
Alexander Kurganov
Alina Chertock
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA. [email protected]
Smadar Karni
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. [email protected]
Alexander Kurganov
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA. [email protected]
Get access

Abstract

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and areseparated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupiedby 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 problembetween its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, andtends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction withone'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.

Keywords

Compressible Euler equationsmulticomponent fluidsmaterial interfacesfinite-volume schemes.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 991 - 1019
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abgrall, R., Generalization of the Roe scheme for the computation of mixture of perfect gases. Rech. Aérosp. 6 (1988) 31–43.
Abgrall, R., How to prevent pressure oscillations in multicomponent flows: A quasi conservative approach. J. Comp. Phys. 125 (1996) 150–160. CrossRef
R. Abgrall and S. Karni, 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.
Abgrall, R. and Karni, S., Computations of compressible multifluids. J. Comp. Phys. 169 (2001) 594–623. CrossRef
Abgrall, R. and Saurel, R., Discrete equations for physical and numerical compressible multiphase flow mixtures. J. Comp. Phys. 186 (2003) 361–396. CrossRef
Abgrall, R., N'Konga, B. and Saurel, R., Efficient numerical approximation of compressible multi-material flow for unstructured meshes. Comput. Fluids 4 (2003) 571–605. CrossRef
Chern, I.-L., Glimm, J., McBryan, O., Plohr, B. and Yaniv, S., Front tracking for gas dynamics. J. Comp. Phys. 62 (1986) 83–110. CrossRef
A. Chertock and A. Kurganov, Conservative locally moving mesh method for multifluid flows. Proceedings of the Fourth International Symposium on Finite Volumes for Complex Applications, Marrakech (2005) 273–284.
Coquel, F., El Amine, K., Godlewski, E., Perthame, B. and Rascle, P., A numerical method using upwind schemes for the resolution of two-phase flows. J. Comp. Phys. 136 (1997) 272–288. CrossRef
Davis, S.F., An interface tracking method for hyperbolic systems of conservation laws. Appl. Numer. Math. 10 (1992) 447–472. CrossRef
Fedkiw, R.P., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comp. Phys. 152 (1999) 457–492. CrossRef
Glimm, J., Grove, J.W., Li, X.L., Shyue, K.-M., Zeng, Y. and Zhang, Q., Three-dimensional front tracking. SIAM J. Sci. Comput. 19 (1998) 703–727. CrossRef
Glimm, J., Li, X.L., Liu, Y. and Zhao, N., Conservative front tracking and level set algorithms. Proc. Natl. Acad. Sci. USA 98 (2001) 14198–14201. CrossRef
Glimm, J., Liu, Y., Xu, Z. and Zhao, N., Conservative front tracking with improved accuracy. SIAM J. Numer. Anal. 41 (2003) 1926–1947. CrossRef
E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996).
Godlewski, E. and Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case. Numer. Math. 97 (2004) 81–130. CrossRef
Godlewski, E., Le Thanh, K.-C., Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II. The case of systems. ESAIM: M2AN 39 (2005) 649–692. CrossRef
Gottlieb, S., Shu, C.-W. and Tadmor, E., High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89–112. CrossRef
Haas, J.-F. and Sturtevant, B., Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181 (1987) 313–336.
Harten, A. and Hyman, J.M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235–269. CrossRef
Harten, A. and Osher, S., Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal. 24 (1987) 279–309. CrossRef
Harten, A., Osher, S., Engquist, B. and Chakravarthy, S.R., Some results on uniformly high order accurate essentially non-oscillatory schemes. Appl. Numer. Math. 2 (1986) 347–377. CrossRef
Jenny, P., Mueller, B. and Thomann, H., Correction of conservative Euler solvers for gas mixtures. J. Comp. Phys. 132 (1997) 91–107. CrossRef
Karni, S., Multicomponent flow calculations by a consistent primitive algorithm. J. Comp. Phys. 112 (1994) 31–43. CrossRef
S. Karni, 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.
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
D. Kröner, Numerical Schemes for Conservation Laws. Wiley, Chichester (1997).
Kurganov, A. and Lin, C.-T., On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2 (2007) 141–163.
Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comp. Phys. 160 (2000) 241–282. CrossRef
Kurganov, A., Noelle, S. and Petrova, G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21 (2001) 707–740. CrossRef
Larrouturou, B., How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comp. Phys. 95 (1991) 59–84. CrossRef
R. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002).
Lie, K.-A. and Noelle, S., 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. CrossRef
Mulder, W., Osher, S. and Sethian, J.A., Computing interface motion in compressible gas dynamics. J. Comp. Phys. 100 (1992) 209–228. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87 (1990) 408–463. CrossRef
Quirk, J.J. and Karni, S., On the dynamics of a shock-bubble interaction. J. Fluid Mech. 318 (1996) 129–163. CrossRef
P.L. Roe, Fluctuations and signals – a framework for numerical evolution problems, in Numerical Methods for Fluid Dynamics, Academic Press, New York (1982) 219–257.
Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comp. Phys. 150 (1999) 425–467. CrossRef
Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comp. Phys. 142 (1998) 208–242. CrossRef
Shyue, K.-M., A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comp. Phys. 156 (1999) 43–88. CrossRef
Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21 (1984) 995–1011. CrossRef
Ton, V., Improved shock-capturing methods for multicomponent and reacting flows. J. Comp. Phys. 128 (1996) 237–253. CrossRef
E.F. Toro, Riemann solvers and numerical methods for fluid dynamics. A practical introduction. Second edition, Springer-Verlag, Berlin (1999).
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. and Jan, Y.-J., A front-tracking method for the computations of multiphase flow. J. Comp. Phys. 169 (2001) 708–759. CrossRef
van Leer, B., Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comp. Phys. 32 (1979) 101–136. CrossRef
J. Wackers and B. Koren, 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
Wang, S.-P., Anderson, M.H., Oakley, J.G., Corradini, M.L. and Bonazza, R., A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows. J. Comp. Phys. 195 (2004) 528–559. CrossRef
A. Wardlaw, Underwater explosion test cases. IHTR 2069 (1998).