Interface tracking method for compressible multifluids

Alina Chertock; Smadar Karni; Alexander Kurganov

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 6, page 991-1019
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Chertock, Alina, Karni, Smadar, and Kurganov, Alexander. "Interface tracking method for compressible multifluids." ESAIM: Mathematical Modelling and Numerical Analysis 42.6 (2008): 991-1019. <http://eudml.org/doc/250405>.

@article{Chertock2008,
abstract = { 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. },
author = {Chertock, Alina, Karni, Smadar, Kurganov, Alexander},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Compressible Euler equations; multicomponent fluids; material interfaces; finite-volume schemes.; compressible Euler equations; finite-volume schemes},
language = {eng},
month = {9},
number = {6},
pages = {991-1019},
publisher = {EDP Sciences},
title = {Interface tracking method for compressible multifluids},
url = {http://eudml.org/doc/250405},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Chertock, Alina
AU - Karni, Smadar
AU - Kurganov, Alexander
TI - Interface tracking method for compressible multifluids
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/9//
PB - EDP Sciences
VL - 42
IS - 6
SP - 991
EP - 1019
AB - 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.
LA - eng
KW - Compressible Euler equations; multicomponent fluids; material interfaces; finite-volume schemes.; compressible Euler equations; finite-volume schemes
UR - http://eudml.org/doc/250405
ER -

References

top
  1. R. Abgrall, Generalization of the Roe scheme for the computation of mixture of perfect gases. Rech. Aérosp.6 (1988) 31–43.  
  2. R. Abgrall, How to prevent pressure oscillations in multicomponent flows: A quasi conservative approach. J. Comp. Phys.125 (1996) 150–160.  
  3. 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.  
  4. R. Abgrall and S. Karni, Computations of compressible multifluids. J. Comp. Phys.169 (2001) 594–623.  
  5. R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase flow mixtures. J. Comp. Phys.186 (2003) 361–396.  
  6. R. Abgrall, B. N'Konga and R. Saurel, Efficient numerical approximation of compressible multi-material flow for unstructured meshes. Comput. Fluids4 (2003) 571–605.  
  7. I.-L. Chern, J. Glimm, O. McBryan, B. Plohr and S. Yaniv, Front tracking for gas dynamics. J. Comp. Phys.62 (1986) 83–110.  
  8. 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.  
  9. 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. Comp. Phys.136 (1997) 272–288.  
  10. S.F. Davis, An interface tracking method for hyperbolic systems of conservation laws. Appl. Numer. Math.10 (1992) 447–472.  
  11. R.P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comp. Phys.152 (1999) 457–492.  
  12. J. Glimm, J.W. Grove, X.L. Li, K.-M. Shyue, Y. Zeng and Q. Zhang, Three-dimensional front tracking. SIAM J. Sci. Comput.19 (1998) 703–727.  
  13. J. Glimm, X.L. Li, Y. Liu and N. Zhao, Conservative front tracking and level set algorithms. Proc. Natl. Acad. Sci. USA98 (2001) 14198–14201.  
  14. J. Glimm, Y. Liu, Z. Xu and N. Zhao, Conservative front tracking with improved accuracy. SIAM J. Numer. Anal.41 (2003) 1926–1947.  
  15. E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws. Springer-Verlag, New York (1996).  
  16. E. Godlewski and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case. Numer. Math.97 (2004) 81–130.  
  17. E. Godlewski, K.-C. Le Thanh, P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II. The case of systems. ESAIM: M2AN39 (2005) 649–692.  
  18. S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property. SIAM Rev.43 (2001) 89–112.  
  19. J.-F. Haas and B. Sturtevant, Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech.181 (1987) 313–336.  
  20. A. Harten and J.M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys.50 (1983) 235–269.  
  21. A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal.24 (1987) 279–309.  
  22. A. Harten, S. Osher, B. Engquist and S.R. Chakravarthy, Some results on uniformly high order accurate essentially non-oscillatory schemes. Appl. Numer. Math.2 (1986) 347–377.  
  23. P. Jenny, B. Mueller and H. Thomann, Correction of conservative Euler solvers for gas mixtures. J. Comp. Phys.132 (1997) 91–107.  
  24. S. Karni, Multicomponent flow calculations by a consistent primitive algorithm. J. Comp. Phys.112 (1994) 31–43.  
  25. 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.  
  26. S. Karni, E. Kirr, A. Kurganov and G. Petrova, Compressible two-phase flows by central and upwind schemes. ESAIM: M2AN38 (2004) 477–493.  
  27. D. Kröner, Numerical Schemes for Conservation Laws. Wiley, Chichester (1997).  
  28. A. Kurganov and C.-T. Lin, On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys.2 (2007) 141–163.  
  29. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comp. Phys.160 (2000) 241–282.  
  30. A. Kurganov, S. Noelle and G. Petrova, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput.21 (2001) 707–740.  
  31. B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comp. Phys.95 (1991) 59–84.  
  32. R. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press (2002).  
  33. K.-A. Lie and S. Noelle, 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.  
  34. W. Mulder, S. Osher and J.A. Sethian, Computing interface motion in compressible gas dynamics. J. Comp. Phys.100 (1992) 209–228.  
  35. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys.87 (1990) 408–463.  
  36. J.J. Quirk and S. Karni, On the dynamics of a shock-bubble interaction. J. Fluid Mech.318 (1996) 129–163.  
  37. 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.  
  38. R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comp. Phys.150 (1999) 425–467.  
  39. K.-M. Shyue, An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comp. Phys.142 (1998) 208–242.  
  40. K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comp. Phys.156 (1999) 43–88.  
  41. P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal.21 (1984) 995–1011.  
  42. V. Ton, Improved shock-capturing methods for multicomponent and reacting flows. J. Comp. Phys.128 (1996) 237–253.  
  43. E.F. Toro, Riemann solvers and numerical methods for fluid dynamics. A practical introduction. Second edition, Springer-Verlag, Berlin (1999).  
  44. G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas and Y.-J. Jan, A front-tracking method for the computations of multiphase flow. J. Comp. Phys.169 (2001) 708–759.  
  45. B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comp. Phys.32 (1979) 101–136.  
  46. J. Wackers and B. Koren, Five-equation model for compressible two-fluid flow. Report MAS-E0414, CWI, Amsterdam (2004). Available at  URIhttp://ftp.cwi.nl/CWIreports/MAS/MAS-E0414.pdf
  47. S.-P. Wang, M.H. Anderson, J.G. Oakley, M.L. Corradini and R. Bonazza, A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows. J. Comp. Phys.195 (2004) 528–559.  
  48. A. Wardlaw, Underwater explosion test cases. IHTR 2069 (1998).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.