Journal of Scientific Computing

, Volume 31, Issue 1–2, pp 99–125 | Cite as

On Boundary Condition Capturing for Multiphase Interfaces

  • Jeong-Mo Hong
  • Tamar Shinar
  • Myungjoo Kang
  • Ronald Fedkiw

This review paper begins with an overview of the boundary condition capturing approach to solving problems with interfaces. Although the authors’ original motivation was to extend the ghost fluid method from compressible to incompressible flow, the elliptic nature of incompressible flow quickly quenched the idea that ghost cells could be defined and used in the usual manner. Instead the boundary conditions had to be implicitly captured by the matrix formulation itself, leading to the novel approach. We first review the work on the variable coefficient Poisson equation, noting that the simplicity of the method allowed for an elegant convergence proof. Simplicity and robustness also allowed for a quick extension to three-dimensional two-phase incompressible flows including the effects of viscosity and surface tension, which is discussed subsequently. The method has enjoyed popularity in both computational physics and computer graphics, and we show some comparisons with the traditional delta function approach for the visual simulation of bubbles. Finally, we discuss extensions to problems where the velocity is discontinuous as well, as is the case for premixed flames, and show an example of multiple interacting liquids that includes all of the aforementioned phenomena.


Multiphase flow two-phase flow interfaces 


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  1. 1.
    Fedkiw R., Aslam T., Merriman B., Osher S. (1999). A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fedkiw R., Aslam T., Xu S. (1999). The Ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154, 393–427MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Nguyen, D., Gibou, F., and Fedkiw, R. (2002). A fully conservative ghost fluid method and stiff detonation waves. In 12th Int. Detonation Symp., San Diego, CA.Google Scholar
  4. 4.
    Liu X.-D., Fedkiw R., Kang M. (2000). A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 154, 151CrossRefMathSciNetGoogle Scholar
  5. 5.
    Liu X.-D., Sideris T.C. (2003). Convergence of the ghost fluid method for elliptic equations with interfaces. Math. Computat. 72(244): 1731–1746MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kang M., Fedkiw R., Liu X.-D. (2000). A boundary condition capturing method for multiphase incompressible flow. J. Sci. Comput. 15, 323–360MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Unverdi S.O., Tryggvason G. (1992). A front-tracking method for viscous, incompressible, multifluid flows. J. Comput. Phys. 100, 25–37MATHCrossRefGoogle Scholar
  8. 8.
    Sussman M., Smereka P., Osher S. (1994). A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159MATHCrossRefGoogle Scholar
  9. 9.
    Brackbill J.U., Kothe D.B., Zemach C. (1992). A continuum method for modelling surface tension. J. Comput. Phys. 100, 335–353MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Tornberg A.-K., Engquist B. (2004). Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200(2): 462–488MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Engquist B., Tornberg A.-K., Tsai R. (2005). Discretization of dirac delta functions in level set methods. J. Comput. Phys. 207(1): 28–51MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Smereka P. (2006). The numerical approximation of a delta function with application to level set methods. J. Comput. Phys. 211, 77–90MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Leveque R.J., Li Z. (1994). The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numerical Anal. 31(4): 1019–1044MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Li Z., Lai M.-C. (2001). The immersed interface method for the navier-stokes equations with singular forces. J. Comput. Phys. 171(2): 822–842MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hong J.-M., Kim C.-H. (2005). Discontinuous fluids. ACM Trans. Graph. (SIGGRAPH Proc.) 24(3): 915–920CrossRefMathSciNetGoogle Scholar
  16. 16.
    Nguyen D., Fedkiw R., Jensen H. (2002). Physically based modeling and animation of fire. ACM Trans. Graph. (SIGGRAPH Proc.) 29, 721–728Google Scholar
  17. 17.
    Nguyen D., Fedkiw R., Kang M. (2001). A boundary condition capturing method for incompressible flame discontinuities. J. Comput. Phys. 172, 71–98MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Losasso F., Shinar T., Selle A., Fedkiw R. (2006). Multiple interacting liquids. ACM Trans. Graph. (SIGGRAPH Proc.) 25(3): 812–819CrossRefGoogle Scholar
  19. 19.
    Landau L.D., Lifshitz E.M. (1998). Fluid Mechanics, 2nd edn. Butterworth-Heinemann, OxfordGoogle Scholar
  20. 20.
    Fedkiw R., Liu X.-D. (1998). The ghost fluid method for viscous flows. In: Haferz M., (eds), Progress in Numerical Solutions of Partial Differential Equations. Arcachon, FranceGoogle Scholar
  21. 21.
    Popinet S., Zaleski S., (1999). A front-tracking algorithm for accurate representation of surface tension. Int. J. Numer. Meth. Fluids 30(6): 775–793MATHCrossRefGoogle Scholar
  22. 22.
    Francois M.M., Cummins S.J., Dendy E.D., Kothe D.B., Sicilian J.M., Williams M.W. (2006). A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213(1): 141–173MATHCrossRefGoogle Scholar
  23. 23.
    Rasmussen N., Enright D., Nguyen D., Marino S., Sumner N., Geiger W., Hoon S., Fedkiw R. (2004). Directable photorealistic liquids. In Proc. of the 2004 ACM SIGGRAPH/Eurographics Symp. on Comput. Anim., pp. 193–202.Google Scholar
  24. 24.
    Li J., Renardy Y., Renardy M. (2000). Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 12(2): 269–282CrossRefGoogle Scholar
  25. 25.
    Juric D., Tryggvason G. (1998). Computations of boiling flows. Int. J. Multiphase Flow 24, 387–410CrossRefGoogle Scholar
  26. 26.
    Qian J., Tryggvason G., Law C.K. (1998). A front method for the motion of premixed flames. J. Comput. Phys. 144, 52–69CrossRefGoogle Scholar
  27. 27.
    Son G., Dir V.K. (1998). Numerical simulation of film boiling near critical pressure with a level set method. J. Heat Transfer 120, 183–192Google Scholar
  28. 28.
    Welch S., Wilson J. (2000). A volume of fluid based method for fluid flows with phase change. J. Comput. Phys. 160, 662–682MATHCrossRefGoogle Scholar
  29. 29.
    Enright D., Fedkiw R., Ferziger J., Mitchell I. (2002). A hybrid particle level set method for improved interface capturing. J. Comput. Phys.. 183, 83–116MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Merriman B., Bence J., Osher S. (1994). Motion of multiple junctions: A level set approach. J. Comput. Phys. 112, 334–363CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Jeong-Mo Hong
    • 1
  • Tamar Shinar
    • 2
  • Myungjoo Kang
    • 3
  • Ronald Fedkiw
    • 4
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  3. 3.Department of Mathematical Sciences and the Research Institute of MathematicsSeoul National UniversitySeoulKorea
  4. 4.Computer Science DepartmentStanford UniversityStanfordUSA

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