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Journal of Scientific Computing

, Volume 63, Issue 1, pp 78–117 | Cite as

A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow

  • John W. Barrett
  • Harald Garcke
  • Robert Nürnberg
Article

Abstract

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier–Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier–Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.

Keywords

Finite elements XFEM Two-phase flow Navier–Stokes  Free boundary problem Surface tension Interface tracking 

References

  1. 1.
    Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22(3), 1150,013 (2012). doi: 10.1142/S0218202511500138 CrossRefGoogle Scholar
  2. 2.
    Aland, S., Voigt, A.: Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 69(3), 747–761 (2012). doi: 10.1002/fld.2611 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. In: Annual Review of Fluid Mechanics, vol. 30, pp. 139–165. Annual Reviews, Palo Alto, CA (1998). doi: 10.1146/annurev.fluid.30.1.139
  4. 4.
    Ausas, R.F., Buscaglia, G.C., Idelsohn, S.R.: A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows. Int. J. Numer. Methods Fluids 70(7), 829–850 (2012). doi: 10.1002/fld.2713 CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bänsch, E.: Finite element discretization of the Navier–Stokes equations with a free capillary surface. Numer. Math. 88(2), 203–235 (2001). doi: 10.1007/PL00005443 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bänsch, E.: Numerical Methods for the Instationary Navier–Stokes Equations with a Free Capillary Surface. University Freiburg, Habilitation (2001)Google Scholar
  7. 7.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222(1), 441–462 (2007). doi: 10.1016/j.jcp.2006.07.026 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Barrett, J.W., Garcke, H., Nürnberg, R.: On the parametric finite element approximation of evolving hypersurfaces in \({\mathbb{R}}^3\). J. Comput. Phys. 227(9), 4281–4307 (2008). doi:  10.1016/j.jcp.2007.11.023 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Barrett, J.W., Garcke, H., Nürnberg, R.: On stable parametric finite element methods for the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth. J. Comput. Phys. 229(18), 6270–6299 (2010). doi: 10.1016/j.jcp.2010.04.039 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Eliminating spurious velocities with a stable approximation of viscous incompressible two-phase Stokes flow. Comput. Methods Appl. Mech. Eng. 267, 511–530 (2013). doi: 10.1016/j.cma.2013.09.023 CrossRefzbMATHGoogle Scholar
  11. 11.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Finite element approximation of one-sided Stefan problems with anisotropic, approximately crystalline, Gibbs–Thomson law. Adv. Differ. Equ. 18(3–4), 383–432 (2013). http://projecteuclid.org/euclid.ade/1360073021
  12. 12.
    Boffi, D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34(2), 664–670 (1997). doi: 10.1137/S0036142994270193 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Boffi, D., Cavallini, N., Gardini, F., Gastaldi, L.: Local mass conservation of Stokes finite elements. J. Sci. Comput. 52(2), 383–400 (2012). doi: 10.1007/s10915-011-9549-4 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991). doi: 10.1007/978-1-4612-3172-1 CrossRefGoogle Scholar
  15. 15.
    Cheng, K.W., Fries, T.P.: XFEM with hanging nodes for two-phase incompressible flow. Comput. Methods Appl. Mech. Eng. 245–246, 290–312 (2012). doi: 10.1016/j.cma.2012.07.011 CrossRefMathSciNetGoogle Scholar
  16. 16.
    Cho, M.H., Choi, H.G., Choi, S.H., Yoo, J.Y.: A Q2Q1 finite element/level-set method for simulating two-phase flows with surface tension. Int. J. Numer. Methods Fluids 70, 468–492 (2012). doi: 10.1002/fld.2696 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005). doi: 10.1017/S0962492904000224 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(6), 603–611 (1991). doi: 10.1007/BF01385643 zbMATHMathSciNetGoogle Scholar
  19. 19.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)Google Scholar
  20. 20.
    Feng, X.: Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006). doi: 10.1137/050638333 CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Ganesan, S.: Finite element methods on moving meshes for free surface and interface flows. Ph.D. thesis, University Magdeburg, Magdeburg, Germany (2006)Google Scholar
  22. 22.
    Ganesan, S., Matthies, G., Tobiska, L.: On spurious velocities in incompressible flow problems with interfaces. Comput. Methods Appl. Mech. Eng. 196(7), 1193–1202 (2007). doi: 10.1016/j.cma.2006.08.018 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006). doi: 10.1093/acprof:oso/9780198566656.001.0001 CrossRefGoogle Scholar
  24. 24.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes. Springer, Berlin (1986)zbMATHGoogle Scholar
  25. 25.
    Groß, S., Reusken, A.: An extended pressure finite element space for two-phase incompressible flows with surface tension. J. Comput. Phys. 224(1), 40–58 (2007). doi: 10.1016/j.jcp.2006.12.021 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Groß, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows, Springer Series in Computational Mathematics, vol. 40. Springer, Berlin (2011)CrossRefGoogle Scholar
  27. 27.
    Grün, G., Klingbeil, F.: Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257, 708–725 (2014). doi: 10.1016/j.jcp.2013.10.028 CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hirt, C.W., Nichols, B.D.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39(1), 201–225 (1981). doi: 10.1016/0021-9991(81)90145-5 CrossRefzbMATHGoogle Scholar
  29. 29.
    Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977). doi: 10.1103/RevModPhys.49.435 CrossRefGoogle Scholar
  30. 30.
    Hughes, T.J.R., Liu, W.K., Zimmermann, T.K.: Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981). doi: 10.1016/0045-7825(81)90049-9 CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Hysing, S., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., Tobiska, L.: Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 60(11), 1259–1288 (2009). doi: 10.1002/fld.1934 CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Jemison, M., Loch, E., Sussman, M., Shashkov, M., Arienti, M., Ohta, M., Wang, Y.: A coupled level set-moment of fluid method for incompressible two-phase flows. J. Sci. Comput. 54(2–3), 454–491 (2013). doi: 10.1007/s10915-012-9614-7 CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn–Hilliard–Navier–Stokes system. Interfaces Free Bound. 10(1), 15–43 (2008). doi: 10.4171/IFB/178 CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    LeVeque, R.J., Li, Z.: Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18(3), 709–735 (1997). doi: 10.1137/S1064827595282532 CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Li, Y., Yun, A., Lee, D., Shin, J., Jeong, D., Kim, J.: Three-dimensional volume-conserving immersed boundary model for two-phase fluid flows. Comput. Methods Appl. Mech. Eng. 257, 36–46 (2013). doi: 10.1016/j.cma.2013.01.009 CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998). doi: 10.1098/rspa.1998.0273 CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Olshanskii, M.A., Reusken, A.: Analysis of a Stokes interface problem. Numer. Math. 103(1), 129–149 (2006). doi: 10.1007/s00211-005-0646-x CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153. Springer, New York (2003)CrossRefGoogle Scholar
  39. 39.
    Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002). doi: 10.1017/S0962492902000077 CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Pilliod Jr, J.E., Puckett, E.G.: Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 199(2), 465–502 (2004). doi: 10.1016/j.jcp.2003.12.023 CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009). doi: 10.1016/j.jcp.2009.04.042 CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Renardy, Y., Renardy, M.: PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method. J. Comput. Phys. 183(2), 400–421 (2002). doi: 10.1006/jcph.2002.7190 CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Scardovelli, R., Zaleski, S.: Interface reconstruction with least-square fit and split Eulerian–Lagrangian advection. Int. J. Numer. Methods Fluids 41(3), 251–274 (2003). doi: 10.1002/fld.431 CrossRefzbMATHGoogle Scholar
  44. 44.
    Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (2005)Google Scholar
  45. 45.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  46. 46.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)zbMATHGoogle Scholar
  47. 47.
    Sussman, M., Ohta, M.: A stable and efficient method for treating surface tension in incompressible two-phase flow. SIAM J. Sci. Comput. 31(4), 2447–2471 (2009). doi: 10.1137/080732122 CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Sussman, M., Semereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114(1), 146–159 (1994). doi: 10.1006/jcph.1994.1155 CrossRefzbMATHGoogle Scholar
  49. 49.
    Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence (2001)Google Scholar
  50. 50.
    Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.J.: A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169(2), 708–759 (2001). doi: 10.1006/jcph.2001.6726 CrossRefzbMATHGoogle Scholar
  51. 51.
    Unverdi, S.O., Tryggvason, G.: A front-tracking method for viscous, incompressible multi-fluid flows. J. Comput. Phys. 100(1), 25–37 (1992). doi: 10.1016/0021-9991(92)90307-K CrossRefzbMATHGoogle Scholar
  52. 52.
    Zahedi, S., Kronbichler, M., Kreiss, G.: Spurious currents in finite element based level set methods for two-phase flow. Int. J. Numer. Methods Fluids 69(9), 1433–1456 (2012). doi: 10.1002/fld.2643 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • John W. Barrett
    • 1
  • Harald Garcke
    • 2
  • Robert Nürnberg
    • 1
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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