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.
Similar content being viewed by others
References
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
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
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
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
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
Bänsch, E.: Numerical Methods for the Instationary Navier–Stokes Equations with a Free Capillary Surface. University Freiburg, Habilitation (2001)
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
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
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
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
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
Boffi, D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34(2), 664–670 (1997). doi:10.1137/S0036142994270193
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
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
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
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
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
Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(6), 603–611 (1991). doi:10.1007/BF01385643
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)
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
Ganesan, S.: Finite element methods on moving meshes for free surface and interface flows. Ph.D. thesis, University Magdeburg, Magdeburg, Germany (2006)
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
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
Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes. Springer, Berlin (1986)
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
Groß, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows, Springer Series in Computational Mathematics, vol. 40. Springer, Berlin (2011)
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
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
Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977). doi:10.1103/RevModPhys.49.435
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
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
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
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
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
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
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
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
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153. Springer, New York (2003)
Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002). doi:10.1017/S0962492902000077
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
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
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
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
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)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)
Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)
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
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
Temam, R.: Navier–Stokes Equations. AMS Chelsea Publishing, Providence (2001)
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
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
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barrett, J.W., Garcke, H. & Nürnberg, R. A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow. J Sci Comput 63, 78–117 (2015). https://doi.org/10.1007/s10915-014-9885-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9885-2