Skip to main content
SpringerLink
Log in

A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

In this paper, a consistent projection-based streamline upwind/pressure stabilizing Petrov-Galerkin (SUPG/PSPG) extended finite element method (XFEM) is presented to model incompressible immiscible two-phase flows. As the application of linear elements in SUPG/PSPG schemes gives rise to inconsistency in stabilization terms due to the inability to regenerate the diffusive term from viscous stresses, the numerical accuracy would deteriorate dramatically. To address this issue, projections of convection and pressure gradient terms are constructed and incorporated into the stabilization formulation in our method. This would substantially recover the consistency and free the practitioner from burdensome computations of most items in the residual. Moreover, the XFEM is employed to consider in a convenient way the fluid properties that have interfacial jumps leading to discontinuities in the velocity and pressure fields as well as the projections. A number of numerical examples are analyzed to demonstrate the complete recovery of consistency, the reproduction of interfacial discontinuities and the ability of the proposed projection-based SUPG/PSPG XFEM to model two-phase flows with open and closed interfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Donea, J., Quartapelle, L.: An introduction to finite element methods for transient advection problems. Comput. Method Appl. Mech. Eng. 95, 169–203 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Donea, J., Quartapelle, L., Selmin, V.: An analysis of time discretization in the finite element solution of hyperbolic problems. J. Comput. Phys. 70, 463–499 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brooks, A., Hughes, T.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Method Appl. Mech. Eng. 32, 199–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ladyzhenskaya, O., Ural’tseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  5. Babuša, I.: The finite element method with Lagrangian multipliers. Numerische Mathematik 20, 179–192 (1973)

    Article  Google Scholar 

  6. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Revue françise d’automatique, informatique, recherche opéationnelle, sommaire: Analyse Numérique 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  7. Hughes, T. J. R., Franca, L. P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the babuša-brezzi condition: A stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations. Comput. Method Appl. Mech. Eng. 59, 85–99 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tezduyar, T. E., Mittal, S., Ray, S. E., et al.: Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Method. Appl. Mech. Eng. 95, 221–242 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jansen, K. E., Collis, S. S., Whiting, C., et al.: A better consistency for low-order stabilized finite element methods. Comput. Method Appl. Mech. Eng. 174, 153–170 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tezduyar, T. E., Osawa, Y.: Finite element stabilization parameters computed from element matrices and vectors. Comput. Method Appl. Mech. Eng. 190, 411–430 (2000)

    Article  MATH  Google Scholar 

  11. Oõte, E., Valls, A., Garcí, J.: FIC/FEM formulation with matrix stabilizing terms for incompressible flows at low and high reynolds numbers. Comput. Mech. 38, 440–455 (2006)

    Article  Google Scholar 

  12. Li, X., Duan, Q.: Meshfree iterative stabilized Taylor-Galerkin and characteristic-based split (CBS) algorithms for incompressible N-S equations. Comput. Method Appl. Mech. Eng. 195, 6125–6145 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Method Eng. 45, 601–620 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Method Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  15. Belytschko, T., Moës, N., Usui, S., et al.: Arbitrary discontinuities in finite elements. Int. J. Numer. Method Eng. 50, 993–1013 (2001)

    Article  MATH  Google Scholar 

  16. Fries, T. P., Belytschko, T.: The extended/generalized finite element method: An overview of the method and its applications. Int. J. Numer. Method Eng. 84, 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Osher, S., Sethian, J.: Fronts propagating with curvaturedependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kang, M., Fedkiw, R. P., Liu, X. D.: A boundary condition capturing method for multiphase incompressible Flow. J. Sci. Comput. 15, 323–360 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sethian, J.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  20. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag, New York (2003)

    MATH  Google Scholar 

  21. Tornberg, A., Engquist, B.: A finite element based level-set method for multiphase flow applications. Comput. Visual. Sci. 3, 93–101 (2000)

    Article  MATH  Google Scholar 

  22. Sauerland, H., Fries, T. P.: The extended finite element method for two-phase and free-surface flows: A systematic study. J. Comput. Phys. 230, 3369–3390 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Reusken, A.: Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Visual. Sci. 11, 293–305 (2008)

    Article  MathSciNet  Google Scholar 

  24. Shakib, F., Hughes, T. J. R., Johan, Z.: A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations. Comput. Method Appl. Mech. Eng. 89, 141–219 (1991)

    Article  MathSciNet  Google Scholar 

  25. Mittal, S.: On the performance of high aspect ratio elements for incompressible flows. Comput. Method Appl. Mech. Eng. 188, 269–287 (2000)

    Article  MATH  Google Scholar 

  26. Oñte, E., Taylor, R. L., Zienkiewicz, O. C., et al.: A residual correction method based on finite calculus. Engineering Computations 20, 629–658 (2003)

    Article  Google Scholar 

  27. Quartapelle, L.: Numerical Solution of the Incompressible Navier-Stokes Equations. Birkhauser Verlag, Basel (1993)

    Book  MATH  Google Scholar 

  28. Smolianski, A.: Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces. Int. J. Numer. Method Fluid 48, 231–269 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gresho, P., Sani, R.: Incompressible Flow and the Finite-Element Method, Vol. 1: Advection-Diffusion. John Wiley & Sons, Ltd, Chichester, UK (1998)

    Google Scholar 

  30. Brackbill, J., Kothe, D., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Clift, R., Grace, J., Weber, M.: Bubbles, Drops, and Particles. Academic press, New York (1978)

    Google Scholar 

  32. Bhaga, D., Weber, M.: Bubbles in viscous liquids: shapes, wakes and velocities. Journal of Fluid Mechanics 105, 61–85 (1981)

    Article  Google Scholar 

  33. Unverdi, S.O., Tryggvason, G.: A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 25–37 (1992)

    Article  MATH  Google Scholar 

  34. Popinet, S., Zaleski, S.: A front-tracking algorithm for accurate representation of surface tension. Int. J. Numer. Method Fluid 30, 775–793 (1999)

    Article  MATH  Google Scholar 

  35. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, England (1961)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aerospace, Tsinghua University, 100084, Beijing, China

    Jian-Hui Liao & Zhuo Zhuang

Authors
  1. Jian-Hui Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhuo Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhuo Zhuang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liao, JH., Zhuang, Z. A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows. Acta Mech Sin 28, 1309–1322 (2012). https://doi.org/10.1007/s10409-012-0103-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-012-0103-x

Keywords

Search

Navigation