Journal of Scientific Computing

, Volume 75, Issue 2, pp 830–858 | Cite as

Divergence-Free H(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics

Article
First Online:
Received:
Revised:
Accepted:
  • 119 Downloads

Abstract

In this article, we consider exactly divergence-free H(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free \(H^1\)-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier–Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin–Helmholtz instabilities and freely decaying two-dimensional turbulence.

Keywords

Incompressible flow Divergence-free H(div)-FEM Pressure/Reynolds-semi-robust error estimates Vortex dynamics 

Mathematics Subject Classification

35Q30 65M15 65M60 76D17 76M10 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

We gratefully acknowledge the comments and suggestions about this paper from the anonymous reviewers.

References

  1. 1.
    Ahmed, N.: On the grad-div stabilization for the steady Oseen and Navier-Stokes equations. Calcolo 54(1), 471–501 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ahmed, N., Chacón Rebollo, T., John, V., Rubino, S.: Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method. IMA J. Numer. Anal. 37(3), 1437–1467 (2017)MathSciNetGoogle Scholar
  3. 3.
    Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations (2017). doi: 10.20347/WIAS.PREPRINT.2368
  4. 4.
    Bazilevs, Y., Michler, C., Calo, V.M., Hughes, T.J.R.: Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Engrg. 196(49–52), 4853–4862 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertozzi, A.L.: Heteroclinic orbits and chaotic dynamics in planar fluid flows. SIAM J. Math. Anal. 19(6), 1271–1294 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boffetta, G., Ecke, R.E.: Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer-Verlag, Berlin (2013)CrossRefMATHGoogle Scholar
  8. 8.
    Botti, L., Di Pietro, D.A.: A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure. J. Comput. Phys. 230(3), 572–585 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer Science & Business Media, New York (2013)CrossRefMATHGoogle Scholar
  10. 10.
    Bracco, A., McWilliams, J.C., Murante, G., Provenzale, A., Weiss, J.B.: Revisiting freely decaying two-dimensional turbulence at millennial resolution. Phys. Fluids 12(11), 2931–2941 (2000)CrossRefMATHGoogle Scholar
  11. 11.
    Burman, E.: Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation. Comput. Methods Appl. Mech. Engrg. 196(41–44), 4045–4058 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39–77 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44(3), 1248–1274 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier-Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp. 74(251), 1067–1095 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31(1), 61–73 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dallmann, H., Arndt, D., Lube, G.: Local projection stabilization for the Oseen problem. IMA J. Numer. Anal. 36(2), 796–823 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Davidson, P.A.: Turbulence: An Introduction for Scientist and Engineers. Oxford University Press, (2004)Google Scholar
  19. 19.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer-Verlag, Berlin (2012)CrossRefMATHGoogle Scholar
  20. 20.
    Durst, F.: Fluid Mechanics: An Introduction to the Theory of Fluid Flows. Springer-Verlag, Berlin (2008)CrossRefMATHGoogle Scholar
  21. 21.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Springer, New York (2004)CrossRefMATHGoogle Scholar
  22. 22.
    Evans, J.A.: Divergence-free B-spline Discretizations for Viscous Incompressible Flows. Ph.D. thesis, The University of Texas at Austin (2011)Google Scholar
  23. 23.
    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations. J. Comput. Phys. 241, 141–167 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements. J. Sci. Comput. 66(3), 991–1024 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gravemeier, V., Wall, W.A., Ramm, E.: Large eddy simulation of turbulent incompressible flows by a three-level finite element method. Int. J. Numer. Meth. Fluids 48(10), 1067–1099 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    van Groesen, E.: Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations. Physica A 148(1–2), 312–330 (1988)CrossRefMATHGoogle Scholar
  27. 27.
    Guzmán, J., Shu, C.W., Sequeira, F.A.: H(div) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal. (2016). doi: 10.1093/imanum/drw054
  28. 28.
    Hillewaert, K.: Development of the Discontinuous Galerkin Method for High-Resolution, Large Scale CFD and Acoustics in Industrial Geometries. Ph.D. thesis, Université catholique de Louvain (2013)Google Scholar
  29. 29.
    Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    John, V.: An assessment of two models for the subgrid scale tensor in the rational LES model. J. Comput. Appl. Math. 173(1), 57–80 (2005)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    John, V.: Finite Element Methods for Incompressible Flow Problems. Springer International Publishing, (2016)Google Scholar
  32. 32.
    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lesieur, M.: Turbulence in Fluids, 4th edn. Springer, Netherlands (2008)CrossRefMATHGoogle Scholar
  34. 34.
    Lesieur, M., Staquet, C., Le Roy, P., Comte, P.: The mixing layer and its coherence examined from the point of view of two-dimensional turbulence. J. Fluid Mech. 192, 511–534 (1988)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 311, 304–326 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, (2002)Google Scholar
  37. 37.
    Melander, M.V., Zabusky, N.J., McWilliams, J.C.: Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340 (1988)Google Scholar
  38. 38.
    Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008)Google Scholar
  39. 39.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, 2nd edn. Springer-Verlag, Berlin (2008)MATHGoogle Scholar
  40. 40.
    San, O., Staples, A.E.: High-order methods for decaying two-dimensional homogeneous isotropic turbulence. Comput. & Fluids 63, 105–127 (2012)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer-Verlag, Berlin (2000)CrossRefMATHGoogle Scholar
  42. 42.
    Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Numer. Math. (2017). doi: 10.1515/jnma-2016-1101
  43. 43.
    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. ESAIM: M2AN 19(1), 111–143 (1985)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Tabeling, P.: Two-dimensional turbulence: a physicist approach. Phys. Rep. 362(1), 1–62 (2002)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Thuburn, J., Kent, J., Wood, N.: Cascades, backscatter and conservation in numerical models of two-dimensional turbulence. Quart. J. Roy. Meteor. Soc. 140(679), 626–638 (2014)CrossRefGoogle Scholar
  46. 46.
    Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Oxford University Press, New York (1988)MATHGoogle Scholar
  47. 47.
    Vemaganti, K.: Discontinuous Galerkin methods for periodic boundary value problems. Numer. Methods Partial Differential Equations 23(3), 587–596 (2007)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Wang, J., Wang, Y., Ye, X.: A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods. SIAM J. Sci. Comput. 31(4), 2784–2802 (2009)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Wang, J., Ye, X.: New finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal. 45(3), 1269–1286 (2007)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Zistl, C., Hilbert, R., Janiga, G., Thévenin, D.: Increasing the efficiency of postprocessing for turbulent reacting flows. Comput. Vis. Sci 12(8), 383–395 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsGeorg-August-University GöttingenGöttingenGermany

Personalised recommendations