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An Implicit High-Order Discontinuous Galerkin Approach for Variable Density Incompressible Flows

  • Francesco Carlo MassaEmail author
  • Francesco Bassi
  • Lorenzo Botti
  • Alessandro Colombo
Conference paper
  • 78 Downloads
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 121)

Abstract

In this work we present a high-order discontinuous Galerkin approach for the simulation of variable density incompressible (VDI) flows. Here, the density is treated as a purely advected property tracking possibly multiple (more than two) components, while the fluids interface is captured in a diffuse fashion by the high-degree polynomial solution thus not requiring any geometrical reconstruction. Specific care is devoted to deal with density over/undershoots, spurious oscillations at flows interfaces and Godunov numerical fluxes at inter-element boundaries. Time integration is performed with high-order implicit schemes thus preventing any time step restriction condition. Promising results with high-degree polynomial representation and relatively coarse meshes are achieved on numerical experiments involving high-density ratios (water–air) and the possible interaction of more than two components.

Notes

Acknowledgements

F. Massa is supported by the Supporting Talented Researchers (STaRS) programm of the University of Bergamo.

References

  1. 1.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008)Google Scholar
  2. 2.
    Bassi, F., Botti, L., Colombo, A., Ghidoni, A., Massa, F.: Linearly implicit Rosenbrock-type Runge-Kutta schemes for the Discontinuous Galerkin solution of compressible and incompressible unsteady flows. Comput. Fluids (2015).  https://doi.org/10.1016/j.compfluid.2015.06.007
  3. 3.
    Elsworth, D.T., Toro, E.F.: Riemann solvers for solving the incompressible Navier-Stokes equations using the artificial compressibility method. College of Aeronautics, Cranfield Institute of Technology, 9208 (1992)Google Scholar
  4. 4.
    Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations. J. Comput. Phys. (2006).  https://doi.org/10.1016/j.jcp.2006.03.006MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bassi, F., Massa, F., Botti, L., Colombo, A.: Artificial compressibility Godunov fluxes for variable density incompressible flows. Comput. Fluids (2018).  https://doi.org/10.1016/j.compfluid.2017.09.010MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Proceedings of 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, pp. 99–108 (1997)Google Scholar
  7. 7.
    Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Meth. Part. D. E. 16, 365–378 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin Methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Nevada (2006)Google Scholar
  10. 10.
    Klöckner, A., Warburton, T., Hesthaven, J.S.: Viscous shock capturing in a time-explicit discontinuous Galerkin Method. Math. Model. Nat. Phenom. (2011).  https://doi.org/10.1051/mmnp/20116303MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jaffre, J., Johnson, C., Szepessy, A.: Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5(3), 286–367 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (2010)Google Scholar
  13. 13.
    Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. V 1–24 (2005)Google Scholar
  14. 14.
    Saad, Y., Shults, M.H.: A generalised minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)CrossRefGoogle Scholar
  15. 15.
    Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  16. 16.
    Ritter, A.: Die Fortpflanzung der Wasserwellen. Z. Ver. Deut. Ing 36, 947–954 (1892)Google Scholar
  17. 17.
    Dressler, F.: Comparison of theories and experiments for the hydraulic dam-break wave. Proc. Int. Assoc. of Sci. Hydrol. Assemblée Générale 3(38), 319–328 (1954)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Francesco Carlo Massa
    • 1
    Email author
  • Francesco Bassi
    • 1
  • Lorenzo Botti
    • 1
  • Alessandro Colombo
    • 1
  1. 1.University of BergamoBergamoItaly

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