The Discontinuous Galerkin Method as an Enabling Technology for DNS and LES of Industrial Aeronautical Applications

  • Koen HillewaertEmail author
  • Corentin Carton de Wiart
Conference paper
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 131)


To enhance prediction capacities and therefore allow more advanced aeronautic and aero-propulsive design, new CFD tools are required. State of the art codes are based on second order accurate finite volume methods and are primarily developed for statistical turbulence modeling approaches. Given the limitations of these models, more direct approaches such as DNS or LES are required for the prediction of off-design aerodynamic performance, noise generation, transitional flows ...

A stumbling block for the industrial use of DNS and LES is the lack of an adequate discretisation strategy. The accurate representation of turbulent flow structures imposes huge resolution and high accuracy requirements, practically unobtainable by industrial codes. Highly accurate academic tools used for the fundamental study of turbulence are on the other hand not able to tackle industrial geometry.

The discontinuous Galerkin method is a relatively new discretisation technique, based on a cell-wise independent polynomial interpolation. This provides high accuracy irrespective of grid irregularity, as well as the possibility to locally adapt both mesh and interpolation order, which could be used to significantly improve solver robustness for automated design. DGM finally provides high computational serial and parallel efficiency, direly needed to reduce restitution time to acceptable levels. These characteristics are important assets for industrial DNS and LES.

The current contribution illustrates the efforts taken at Cenaero towards proving this potential. Following numerical aspects, the assessment of DGM for wall-resolved and wall-modeled implicit LES modeling is discussed, thereby illustrating the obtention of high accuracy, on par with academic codes. Finally some practical case studies are included.


Computational Fluid Dynamics Discontinuous Galerkin Method Quadrature Point Spectral Element Method Homogeneous Isotropic Turbulence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Altmann, C., Beck, A.D., Hindenlang, F., Staudenmaier, M., Gassner, G.J., Munz, C.-D.: An efficient high performance parallelization of a discontinuous galerkin spectral element method. In: Keller, R., Kramer, D., Weiss, J.-P. (eds.) Facing the Multicore-Challenge III. LNCS, vol. 7686, pp. 37–47. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  2. 2.
    Andersson, N., Eriksson, L.E., Davidson, L.: Large-Eddy Simulation of a Mach 0.75 jet. AIAA paper 2003–3312/. In: 9th AIAA/CEAS Aeroacoustics Conference (2003)Google Scholar
  3. 3.
    Arnold, D., Brezzi, F., Cockburn, B., Marini, L.: Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM Journal of Numerical Analysis 39, 1749–1779 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bricteux, L., Duponcheel, M., Winckelmans, G.: A multiscale subgrid model for both free vortex flows and wall-bounded flows. Physics of Fluids 21, 105,102 (2009). doi: 10.1063/1.3241991 Google Scholar
  5. 5.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer-Verlag (1988)Google Scholar
  6. 6.
    Cockburn, B.: Discontinuous Galerkin Methods for Convection-Dominated Problems. In: Barth, T.J., Deconinck, H. (eds.) High-Order Methods for Computational Physics. LNCSE, vol. 9, pp. 69–224. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  7. 7.
    Deville, M., Fischer, P., Mund, E.: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press (2002)Google Scholar
  8. 8.
    Gassner, G., Kopriva, D.A.: A comparison of the dispersion and dissipation errors of gauss and gauss-lobatto discontinuous galerkin spectral element methods. SIAM J. Scientific Computing 33(5), 2560–2579 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), 1309–1331 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods; Algorithms, Analysis and Applications. Text in Applied Mathematics. Springer (2008)Google Scholar
  11. 11.
    Hillewaert, K.: Development of the Discontinuous Galerkin Method for high-resolution, large scale CFD and acoustics in industrial geometries. Ph.D. thesis, Ecole polytechnique de Louvain/iMMC (2013)Google Scholar
  12. 12.
    Hoyas, S., Jimenez, J.: Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Physics of Fluids 20, 101, 511 (2008). doi: 10.1063/1.3005862 Google Scholar
  13. 13.
    Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference, AIAA-2007-4079 (2007)Google Scholar
  14. 14.
    Jiang, G., Shu, C.W.: On a cell entropy inequality for discontinuous Galerkin methods. Mathematics of Computation 62(206), 531–538 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jordan, P., Gervais, Y., Valiére, J.C., Foulon, H.: Final results from single point measurements. Project deliverable D3.4, JEAN - EU 5th Framework Programme, G4RD-CT2000-00313 (2002)Google Scholar
  16. 16.
    Karniadakis, G., Sherwin, S.: Spectral/hp element methods for computational fluid dynamics. Oxford University Press (2005)Google Scholar
  17. 17.
    Lodato, G., Castonguay, P., Jameson, A.: Discrete filter for large-eddy simulation using high-order spectral difference methods. International Journal for Numerical Methods in Fluids 72(2), 231–258 (2012). doi: 10.1002/fld.3740 Google Scholar
  18. 18.
    Lodato, G., Castonguay, P., Jameson, A.: Structural wall-modelled LES using a high-order spectral difference scheme for unstructured mesh. Flow, turbulence and combustion 92(593–606) (2014)Google Scholar
  19. 19.
    Moser, R., Kim, J., Mansour, N.: Direct Numerical Simulation of turbulent channel flow up to \(Re_\tau =590\). Physics of Fluids 11(4), 943–945 (1998)CrossRefGoogle Scholar
  20. 20.
    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Shahbazi, K.: An explicit expression for the penalty parameter of the interior penalty method. Journal of Computational Physics 205, 401–407 (2005)CrossRefGoogle Scholar
  22. 22.
    Sun, Y., Wang, Z., Lia, Y.: High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Communication in Computational Physics 2(2), 310–333Google Scholar
  23. 23.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer-Verlag (1999)Google Scholar
  24. 24.
    Tucker, P.: Computation of unsteady turbomachinery flows: Part 1 - progress and challenges. Progress in Aerospace Sciences 47, 522–545 (2011)CrossRefGoogle Scholar
  25. 25.
    Tucker, P.: Computation of unsteady turbomachinery flows: Part 2 - LES and hybrids. Progress in Aerospace Sciences 47, 546–569 (2011)CrossRefGoogle Scholar
  26. 26.
    Vasilyev, O.: High order finite difference schemes on non-uniform meshes with good conservation properties. Journal of Computational Physics 157(2), 746–761 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H., Kroll, N., May, G., Persson, P.O., van Leer, B., Visbal, M.: High-order CFD methods: Current status and perspectives. International Journal for Numerical Methods in Fluids 72(8), 811–845 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    de Wiart, C.C.: Towards a discontinuous Galerkin solver for scale-resolving simulations of moderate reynolds number flows, and application to industrial cases. Ph.D. thesis, Ecole polytechnique de Louvain/iMMC (2014)Google Scholar
  29. 29.
    de Wiart, C.C., Hillewaert, K., Bricteux, L., Winckelmans, W.: Implicit LES of free and wall bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method. Accepted in Int. J. Numer. Meth. Fluids (early view) (2015). doi: 10.1002/fld.4021 Google Scholar
  30. 30.
    de Wiart, C.C., Hillewaert, K., Duponcheel, M., Winckelmans, G.: Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. Int. J. Numer. Meth. Fluids 74, 469–493 (2014). doi: 10.1002/fld.3859 Google Scholar
  31. 31.
    Winckelmans, G., Jeanmart, H., Carati, D.: On the comparison of turbulence intensities from large-eddy simulation with those from experiment or direct numerical simulation. Physics of Fluids 14(5), 1809 (2002). doi: 10.1063/1.1466824 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Centre de recherche en Aéronautique (Cenaero)GosseliesBelgium

Personalised recommendations