The Multicore Challenge: Petascale DNS of a Spatially-Developing Supersonic Turbulent Boundary Layer Up to High Reynolds Numbers Using DGSEM

  • Muhammed AtakEmail author
  • Johan Larsson
  • Claus-Dieter Munz


With increasing computational power on modern supercomputing systems, direct numerical simulations (DNS) have been gaining in importance for the investigation of wall-bounded turbulent flows. The applied numerical method, however, has to enable an efficient usage of high performance computing systems to satisfy the involved computational costs. In this context, discontinuous Galerkin (DG) methods have become a promising candidate to conduct DNS in an efficient way as they offer an excellent scaling combined with arbitrary high spatial accuracy in complex geometries. On the other hand, the DG method has been also suffering from being considered as inefficient and slow within the computational turbulence community, which doubted the suitability of the DG method when applied to turbulent flows. In this work, we performed a DNS of a compressible spatially-developing supersonic flat plate turbulent boundary layer up to Re θ  = 3878 using the discontinuous Galerkin spectral element method (DGSEM). To our knowledge, the present simulation is currently the biggest computation within the DG community and enabled us to generate a reliable high-fidelity database for further complex studies. The usage of the DGSEM approach allowed an efficient exploitation of the whole computational power available on the HLRS Cray XC40 supercomputer and to run the simulation with a near-perfect scaling up to 93,840 processors. The obtained results demonstrate the strong potential of the DGSEM at conducting sustainable and efficient DNS of high Reynolds number wall-bounded turbulent flows.


Direct Numerical Simulation Turbulent Boundary Layer Discontinuous Galerkin Discontinuous Galerkin Method Skin Friction Coefficient 
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.



The research presented in this paper was supported in parts by the Deutsche Forschungsgemeinschaft (DFG) and we appreciate the ongoing kind support provided by HLRS and Cray in Stuttgart, Germany.


  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babinsky, H., Harvey, J.K.: Shock Wave-Boundary-Layer Interactions. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Babucke, A.: Direct Numerical Simulation of Noise-Generation Mechanisms in the Mixing Layer of a Jet. Dissertation, University of Stuttgart (2009)Google Scholar
  4. 4.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bernardini, M., Pirozzoli, S.: Wall pressure fluctuations beneath supersonic turbulent boundary layers. Phys. Fluids 23(8), 085102 (2011)CrossRefGoogle Scholar
  6. 6.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Coles, D.: Measurements of Turbulent Friction on a Smooth Flat Plate in Supersonic Flow. J. Aerosol Sci. 21(7), 433–448 (1954)CrossRefGoogle Scholar
  8. 8.
    Colonius, T., Lele, S.K., Moin, P.: Boundary conditions for direct computations of aerodynamic sound generation. AIAA J. 31, 1574–1582 (1991)CrossRefGoogle Scholar
  9. 9.
    Del Alamo, J.C., Jimenez, J.: Estimation of turbulent convection velocities and corrections to Taylor’s approximation. J. Fluid Mech. 640, 5–26 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eckert, E.R.G.: Engineering relations for friction and heat transfer to surfaces in high velocity flow. J. Aerosol Sci. 8, 585–587 (1955)Google Scholar
  11. 11.
    Gassner, G., Lörcher, F., Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion. II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34, 260–286 (2008)zbMATHGoogle Scholar
  12. 12.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)CrossRefGoogle Scholar
  13. 13.
    Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kopriva, D.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer, New York (2009)CrossRefGoogle Scholar
  15. 15.
    Mack, L.M.: Boundary layer linear stability theory. Special course on stability and transition of laminar flow, AGARD Report 709 (1984)Google Scholar
  16. 16.
    Mayer, C.S.J., von Terzi, D.A., Fasel, H.F.: Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3. J. Fluid Mech. 674, 5–42 (2001)CrossRefGoogle Scholar
  17. 17.
    Piponniau, S., Dussauge, J.P., Debieve, F.F., Dupont, P.: A simple model for low-frequency unsteadiness in shock-induced separation. J. Fluid Mech. 629, 87–108 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Pirozzoli, S., Grasso, F., Gatsky, T.B.: Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M = 2.25. Phys. Fluids 16, 530–545 (2004)CrossRefGoogle Scholar
  19. 19.
    Schlatter, P., Örlü, R.: Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126 (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    Smits, A.J., Dussauge, J.P.: Turbulent Shear Layers in Supersonic Flow. Springer, New York (2006)Google Scholar
  21. 21.
    Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (1991)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Muhammed Atak
    • 1
    Email author
  • Johan Larsson
    • 2
  • Claus-Dieter Munz
    • 1
  1. 1.Institute of Aerodynamics and Gas DynamicsUniversity of StuttgartStuttgartGermany
  2. 2.Department of Mechanical EngineeringUniversity of MarylandCollege ParkUSA

Personalised recommendations