Flow, Turbulence and Combustion

, Volume 60, Issue 1, pp 47–85 | Cite as

Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients

  • M. Skote
  • D.S. Henningson
  • R.A.W.M. Henkes


Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.

turbulent boundary layers DNS self-similarity 


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  1. 1.
    Bradshaw, P., The turbulent structure of equilibrium boundary layers. J. Fluid Mech. 29 (1967) 625–645.Google Scholar
  2. 2.
    Clauser, F.H., Turbulent boundary layers in adverse pressure gradients. J. Aero. Sci. 21 (1954) 91–108.Google Scholar
  3. 3.
    Coleman, G.N. and Spalart, P.R., Direct numerical simulation of a small separation bubble. In: So, R.M.C., Speziale, C.G., Launder, B.E. (eds), Near-Wall Turbulent Flows. Elsevier, Amsterdam (1993) pp. 277–286.Google Scholar
  4. 4.
    Durbin, P.A. and Belcher, S.E., Scaling of adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 238 (1992) 699–722.Google Scholar
  5. 5.
    Fernholz, H.H. and Finley, P.J., The incompressible zero-pressure-gradient turbulent boundary layer: An assessment of the data. Prog. Aerospace Sci. 32 (1996) 245–311.CrossRefGoogle Scholar
  6. 6.
    George, W.K. and Castillo, L., Boundary layers with pressure gradient: Another look at the equilibrium boundary layer. In: So, R.M.C., Speziale, C.G. and Launder, B.E. (eds), Near-Wall Turbulent Flows. Elsevier, Amsterdam (1993) pp. 901–910.Google Scholar
  7. 7.
    Hanjalić, K., Jakirlić, S. and Hadžć, I., Computation of oscillating turbulent flows at transitional re-numbers. In: Durst, F. et al. (eds), Turbulent Shear Flows, Vol. 9. Springer-Verlag, Berlin (1995) pp. 323–342.Google Scholar
  8. 8.
    Hanjalić, K. and Launder, B.E., Sensitizing the dissipation equation to irrotational strains. J. Fluids Engrg. 102 (1980) 34–40.Google Scholar
  9. 9.
    Head, M.R., Equilibrium and near-equilibrium turbulent boundary layers. J. Fluid Mech. 73 (1976) 1–8.Google Scholar
  10. 10.
    Henkes, R.A.W.M., Comparison of turbulence models for attached boundary layers relevant to aeronautics. Appl. Sci. Res. 57 (1997) 43–65.Google Scholar
  11. 11.
    Henkes, R.A.W.M., Scaling of equilibrium boundary layers under adverse pressure gradient using turbulence models. AIAA J. 36 (1998) 320–326.Google Scholar
  12. 12.
    Henkes, R.A.W.M., Skote, M. and Henningson, D.S., Application of turbulence models to equilibrium boundary layers under adverse pressure gradient. In: Eleventh Symposium on Turbulent Shear Flows, Grenoble, France (1997) 33:13–18.Google Scholar
  13. 13.
    Lundbladh, A. and Henningson, D.S., Evaluation of turbulence models from direct numerical simulations of turbulent boundary layers. FFA-TN 1995-09, Aeronautical Research Institute of Sweden, Bromma (1995).Google Scholar
  14. 14.
    Lundbladh, A., Henningson, D.S. and Johansson, A.V., An efficient spectral integration method for the solution of the Navier—Stokes equations. FFA-TN 1992-28, Aeronautical Research Institute of Sweden, Bromma (1992).Google Scholar
  15. 15.
    Lundbladh, A., Schmid, P.J., Berlin, S. and Henningson, D.S., Simulation of bypass transition in spatially evolving flows. In: Proceedings of the AGARD Symposium on Application of Direct and Large Eddy Simulation to Transition and Turbulence, AGARD-CP-551 (1994) 18:1–13.Google Scholar
  16. 16.
    Mellor, G.L. and Gibson, D.M., Equilibrium turbulent boundary layers. J. Fluid Mech. 24 (1966) 225–253.Google Scholar
  17. 17.
    Na, Y. and Moin, P., Direct numerical simulation of studies of turbulent boundary layers with adverse pressure gradient and separation. Technical Report TF-68, Thermosciences Division, Department of Mechanical Engineering, Stanford University (1996).Google Scholar
  18. 18.
    Nagano, Y., Tagawa, M. and Tsuji, T., Effects of adverse pressure gradients on mean flows and turbulence statistics in a boundary layer. In: Durst, F. et al. (eds), Turbulent Shear Flows, Vol. 8. Springer-Verlag, Berlin (1992) pp. 7–21.Google Scholar
  19. 19.
    Nordström, J., Nordin, N. and Henningson, D.S., The fringe region technique used in the direct numerical simulation of the incompressible Navier—Stokes equations. SIAM J. Sci. Comp. (1998) (to appear).Google Scholar
  20. 20.
    Rotta, J.C., Turbulent boundary layers in incompressible flow. Prog. Aerospace Sci. 2 (1962) 3–219.Google Scholar
  21. 21.
    Samuel, A.E. and Joubert, P.N., A boundary layer developing in an increasingly adverse pressure gradient. J. Fluid Mech. 66 (1974) 481–505.Google Scholar
  22. 22.
    Schofield, W.H., Equilibrium boundary layers in moderate to strong adverse pressure gradients. J. Fluid Mech. 113 (1981) 91–122. APPENDGoogle Scholar
  23. 23.
    Skåre, P.E. and Krogstad, P.-Å., A turbulent equilibrium boundary layer near separation. J. Fluid Mech. 272 (1994) 319–348.Google Scholar
  24. 24.
    Spalart, P.R., Direct simulation of a turbulent boundary layer up to Reθ = 1410. J. Fluid Mech. 187 (1988) 61–98.Google Scholar
  25. 25.
    Spalart, P.R. and Coleman, G.N., Numerical study of a separation bubble with heat transfer. European J. Mechanics B/Fluids 16 (1997) 169.Google Scholar
  26. 26.
    Spalart, P.R. and Leonard, A., Direct numerical simulation of equilibrium turbulent boundary layers. In: Durst, F. et al. (eds), Turbulent Shear Flows, Vol. 5. Springer-Verlag, Berlin (1987) pp. 234–252.Google Scholar
  27. 27.
    Spalart, P.R. and Watmuff, J.H., Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249 (1993) 337–371.Google Scholar
  28. 28.
    Stratford, B.S., An experimental flow with zero skin friction throughout its region of pressure rise. J. Fluid Mech. 5 (1959) 17–35.Google Scholar
  29. 29.
    Stratford, B.S., The prediction of separation of the turbulent boundary layer. J. Fluid Mech. 5 (1959) 1–16.Google Scholar
  30. 30.
    Tennekes, H. and Lumley, J.L., A First Course in Turbulence. The MIT Press, Cambridge, MA (1972).Google Scholar
  31. 31.
    Townsend, A.A., The development of turbulent boundary layers with negligible wall stress. J. Fluid Mech. 8 (1960) 143–155.Google Scholar
  32. 32.
    Townsend, A.A., The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge (1976). Google Scholar

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© Kluwer Academic Publishers 1998

Authors and Affiliations

  • M. Skote
  • D.S. Henningson
  • R.A.W.M. Henkes

There are no affiliations available

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