Two-Dimensional Nonlinear Saturation Behaviour of Instability Waves in a Boundary Layer at Mach 5

  • N. A. Adams
  • L. Kleiser


The two-dimensional nonlinear evolution of a second-mode instability wave in a flat plate boundary layer at a free-stream Mach number of M = 5 is investigated by direct numerical simulation. An explicit spectral/finite-difference scheme employing the temporal model is used. A nonlinear saturation process is found, during which initially a weak viscous shock develops near the wall. A comparison of the numerical result obtained in the shock region with a one-dimensional analytic weak-shock solution is made and a good agreement of the shock-normal velocity distribution and shock thickness is found. The nonlinear saturation is characterized by a cascade of states with alternating high and low energy levels of the higher Fourier-modes. The system evolves on a slow time scale towards a steady state which appears to be different from the undisturbed laminar state.


Direct Numerical Simulation Spanwise Direction Fourier Mode Secondary Instability Nonlinear Saturation 
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  1. 1.
    Adams, N.A., 1993 Numerische Simulation von Transitionsmechanismen in kompressiblen Grenzschichten, DLR-FB 93–29 (also doctoral thesis, Technische Universität München) (in German).Google Scholar
  2. 2.
    Adams, N.A., Kleiser, L., 1993 Numerical simulation of transition in a compressible flat plate boundary layer, ASME FED-Vol. 151, Transitional and Turbulent Compressible Flows, Kral, L.D., Zang, T.A. (eds.), pp. 101–110.Google Scholar
  3. 3.
    Adams, N.A., Kleiser, L., 1993 Numerical simulation of fundamental breakdown of a laminar boundary-layer at Mach 4.5, AIAA-paper 93–5027.Google Scholar
  4. 4.
    Adams, N.A., Sandham, N.D., and Kleiser, L., 1992 A method for direct numerical simulation of compressible boundary-layer transition, NNFM 35, Vos, J.B., Rizzi, A., Ryhming, I. (editors), Vieweg-Verlag, Braunschweig, pp. 523–532.Google Scholar
  5. 5.
    Anderson, D.A., Tannehill, J.C., Pletcher, R.H., 1984 Computational Fluid Mechanics and Heat Transfer, Hemisphere Publ. Corp., New York.MATHGoogle Scholar
  6. 6.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., 1988 Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin.MATHGoogle Scholar
  7. 7.
    Eißler, W., Bestek, H., 1993 Spatial numerical simulations of nonlinear transition phenomena in supersonic boundary layers, ASME FED-Vol. 151, Transitional and Turbulent Compressible Flows, Kral, L.D., Zang, T.A. (eds.), pp. 69–76.Google Scholar
  8. 8.
    Erlebacher,G., and Hussaini, M.Y., 1990 Numerical experiments in supersonic boundary-layer stability, Phys. Fluids A 2, pp. 94–104.MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Fasel, H., Thumm, A., Bestek, H., 1993 Direct numerical simulation of transition in supersonic boundary layers: oblique breakdown, ASME FED-Vol. 151, Transitional and Turbulent Compressible Flows, Kral, L.D., Zang, T.A. (eds.), pp. 77–92.Google Scholar
  10. 10.
    Henningson, D., 1992 Private communication.Google Scholar
  11. 11.
    Herbert, T., 1988 Secondary instability of boundary layers, Ann. Rev. Fluid Mech. 20, pp. 487–526.ADSCrossRefGoogle Scholar
  12. 12.
    Kleiser, L., and Zang, T.A., 1991 Numerical simulation of transition in wallbounded shear flows, Ann. Rev. Fluid Mech. 23, pp. 495–537.ADSCrossRefGoogle Scholar
  13. 13.
    Koch, W., 1992 On a degeneracy of temporal secondary instability modes in Blasius boundary-layer flow, J. Fluid Mech. 243, pp. 319–351.MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Lele, S.K., 1989 Direct numerical simulation of compressible free shear flows, AIAA-paper 89–0374.Google Scholar
  15. 15.
    Lele, S.K., 1992 Compact finite difference schemes with spectral-like resolution, J. Comp. Phys. 103, pp. 16–42.MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Lee, S., Lele, S.K., Moin, P., 1991 Eddy shocklets in decaying compressible turbulence, Phys. Fluids A 3 (4), pp. 657 - 664.ADSCrossRefGoogle Scholar
  17. 17.
    Mack, L.M., 1984 Boundary-layer linear stability theory, in Special Course on Stability and Transition of Laminar Flow, AGARD Report No. 709, pp. 3–1–3–81.Google Scholar
  18. 18.
    Ng, L., and Erlebacher, G., 1992 Secondary instabilities in compressible boundary layers, Phys. Fluids A 4, pp. 710–726.ADSMATHCrossRefGoogle Scholar
  19. 19.
    Pruett, C.D., Chang, C.L, 1993 A comparison of PSE and DNS for high-speed boundary-layer flows, ASME FED-Vol. 151, Transitional and Turbulent Compressible Flows, Kral, L.D., Zang, T.A. (eds.), pp. 57–67.Google Scholar
  20. 20.
    Pruett, C.D., and Zang, T.A., 1992 Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers, Theor. Comp. Fluid Dyn. 3, pp. 345–367.MATHCrossRefGoogle Scholar
  21. 21.
    Sandham, N.D., and Adams, N.A., 1993 Numerical simulation of boundarylayer transition at Mach two, Applied Scientific Research 51, pp. 371–375.CrossRefGoogle Scholar
  22. 22.
    Sandham, N.D., Reynolds, W.C., 1991 Three-dimensional simulations of large eddies in the compressible mixing layer, J. Fluid Mech. 224, pp. 133–158.ADSMATHCrossRefGoogle Scholar
  23. 23.
    Stewartson, K., 1964 The Theory of Laminar Boundary Layers in Compressible Fluids, Oxford University Press, Amen House, London, pp. 33–60.MATHGoogle Scholar
  24. 24.
    Thompson, K.W., 1987 Time dependent boundary conditions for hyperbolic systems, J. Comp. Phys. 68, pp. 1–24.ADSMATHCrossRefGoogle Scholar
  25. 25.
    Thumm, A., Wolz, W., and Fasel, H., 1990 Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers, in Laminar-Turbulent Transition, D. Arnal and R. Michel (eds.), Springer, pp. 303–308.Google Scholar
  26. 26.
    Vincenti, W.G., Kruger C.H., 1965 Introduction to Physical Gas Dynamics, John Wiley & Sons Inc., pp. 412–424.Google Scholar
  27. 27.
    Wray, A.A., 1986 Very low storage time-advancement schemes, Internal Report, NASA Ames Research Center, Moffet Field, CA.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • N. A. Adams
    • 1
  • L. Kleiser
    • 1
  1. 1.Institute for Theoretical Fluid MechanicsDLRBunsenstraße 10GöttingenGermany

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