Floquet Analysis of Secondary Instability in Shear Flows

  • Thorwald Herbert
  • Fabio P. Bertolotti
  • German R. Santos
Conference paper
Part of the ICASE NASA LaRC Series book series (ICASE/NASA)

Abstract

In previous work, the parametric excitation of secondary, three-dimensional disturbances in the Blasius boundary layer was investigated using the concept of temporal growth. The analysis was restricted to fundamental (peak-valley splitting) modes and subharmonic modes, i.e. to disturbances with the same or twice the streamwise wavelength of the primary TS wave. Here, we generalize these studies in two directions. First, we analyze the spatial growth of secondary disturbances and develop an analogue of Gaster’s transformation. Second, we study the secondary instability with respect to detuned modes, i.e. disturbances with arbitrary streamwise wavelength, and relate the results to observations of subharmonic and combination resonance in the flat-plate boundary layer. As a by-product, we show the rapid convergence of the Fourier series that governs the streamwise structure of the disturbances.

Keywords

Vortex Vorticity Expense Eter Triad 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow, Phys. Fluids 3, 656–657.ADSCrossRefGoogle Scholar
  2. Bertolotti, F. P. 1985 Temporal and spatial growth of subharmonic disturbances in Falkner-Skan flows, M. S. Thesis, Virginia Polytechnic Institute, Blacksburg, Virginia.Google Scholar
  3. Blackwelder, R. F. 1979 Boundary-layer transition, Phys. Fluids 22, 583–584.ADSCrossRefGoogle Scholar
  4. Bridges, T. J. & Morris, P.O. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly, J. Comp. Physics 55, 437–460.MathSciNetADSMATHCrossRefGoogle Scholar
  5. Craik, A.D.D. 1971 Nonlinear resonant instability in boundary layers, J. Fluid Mech. 50, 393–413.ADSMATHCrossRefGoogle Scholar
  6. Davis, S. H. 1976 The stability of time-periodic flows, Ann. Rev. Fluid Mech. 8, 57–74.ADSCrossRefGoogle Scholar
  7. Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability, J. Fluid Mech. 14, 222–224.MathSciNetADSMATHCrossRefGoogle Scholar
  8. Herbert, Th. 1977 Finite-amplitude stability of plane parallel flows, AGARD CP-224, 3/1–10.Google Scholar
  9. Herbert, Th. & Morkovin, M. V. 1980 Dialogue on bridging some gaps in stability and transition research, in: Laminar-Turbulent Transition (eds. R. Eppler & H. Fasel), 47–72, Springer-Verlag.CrossRefGoogle Scholar
  10. Herbert, Th. 1981a Stability of plane Poiseuille flow-theory and experiment, VPI-E-81–35, Blacksburg, VA. Published in: Fluid Dyn. Trans. 11, 77–126 (1983).Google Scholar
  11. Herbert, Th. 1981b A secondary instability mechanism in plane Poiseuille flow, Bull. Amer. Phys. Soc 26, 1257.Google Scholar
  12. Herbert, Th. 1983a Secondary instability of plane channel flow to subharmonic three-dimensional disturbances, Phys. Fluids. 26, 871–874.ADSCrossRefGoogle Scholar
  13. Herbert, Th. 1983b Subharmonic three-dimensional disturbances in unstable shear flows, AIAA Paper No. 83–1759.Google Scholar
  14. Herbert, Th. 1984a Analysis of the subharmonic route to transition in boundary layers, AIAA Paper No. 84–0009.Google Scholar
  15. Herbert, Th. 1984b Modes of secondary instability in plane Poiseuille flow, in: Turbulence and Chaotic Phenomena in Fluids (ed T. Tatsumi), 53–58, North-Holland.Google Scholar
  16. Herbert, Th. 1985a Three-dimensional phenomena in the transitional flat-plate boundary layer, AIAA Paper No. 85–0489.Google Scholar
  17. Herbert, Th. 1985b Secondary instability of plane shear flows-theory and applications, in: Laminar-Turbulent Transition (ed. V. V. Kozlov), 9–20, Springer-Verlag.CrossRefGoogle Scholar
  18. Kachanov, Yu. S. & Levchenko, V. Ya 1982 Resonant interactions of disturbances in transition to turbulence in a boundary layer (in Russian), Preprint No. 10–82, I.T.A.M., USSR Academy of Sciences, Novosibirsk.Google Scholar
  19. Kachanov, Yu. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminarturbulent transition in a boundary layer, J. Fluid Mech. 138, 209–247.ADSCrossRefGoogle Scholar
  20. Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time, J. Fluid Mech. 27, 657–689.ADSMATHCrossRefGoogle Scholar
  21. Kerczek, C. von 1985 Stability characteristics of some oscillatory flows-Poiseuille, Ekman and Films, in these Proceedings.Google Scholar
  22. Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability, J. Fluid Mech. 12, 1–34.ADSMATHCrossRefGoogle Scholar
  23. Kleiser, L. 1982 Numerische Simulationen zum laminar-turbulenten Umschlagsprozess der ebenen Poiseuille-Strömung, Dissertation, Universität Karlsruhe. See also: Spectral simulations of laminarturbulent transition in plane Poiseuille flow and comparison with experiments, Springer Lecture Notes in Physics 170, 280–287 (1982).Google Scholar
  24. Kozlov, V. V. & Ramasanov, M. P. 1983 Development of finite-amplitude disturbance in Poiseuille flow, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, 43–47.Google Scholar
  25. Maseev, L. M. 1968 Occurrence of three-dimensional perturbations in a boundary layer, Fluid Dyn. 3, 23–24.ADSCrossRefGoogle Scholar
  26. Nagata, M. & Busse, F. H. 1983 Three-dimensional tertiary motions in a plane shear layer, J. Fluid Mech. 135, 1–28.ADSMATHCrossRefGoogle Scholar
  27. Nayfeh, A. H. 1985 Three-dimensional spatial secondary instability in boundary-layer flows, AIAA Paper No. 85–1697.Google Scholar
  28. Nayfeh, A. H. & Bozatli, A. N. 1979 Secondary instability in boundary-layer flows, Phys. Fluids 22, 805–813.ADSMATHCrossRefGoogle Scholar
  29. Nishioka, M., Iida, S. & Ichikawa, J. 1975 An experimental investigation on the stability of plane Poiseuille flow, J. Fluid Mech. 72, 731–751.ADSCrossRefGoogle Scholar
  30. Nishioka, M. & Asai, M. 1983 Evolution of Tollmien-Schlichting waves into wall turbulence, in: Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), 87–92, North-Holland.Google Scholar
  31. Nishioka, M. & Asai, M. 1985 3-D wave disturbances in plane Poiseuille flow, in: Laminar-Turbulent Transition (ed. V.V. Kozlov), 173–182, Springer-Verlag.CrossRefGoogle Scholar
  32. Orszag, S. A. & Kells, L. C., 1980 Transition to turbulence in plane Poiseuille flow and plane Couette flow, J. Fluid Mech. 96, 159–206.ADSMATHCrossRefGoogle Scholar
  33. Orszag, S. A. & Patera, A. T. 1980 Sub critical transition to turbulence in plane channel flows, Phys. Rev. Lett. 45, 989–993.ADSCrossRefGoogle Scholar
  34. Orszag, S. A. & Patera, A. T. 1981 Subcritical transition to turbulence in plane shear flows, in: Transition and Turbulence (ed. R.E. Meyer), 127–146, Academic Press.Google Scholar
  35. Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows, J. Fluid Mech. 128, 347–385.ADSMATHCrossRefGoogle Scholar
  36. Pierrehumbert, R. T. & Widnall, S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer, J. Fluid Mech. 114, 59–82.ADSMATHCrossRefGoogle Scholar
  37. Raetz, G.S. 1959 A new theory of the cause of transition in fluid flows, NORAIR Report NOR-59–383, Hawthorne, CA.Google Scholar
  38. Raetz, G.S. 1964 Current status of resonance theory of transition, NORAIR Report NOR-64–111, Hawthorne, CA.Google Scholar
  39. Saric, W. S. & Thomas, A. S. W. 1984 Experiments on the subharmonic route to turbulence in boundary layers, in: Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), 117–122, North-Holland.Google Scholar
  40. Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech. 4, 1–21.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Thorwald Herbert
  • Fabio P. Bertolotti
  • German R. Santos

There are no affiliations available

Personalised recommendations