Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-29T11:12:53.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Tollmien-Schlichting-wave resonant mechanism for subharmonic-type transition

Published online by Cambridge University Press:  26 April 2006

M. B. Zelman  and
I. I. Maslennikova
M. B. Zelman
Affiliation:
Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia
I. I. Maslennikova
Affiliation:
Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

Disturbance interactions in wave triads and multiwave systems of various configurations are investigated to reveal the mechanism of laminar-turbulent transition in Blasius and pressure-gradient boundary layers. The averaging method of weakly nonlinear instability theory in quasi-parallel flows is applied. Tollmien-Schlichting-wave resonant interaction is shown to be the only leading mechanism of subharmonic (S)-type transition. The mechanism universally dominates in boundary layers excited by sufficiently small initial disturbances. The role of any other mode is inefficient. Weakly nonlinear models are concluded not to explain the K-type transition scenario. The results of the study are employed to interpret physical and numerical experimental data.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 449 - 478
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bayly, B. J., Orszag, S. A. & Herbert, Th. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359.Google Scholar
Benney, D. J. 1964 Finite amplitude effects in an unstable laminar boundary layer. Phys. Fluids 7, 319.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3, 656.Google Scholar
Bouthier, M. 1973 Stabilité linéaire des écoulements presque parallels. La couche limite de Blasius. J. Méc. 72, 75.Google Scholar
Corke, T. C. & Mangano, R. A. 1987 Transition of a boundary layer: controlled fundamental- subharmonic interactions. In Turbulent Management and Relaminarization, IUTAM Symp. Bangalore, India, pp. 199–213. Springer.
Corke, T. C. & Mangano, R. A. 1989 Resonant growth of three-dimensional modes in transitioning Blasius boundary layers. J. Fluid Mech. 209, 93.Google Scholar
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393.Google Scholar
Craik, A. D. D. 1980 Nonlinear evolution and breakdown in unstable boundary layers. J. Fluid Mech. 99, 247.Google Scholar
Craik, A. D. D. 1985 Wave Interactions in Fluid Flows. Cambridge University Press.
Fasel, H. F., Rist, U. & Konzelman, U. 1987 Numerical investigation of the three-dimensional development in boundary layer transition. AIAA Paper 87-1203.
Fischer, T. M. & Dallmann, U. 1991 Primary and secondary stability analysis of a three-dimensional boundary layer. In Boundary Layer Transition & Control, Cambridge, UK, pp. 5.1–5.14.
Gaster, M. 1974 On the effect of boundary layer growth on flow stability. J. Fluid Mech. 66, 465.Google Scholar
Gaster, M. 1984 On transition to turbulence in boundary layers. In Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena in Fluids, Kyoto, Japan, pp. 99–106. North-Holland.
Goldshtik, M. A., Lifshits, A. M. & Shtern, V. N. 1984 Threshold regimes in the plane channel flow. In Laminar-Turbulent Transition (ed. V. Kozlov), pp. 191–198. Springer.
Herbert, Th. 1983 Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys. Fluids 26, 871.Google Scholar
Herbert, Th. 1984 Analysis of the subharmonic route to transition in boundary layers. AIAA Paper 84-0009.
Herbert, Th. 1985 Three-dimensional phenomena in transitional flat-plate boundary layer. AIAA paper 85-049.
Herbert, Th. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487.Google Scholar
Herbert, Th. & Bertolotti, F. 1985 The effects of pressure gradients on the growth of subharmonic disturbances in boundary layers. In Proc. Conf. Low Reynolds Number Airfoil Aerodyn. (ed. T. Müller), pp. 65–76. Notre Dame University.
Itoh, N. 1975 Spatial growth of finite wave disturbances in parallel and nearly parallel flows. Trans. Japan Soc. Aeronaut. Sci. 17, 166.Google Scholar
Itoh, N. 1987 Another route to the three-dimensional development of Tollmien-Schlichting waves with finite amplitude. J. Fluid Mech. 181, 1.Google Scholar
Jaffe, N. A., Okamur, T. T. & Smith, A. M. O. 1970 Determination of spatial amplification factors and their application to predicting transition. AIAA J. 8 No. 2, 301.Google Scholar
Kachanov, Yu. S. 1987 On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 184, 43.Google Scholar
Kachanov, Yu. S., Kozlov, V. V. & Levchenko, V. Ya. 1977 Nonlinear development of wave in boundary layer. Izv. Akad. Nauk SSSR, Mech. Zhidk. i Gaza 3, 49 (in Russian). (See also Fluid Dyn. 12, 1978, 383.)Google Scholar
Kachanov, Yu. S., Kozlov, V. V., Levchenko, V. Ya. & Ramazanov, M. P. 1984 On nature of K-breakdown of laminar boundary layer. In Laminar-Turbulent Transition (ed. V. Kozlov), pp. 61–74. Springer.
Kachanov, Yu. S. & Levchenko, V. Ya.1982 Resonant interactions of disturbances in transition to turbulence in boundary layer. Preprint N10-82, ITPM SO AN SSSR, Novosibirsk (in Russian).
Kachanov, Yu. S. & Levchenko, V. Ya.1984 The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech. 138, 209.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657.Google Scholar
Klebanoff, P. S. & Tidstrom, K. D. 1959 Evolution of amplified waves leading to transition in a boundary layer with zero pressure gradient. NASA Tech. NoteD-195.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1.Google Scholar
Kleiser, L. 1985 Three-dimensional processes in laminar-turbulent transition. In Nonlinear Dynamics in Transcritical Flows (ed. H. L. Jordan et al.). Lecture Notes in Engineering. Springer.
Maslennikova, I.I. & Zelman, M. B. 1985 On subharmonic-type laminar-turbulent transition in boundary layers. In Laminar-Turbulent Transition (ed. V. Kozlov), pp. 21–28. Springer.
Morkovin, M. & Reshotko, E. 1989 Dialog on progress and issues in stability and transition research. In Laminar-Turbulent Transition (ed. D. Arnal & R. Michel), pp. 3–30. Springer.
Nayfeh, A. H. 1987 Nonlinear stability of boundary layers. AIAA Paper 87-0044.
Nayfeh, A. H. & Bozatli, A. N. 1979 Nonlinear interactions in boundary layer. AIAA Paper 79-1497.
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347.Google Scholar
Raetz, G. S. 1959 New theory of the cause of transition in fluid flows. Norair Rep. NOR 59-383. Howpton, CA.
Rist, U. & Fasel, H. 1991 Spatial three-dimensional numerical simulation of laminar-turbulent transition in a flat-plate boundary layer. In Proc. Conf. Boundary Layer Transition & Control, Cambridge, UK, pp. 25.1–25.9.
Saric, W. S., Kozlov, V. V. & Levchenko, V. Ya. 1984 Forced and unforced subharmonic resonance in boundary layer transition. AIAA Paper 84-0007.
Saric, W. S. & Nayfeh, A. H. 1975 Nonparallel stability of boundary-layer flows. Phys. Fluids 18, 945.Google Scholar
Saric, W. S. & Reynolds, G. A. 1980 Experiments on the stability of nonlinear waves in a boundary layer. In Laminar-Turbulent Transition (ed. R. Eppler & H. Fasel), pp. 125–134. Springer.
Saric, W. S. & Thomas, A. S. W. 1984 Experiments on the subharmonic route to turbulence in boundary layers. In Proc. IUTAM Symp. on Turbulence and Chaotic Phenomena in Fluids, Kyoto, Japan, pp. 117–122. North-Holland.
Singer, B. A., Reed, H. L. & Ferziger, J. H.1989 The effects of streamwise vortices on transition in the plane channel. Phys. Fluids A 1, 1960.Google Scholar
Smith, F. T. & Stewart, P. A. 1987 The resonant-triad nonlinear interaction in boundary layer transition. J. Fluid Mech. 179, 227.Google Scholar
Spalart, Ph. & Yang, K. S. 1987 Numerical study of ribbon-induced transition in Blasius flow. J. Fluid Mech. 178, 345.Google Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353.Google Scholar
Stuart, J. T. 1962 On three-dimensional nonlinear effects in the stability of parallel flows. Adv. Aero. Sci. 3–4, 121.
Thomas, A. S. W. 1983 The control of boundary-layer transition using a wave-superposition principle. J. Fluid Mech. 137, 233.Google Scholar
Thomas, A. S. W. & Saric, W. S. 1981 Harmonic and subharmonic waves during boundary-layer transition. Bull. Am. Phys. Soc. 26, 1252.Google Scholar
Usher, J. R. & Craik, A. D. D. 1975 Nonlinear wave interactions in shear flows. Part 2. J. Fluid Mech. 70, 437.Google Scholar
Volodin, A. G. & Zelman, M. B. 1978 Three-wave resonant interaction of disturbances in boundary layer. Izv. Akad. Nauk SSSR, Mech. Zhidk. i Gaza, No. 5, 78 (in Russian).
Volodin, A. G. & Zelman, M. B. 1981 On the nature of differences in some forms of transition in boundary layers. AIAA J. 19, 950.Google Scholar
Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. J. Fluid Mech. 9, 371.Google Scholar
Zelman, M. B. 1974 On the nonlinear development of disturbances in parallel flows. Izv. Sib. Otd. Akad. Nauk SSSR 13 16 (in Russian).Google Scholar
Zelman, M. B. 1989 The development of nonlinear disturbances in Blasius boundary layer. In Models in the Mechanics of Nonhomogeneous Systems (ed. G. Gadijak), pp. 117–136. Novosibirsk, USSR (in Russian). (See also Russian J. Theor. Appl. Mech. (1991, No. 2.)
Zelman, M. B. & Kakotkin, A. Ph. 1982 On the weakly nonlinear stability of nonparallel boundary layers. Chisl. Met. Mech. Spl. Sred 13 No. 3 (in Russian).Google Scholar
Zelman, M. B. & Maslennikova, I.I. 1984 On effects of wave disturbance resonant interaction in boundary layer. Izv. Akad. Nauk SSSR, Mech. Zhidk. i Gaza, No. 4, 23.
Zelman, M. B. & Maslennikova, I.I. 1985 Resonant interaction of spatial disturbances in a boundary layer. J. Appl. Mech. Tech. Phys. 26 3)1 378.Google Scholar
Zelman, M. B. & Maslennikova, I.I. 1987 On the mechanism of active control of subharmonic transition in boundary layer. J. Appl. Mech. Tech. Phys. 28 1), 25. (See also Proc. 3rd Asian Cong. Fluid Mechanics, 1987, Tokyo, Japan, pp. 729–732.)Google Scholar
Zelman, M. B. & Maslennikova, I.I. 1989 On the formation of spatial structure of subharmonic transition in Blasius flow. Izv. Akad. Nauk SSSR, Mech. Zhidk. i Gaza, No. 3, 77. (See also in Laminar-Turbulent Transition 1990 (ed. D. Arnal & R. Michel), pp. 142–142. Springer.)
Zelman, M. B. & Maslennikova, I.I. 1992 Subharmonic transition spectrum generation in boundary layer. J. Appl. Mech. Tech. Phys. 33 2)1 197.Google Scholar
Zelman, M. B. & Smorodsky, B. V. 1988 On the resonant interaction of wave packets in boundary layer. Izv. Akad. Nauk SSSR, Mech. Zhidk. i Gaze, No. 6, 67 (in Russian). (See also in Laminar-Turbulent Transition (1990) (ed. D. Arnal & R. Michel), pp. 290–290. Springer.)
Zhao, G.-F. & Zhao, H. 1988 Weakly nonlinear theory for plane Poiseuille flow. Appl. Math. Mech. (English edn) 9, 617.Google Scholar