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On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition

Published online by Cambridge University Press:  15 October 2007

XUESONG WU ,
P. A. STEWART  and
S. J. COWLEY
XUESONG WU
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
P. A. STEWART
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
S. J. COWLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
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Abstract

This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 265 - 294
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Bodonyi, R. J. & Smith, F. T. 1981 The upper-branch stability of the Blasius boundary layer, including non-parallel flow effects. Proc. R. Soc. Lond. A 375, 65–92.Google Scholar
Borodulin, V. I., Kachanov, Y. S. & Koptsev, D. B. 2002a Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient: I. Tuned resonance. J. Turbulence 3 (62), 1–38.Google Scholar
Borodulin, V. I., Kachanov, Y. S., Koptsev, D. B. & Roschektayev, A. P. 2002b Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient: II. Detuned resonance. J. Turbulence 3 (63), 1–32.Google Scholar
Borodulin, V. I., Kachanov, Y. S. & Koptsev, D. B. 2002c Experimental study of resonant interactions of instability waves in a self-similar boundary layer with an adverse pressure gradient: III. Broadband disturbances. J. Turbulence 3 (64), 1–19.Google Scholar
Corke, T. C. & Mangano, R. A. 1989 Resonant growth of three-dimensional modes in transitioning Blasius boundary layers. J Fluid Mech. 209, 93–150.CrossRefGoogle Scholar
Corke, T. C., Krull, J. D. & Ghassemi, M. 1992 Three-dimensional-mode resonance in far wakes. J. Fluid Mech. 239, 99–132.CrossRefGoogle Scholar
Craik, A. D. D. 1971 Non-linear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.CrossRefGoogle Scholar
Goldstein, M. E. 1994 Nonlinear interactions between oblique instability waves on nearly parallel shear flows. Phys. Fluids A6, 724–735.CrossRefGoogle Scholar
Goldstein, M. E. 1995a The effect of nonlinear critical layers on boudary layer transition. In Proc. IUTAM Symp. on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers (ed. P. Duck & P. Hall), pp. 1–10. Springer.CrossRefGoogle Scholar
Goldstein, M. E. 1995b The role of nonlinear critical layers in boundary layer transition. Phil. Trans. R. Soc. Lond. A 352, 425–442.Google Scholar
Goldstein, M. E. & Choi, S.-W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120. Corrigendum J. Fluid Mech. 216, 1990, 659.CrossRefGoogle Scholar
Goldstein, M. E. & Durbin, P. A. 1986 Nonlinear critical layers eliminate the upper branch of spatially growing Tollmien–Schlichting waves. Phys. Fluids 29, 2344–2345.CrossRefGoogle Scholar
Goldstein, M. E. & Lee, S. S. 1992 Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer. J. Fluid Mech. 245, 523–551.CrossRefGoogle Scholar
Healey, J. J. 1994 On the neutral curve of the flat-plate boundary layer: comparison between experiment, Orr–Sommerfeld theory and asymptotic theory. J. Fluid Mech. 288, 59–83.CrossRefGoogle Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487–526.CrossRefGoogle Scholar
Jennings, M. J. 1997 On the weakly nonlinear evolution of Tollmien–Schlichting waves in shear flow. PhD thesis, University of Cambridge, UK.Google Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.CrossRefGoogle Scholar
Kachanov, Y. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminar–turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.CrossRefGoogle Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
Lee, S. S. 1997 Critical-layer analysis of fully coupled resonant-triad interaction in a boundary layer. J. Fluid Mech. 347, 71–103.CrossRefGoogle Scholar
Mankbadi, R. R., Wu, X. & Lee, S. S. 1993 A critical-layer analysis of the resonant triad in Blasius boundary-layer transition: nonlinear interactions. J. Fluid Mech. 256, 85–106.CrossRefGoogle Scholar
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Norair Rep. NOR-59-383. Hawthorne, California.Google Scholar
Reid, W. H. 1965 The stability of parallel flows. In Basic Developments in Fluid Dynamics, ed. Holt, M., pp. 249–308. Academic.Google Scholar
Smith, F. T. & Daniels, P. G. 1981 Removal of Goldstein's singularity at separation in flow past obstacles in wall layers. J. Fluid Mech. 110, 1–37.CrossRefGoogle Scholar
Smith, F. T. & Stewart, P. A. 1987 The resonant-triad nonlinear interaction in boundary-layer transition. J. Fluid Mech. 179, 227–252.CrossRefGoogle Scholar
Spalart, P. R. & Yang, K. S. 1987 Numerical study of ribbon-induced transition in Blasius boundary layer. J. Fluid Mech. 178, 345–365.CrossRefGoogle Scholar
Van Dommelen, L. L. & Cowley, S. J. 1990 On the Lagrangian description of unsteady boundary-layer separation. Part 1. General theory. J. Fluid Mech. 210, 593–626.CrossRefGoogle Scholar
Van Dommelen, L. L. & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38, 125–140.CrossRefGoogle Scholar
Williamson, C. H. K. & Prasad, A. 1993 a A new mechanism for oblique wave resonance in the ‘natural’ far wake. J. Fluid Mech. 256, 269–313.CrossRefGoogle Scholar
Williamson, C. H. K. & Prasad, A. 1993 b Acoustic forcing of oblique wave resonance in the far wake. J. Fluid Mech. 256, 315–341.CrossRefGoogle Scholar
Wu, X, 1993 On critical-layer and diffusion-layer nonlinearity in the three-dimensional stage of boundary-layer transition. Proc. R. Soc. Lond. A 443, 95–106.Google Scholar
Wu, X. 1995 Viscous effects on fully coupled resonant-triad interactions: an analytical approach. J. Fluid Mech. 292, 377–407.CrossRefGoogle Scholar
Wu, X. 1996 On an active resonant triad of mixed modes in symmetric shear flows: a plane wake as a paradigm. J. Fluid Mech. 317, 337–368.CrossRefGoogle Scholar
Wu, X. & Stewart, P. A. 1996 Interaction of phase-locked modes: a new mechanism for the rapid growth of three-dimensional disturbances. J. Fluid Mech. 316, 335–372.CrossRefGoogle Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear flows to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–721.CrossRefGoogle Scholar
Wu, X., Stewart, P. A. & Cowley, S. J. 1996 On the weakly development of Tollmien-Schlichting wave-trains in boundary layers. J. Fluid Mech. 323, 133–171.CrossRefGoogle Scholar
Wu, X., Leib, S. J. & Goldstein, M. E. 1997 On the nonlinear evolution of a pair of oblique Tollmien-Schlichting waves in boundary layers. J. Fluid Mech. 340, 361–394.CrossRefGoogle Scholar
Wundrow, D. W., Hultgren, L. S. & Goldstein, M. E. 1994 Interaction of oblique instability waves with a nonlinear plane wave. J. Fluid Mech. 264, 343–372.CrossRefGoogle Scholar
Zhuk, V. I. & Ryzhov, O. S. 1989 On 3d inviscid disturbances inducing their own pressure gradient in a boundary layer. Dokl. Akad. Nauk. SSSR 301 (1), 52–56. See also Sov. Phys. Dokl. 34(11), 949–951.Google Scholar