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Nonlinear interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition

Part of: Incompressible viscous fluids

Published online by Cambridge University Press:  26 February 2010

F. T. Smith  and
A. G. Walton
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WC1E 6BT.
A. G. Walton
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WC1E 6BT.
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Abstract

The nonlinear interactions that evolve between a planar or nearly planar Tollmien-Schlichting (TS) wave and the associated longitudinal vortices are considered theoretically, for a boundary layer at high Reynolds numbers. The vortex flow is either induced by the TS nonlinear forcing or is input upstream, and similarly for the nonlinear wave development. Three major kinds of nonlinear spatial evolution, Types I-III, are found. Each can start from secondary instability and then becomes nonlinear, Type I proving to be relatively benign but able to act as a pre-cursor to the Types II, III which turn out to be very powerful nonlinear interactions. Type II involves faster streamwise dependence and leads to a finite-distance blow-up in the amplitudes, which then triggers the full nonlinear three-dimensional triple-deck response, thus entirely altering the mean-flow profile locally. In contrast, Type III involves slower streamwise dependence but a faster spanwise response, with a small TS amplitude thereby causing an enhanced vortex effect which, again, is substantial enough to entirely alter the mean-flow profile, on a more global scale. Concentrated spanwise formations in which there is localized focussing of streamwise vorticity and/or wave amplitude can appear, and certain of the nonlinear features also suggest by-pass processes for transition and significant change in the flow structure downstream. The powerful nonlinear 3D interactions II, III seem potentially very relevant to experimental and computational findings in fully fledged transition; in particular, it is suggested in an appendix that the Type-Ill interaction can terminate in a form of 3D boundary-layer separation which appears possibly connected with the formation of lambda vortices in practice.

MSC classification

Secondary: 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 262 - 289
Copyright
Copyright © University College London 1989

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References

Arnal, D. (1984). Description and prediction of transition in 2D incompressible flow. Special Course on Stability and Transition of Laminar Flow, AGARD-R-709, 2–1 to 2–71.Google Scholar
Benney, D. J. and Lin, C. C. (1960). On the secondary motion induced by oscillation in a shear flow. Phys. Fluids, 3, 656.CrossRefGoogle Scholar
Craik, A. D. D. (1971). Nonlinear resonant instability in boundary layers. J. Fluid Mech., 50, 393.CrossRefGoogle Scholar
Hocking, L. M. and Stewartson, K. (1972). On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. Roy. Soc. A, 326, 289.Google Scholar
Hall, P. and Smith, F. T. (1984). On the effects of nonparallelism, three-dimensionality and mode interaction in nonlinear boundary layer stability. Studies in Appl. Math., 70, 91.CrossRefGoogle Scholar
Hall, P. and Smith, F. T. (1987). The nonlinear interaction of Tollmien-Schlichting waves and Taylor-Gortler vortices in curved channel flows. ICASE Report 87–25; and Proc. Roy. Soc. A, 417 (1988), 255.Google Scholar
Hall, P. and Smith, F. T. (1988a). Nonlinear Tollmien-Schlichting/vortex interaction in boundary layers. ICASE Report 88–46; and European J. Mech., 1989, B8, 179.Google Scholar
Hall, P. and Smith, F. T. (1988b). The strong nonlinear interaction of Tollmien-Schlichting waves and Taylor-Görtler vortices in curved channel flow. ICASE Report 88–45; and submitted to j Fluid Mech, 1989 (with J. Bennett).Google Scholar
Hall, P. and Smith, F. T. (1989). Strong nonlinear interactions between waves and longitudinal vortices in boundary layers, submitted for publication.Google Scholar
Herbert, T. (1984). Analysis of the subharmonic route to transition in boundary layers. AIAA Paper 84–0009, presented in Reno, Nevada.CrossRefGoogle Scholar
Smith, F. T. (1979a). Nonlinear stability of boundary layers for disturbances of various sizes. Proc. Roy. Soc. A, 366, 91 and 368, 573.Google Scholar
Smith, F. T. (1979b). Instability of flow through pipes of general cross-section, Parts 1 and 2. Mathematika, 26, 187–210 and 211–223.CrossRefGoogle Scholar
Smith, F. T. (1987). On the first-mode instability in subsonic, supersonic and hypersonic boundary layers. Utd. Tech. Res. Cent. Report UT 87–52 and j. Fluid Mech, 1989, 198, 127–153.Google Scholar
Smith, F. T. (1988). Finite-time break-up can occur in any unsteady interacting boundary layer. Mathematika, 35, 256.CrossRefGoogle Scholar
Smith, F. T. (1989). Steady and unsteady 3D interacting boundary layers. Computers and Fluids, to appear.Google Scholar
Smith, F. T. and Stewart, P. A. (1987). The resonant-triad nonlinear interaction in boundary-layer transition. j. Fluid Mech, 179, 227.CrossRefGoogle Scholar
Srivastava, K. M. and Dallmann, U. (1987). Effect of streamwise vortices on Tollmien-Schlichting waves in growing boundary layers. Phys. Fluids, 30, 1005.CrossRefGoogle Scholar
Walton, A. G. (1990). Ph.D. thesis (Univ. of London)Google Scholar