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On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

Published online by Cambridge University Press:  26 April 2006

Xuesong Wu ,
Philip A. Stewart  and
Stephen J. Cowley
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK
Philip A. Stewart
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Stephen J. Cowley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 323 , 25 September 1996 , pp. 133 - 171
Copyright
© 1996 Cambridge University Press

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