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Three-dimensional nonlinear blow-up from a nearly planar initial disturbance, in boundary-layer transition: theory and experimental comparisons

Published online by Cambridge University Press:  26 April 2006

P. A. Stewart  and
F. T. Smith
P. A. Stewart
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK Current address: School of Mathematics, The University, Leeds, LS2 9JT, UK.
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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Abstract

This theoretical study describes how three-dimensional nonlinear distortion may soon take effect, following a small initial input disturbance that is nearly planar, in an otherwise two-dimensional boundary layer at high Reynolds number. The mechanism involved is a form of vortex-wave interaction, the first such to be examined in the so-called high-frequency range. The interaction is powerful, in that three-dimensional disturbances of relatively low amplitude (the wave part) interact nonlinearly with the three-dimensional corrections to the mean flow (the vortex part) at a stage where the purely two-dimensional case alone would still be linear. A coupled nonlinear partial-differential system is derived, governing the vortex and wave parts. Computations and analysis of the system are then presented. These point to a finite-time singularity arising in the solution, involving blow-up of both the vortex and the wave amplitudes (but particularly the former), accompanied by spanwise focusing into streets. This is believed to be the first nonlinear interaction in the high-frequency range to produce a finite-time (or-distance) blow-up. The blow-up is such that the local flow soon enters a strongly nonlinear three-dimensional stage in which the total mean flow is altered. The implications of this blow-up and focusing for one of the classic paths of boundary-layer transition are also discussed, and here quantitative and/or order-of-magnitude comparisons suggest that the theory is in line with the findings of Klebanoff & Tidstrom (1959) and later experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 79 - 100
Copyright
© 1992 Cambridge University Press

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