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Two-dimensional disturbance travel, growth and spreading in boundary layers

Published online by Cambridge University Press:  21 April 2006

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

The nonlinear growth of Tollmien-Schlichting disturbances in a boundary layer is considered as an initial-value problem, for the unsteady two-dimensional triple deck, and computational and analytical solutions are presented. On the analytical side, the nonlinear properties of relatively high-frequency/high-speed disturbances are discussed. The disturbances travel at the group velocity and their amplitude is controlled by a generalized cubic Schrödinger equation, during a first stage of the nonlinear development. The equation, which has been studied in other contexts also, is integrated numerically here, and the resulting large-time/far-downstream behaviour is then deduced analytically. This behaviour comprises an exponentially fast growth and spreading of the disturbance, the spreading being governed only by an integral property of the initial disturbance. Secondary sideband instability does not occur, and there is no conclusive sign of a chaotic response, during this stage, although the three-dimensional counterpart could well yield both phenomena. In the subsequent (and more nonlinear) second stage further downstream, however, where the amplitude is larger, spiked behaviour and spectrum broadening can occur because of vorticity bursts from the viscous sublayer. Computationally, two forms of numerical solution of the triple-deck problem, one spectral, the other finite-difference, are given. The results from each form tend to support the conclusions of the high-frequency analysis for initial-value problems, and recent calculations of the two-dimensional unsteady Navier-Stokes equations also provide some backing. One implication is that the unsteady planar interacting-boundary-layer equations, or a composite version, can capture much of the physics involved in the beginnings of boundary-layer transition although, again, three-dimensionality is undoubtedly an important element which will need to be incorporated eventually.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 169 , August 1986 , pp. 353 - 377
Copyright
© 1986 Cambridge University Press

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