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The mechanics of the Tollmien-Schlichting wave

Published online by Cambridge University Press:  26 April 2006

Peter G. Baines ,
Sharan J. Majumdar  and
Humio Mitsudera
Peter G. Baines
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia
Sharan J. Majumdar
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
Humio Mitsudera
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia Present address: JAMSTEC, Yokosuka, Japan.
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Abstract

We describe a mechanistic picture of the essential dynamical processes in the growing Tollmien-Schlichting wave in a Blasius boundary layer and similar flows. This picture depends on the interaction between two component parts of a disturbance (denoted ‘partial modes’), each of which is a complete linear solution in some idealization of the system. The first component is an inviscid mode propagating on the vorticity gradient of the velocity profile with the free-slip boundary condition, and the second, damped free viscous modes in infinite uniform shear with the no-slip condition. There are two families of these viscous modes, delineated by whether the phase lines of the vorticity at the wall are oriented with or against the shear, and they are manifested as resonances in a forced system. The interaction occurs because an initial ‘inviscid’ disturbance forces a viscous response via the no-slip condition at the wall. This viscous response is large near the resonance associated with the most weakly damped viscous mode, and in the unstable parameter range it has suitable phase at the outer part of the boundary layer to increase the amplitude of the inviscid partial mode by advection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 312 , 10 April 1996 , pp. 107 - 124
Copyright
© 1996 Cambridge University Press

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