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The generation of Tollmien-Schlichting waves by free-stream turbulence

Published online by Cambridge University Press:  26 April 2006

P. W. Duck ,
A. I. Ruban  and
C. N. Zhikharev
P. W. Duck
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
A. I. Ruban
Affiliation:
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky-3, Moscow reg., 140160, Russia
C. N. Zhikharev
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA
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Abstract

The phenomenon of Tollmien-Schlichting wave generation in a boundary layer by free-stream turbulence is analysed theoretically by means of asymptotic solution of the Navier-Stokes equations at large Reynolds numbers (Re → ∞). For simplicity the basic flow is taken to be the Blasius boundary layer over a flat plate. Free-stream turbulence is taken to be uniform and thus may be represented by a superposition of vorticity waves. Interaction of these waves with the flat plate is investigated first. It is shown that apart from the conventional viscous boundary layer of thickness O(Re−1/2), a ‘vorticity deformation layer’ of thickness O(Re−1/4) forms along the flat-plate surface. Equations to describe the vorticity deformation process are derived, based on multiscale asymptotic techniques, and solved numerically. As a result it is shown that a strong singularity (in the form of a shock-like distribution in the wall vorticity) forms in the flow at some distance downstream of the leading edge, on the surface of the flat plate. This is likely to provoke abrupt transition in the boundary layer. With decreasing amplitude of free-stream turbulence perturbations, the singular point moves far away from the leading edge of the flat plate, and any roughness on the surface may cause Tollmien-Schlichting wave generation in the boundary layer. The theory describing the generation process is constructed on the basis of the ‘triple-deck’ concept of the boundary-layer interaction with the external inviscid flow. As a result, an explicit formula for the amplitude of Tollmien-Schlichting waves is obtained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 312 , 10 April 1996 , pp. 341 - 371
Copyright
© 1996 Cambridge University Press

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