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On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm

Published online by Cambridge University Press:  26 April 2006

Xuesong Wu ,
Sangsoo Lee  and
Stephen J. Cowley
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, 180 Queens Gate, London SW7 2BZ, UK
Sangsoo Lee
Affiliation:
Sverdrup Technology, Inc., Lewis Research Center Group, Cleveland, OH 44135, USA
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 evolution of a pair of initially linear oblique waves in a high-Reynolds-number shear layer is studied. Attention is focused on times when disturbances of amplitude ε have O($\epsilon^{\frac{1}{3}} R $) growth rates, where R is the Reynolds number. The development of a pair of oblique waves is then controlled by nonlinear critical-layer effects (Goldstein & Choi 1989). Viscous effects are included by studying the distinguished scaling ε = O(R-1). When viscosity is not too large, solutions to the amplitude equation develop a finite-time singularity, indicating that an explosive growth can be induced by nonlinear effects; we suggest that such explosive growth is the precursor to certain of the bursts observed in experiments on Stokes layers and other shear layers. Increasing the importance of viscosity generally delays the occurrence of the finite-time singularity, and sufficiently large viscosity may lead to the disturbance decaying exponentially. For the special case when the streamwise and spanwise wavenumbers are equal, the solution can evolve into a periodic oscillation. A link between the unsteady critical-layer approach to high-Reynolds-number flow instability, and the wave/vortex approach of Hall & Smith (1991), is identified.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 681 - 721
Copyright
© 1993 Cambridge University Press

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