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On the secondary instability of Taylor-Görtler vortices to Tollmien-Schlichting waves in fully developed flows

Published online by Cambridge University Press:  21 April 2006

James Bennett  and
Philip Hall
James Bennett
Affiliation:
Mathematics Department, Exeter University, Exeter EX4 4QE, UK
Philip Hall
Affiliation:
Mathematics Department, Exeter University, Exeter EX4 4QE, UK
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Abstract

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Görtler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor-Görtler vortices on the growth of small-amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor-Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 186 , January 1988 , pp. 445 - 469
Copyright
© 1988 Cambridge University Press

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