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Nonlinear instability of viscous plane Couette flow Part 1. Analytical approach to a necessary condition

Published online by Cambridge University Press:  26 April 2006

Bérengére Dubrulle  and
Jean-Paul Zahn
Bérengére Dubrulle
Affiliation:
CERFACS, 42 avenue Coriolis, 31057 Toulouse, France Observatoire Midi Pyrénées, 14 avenue Belin, 31400 Toulouse. France
Jean-Paul Zahn
Affiliation:
Observatoire Midi Pyrénées, 14 avenue Belin, 31400 Toulouse. France Astronomy Department, Columbia University, New York, NY 10027, USA
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Abstract

We perform a two-dimensional analytical stability analysis of a viscous, unbounded plane Couette flow perturbed by a finite-amplitude defect and generalize the results obtained in the inviscid limit by Lerner and Knobloch. The dispersion relation is derived and is used to establish the condition of marginal stability, as well as the growth rates at different Reynolds numbers. We confirm that instability occurs at wavenumbers of the order of ε, the non-dimensional amplitude of the defect. For large enough εR (R being the Reynolds number based on the width of the defect), the maximum growth rate is about ½ε, at approximately half the critical wavenumber. We formulate the instability conditions in the case where the flow has a finite extension in the downstream direction. Instability appears when ε is greater than RL1/3, where RL is the Reynolds number based on the downstream scale, and when the ratio of the defect width to the downstream scale lies in the interval [(εRL}−½, ε].

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 231 , October 1991 , pp. 561 - 573
Copyright
© 1991 Cambridge University Press

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