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The stability of three-dimensional time-periodic flows with ellipsoidal stream surfaces

Published online by Cambridge University Press:  26 April 2006

G. K. Forster  and
A. D. D. Craik
G. K. Forster
Affiliation:
School of Mathematical and Computational Sciences, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, UK
A. D. D. Craik
Affiliation:
School of Mathematical and Computational Sciences, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, UK
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Abstract

Most steady flows with constant vorticity and elliptical streamlines are known to be unstable. These, and certain axisymmetric time-periodic flows, can be analysed by Floquet theory. However, Floquet theory is inapplicable to other time-periodic flows that yield disturbance equations containing a quasi-periodic, rather than periodic, function. A practical method for surmounting this difficulty was recently given by Bayly, Holm & Lifschitz. Employing their method, we determine the stability of a clas of three-dimensional time-periodic flows: namely, those unbounded flows with fixed ellipsoidal stream surfaces and spatially uniform but time-periodic strain rates. Corresponding, but bounded, flows are those within a fixed ellipsoid with three different principal axes. This is perhaps the first exact stability analysis of non-reducibly three-dimensional and time-dependent flows. Though the model has some artificial features, the results are likely to shed light on more complex systems of practical interest.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 324 , 10 October 1996 , pp. 379 - 391
Copyright
© 1996 Cambridge University Press

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