_ Vol. 25 (1) (1999) pp. 19-36 _
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Fluid Dynamics Research, Vol. 25 (1) (1999) pp. 19-36
Copyright (c) 1999 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
PII: S0169-5983(99)00007-6

The stability of non-axisymmetric time-periodic vortical flows

A.D.D. Craik and G.K. Forster

School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland, UK

Received 19 June 1998; revised 9 October 1998; accepted 21 November 1998

Abstract

Steady inviscid fluid flows with constant vorticity and elliptical streamlines are known to be unstable. So too are axisymmetric flows with periodically strained vorticity. But elliptical instability may be suppressed by constant vortex stretching. Here we examine the stability of periodically strained, unbounded, non-axisymmetric flows with spatially uniform vorticity. Because of asymmetry, the analysis requires recently developed techniques. We find that the flows are normally highly unstable, with sensitive dependence on disturbance wavenumber orientation. Our results reveal a wealth of fine structure that is absent in axisymmetric cases, but which was recently found for other non-axisymmetric periodic flows. However, when the frequency of the imposed periodic motion is sufficiently large, instability is suppressed, as the requisite internal resonances are impossible. We also briefly examine the limiting case of continuous stretching. Though elliptical instability is inhibited, other exponentially growing disturbances are present. In fluctuating environments, elliptical instability is just one of many possible types of parametric instability; and its inhibition by stretching is likely to be overshadowed by other periodically driven instabilities. Such unbounded-flow models may shed light on the local interaction between small and large scales within turbulent flows.

Correspondence address

A.D.D. Craik: Corresponding author. Research Institute for Mathematical Sciences, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
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