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Modulated, frequency-locked, and chaotic cross-waves

Published online by Cambridge University Press:  26 April 2006

William B. Underhill ,
Seth Lichter  and
Andrew J. Bernoff
William B. Underhill
Affiliation:
Department of Aerospace and Mechanical Engineering. The University of Arizona, Tucson, AZ 85721, USA
Seth Lichter
Affiliation:
Department of Aerospace and Mechanical Engineering. The University of Arizona, Tucson, AZ 85721, USA
Andrew J. Bernoff
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, AZ 85721, USA
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Abstract

Measurements were made of the wave height of periodic, quasi-periodic, and chaotic parametrically forced cross-waves in a long rectangular channel. In general, three frequencies (and their harmonics) may be observed: the subharmonic frequency and two slow temporal modulations — a one-mode instability associated with streamwise variation and a sloshing motion associated with spanwise variation. Their interaction, as forcing frequency, f, and forcing amplitude, a, were varied, produced a pattern of Arnold tongues in which two or three frequencies were locked. The overall picture of frequency-locked and -unlocked regions is explained in terms of the Arnold tongues predicted by the circle-map theory describing weakly coupled oscillators. Some of the observed tongues are apparently folded by a subcritical bifurcation, with the tips of the tongues lying on the unstable manifold folded under the observed stable manifold. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed in agreement with the coupled-mode analysis of Ayanle, Bernoff & Lichter (1990). At large cross-wave amplitudes, the spanwise wave structure apparently breaks up, because of modulational instability, into coherent soliton-like structures that propagate in the spanwise direction and are reflected by the sidewalls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 371 - 394
Copyright
© 1991 Cambridge University Press

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