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Call for Papers: Special Volume 2013 of the Far East Journal of Mathematical Sciences (FJMS) devoted to articles on Computer Sciences, Information Sciences, Financial Management and Biological Sciences. ***** Foreign Subscriber Information: Avail free access to the Electronic Version of Special Volume 2013 of the Far East Journal of Mathematical Sciences (FJMS) on its Foreign 2013 Subscription.

Advances and Applications in Fluid Mechanics

Advances and Applications in Fluid Mechanics
Volume 6, Issue 2, Pages 141 - 165 (October 2009)

SPATIOTEMPORAL CASCADES OF EXPOSED AND HIDDEN PERTURBATIONS OF THE COUETTE FLOW
Victor A. Miroshnikov (U.S.A.)
Received August 18, 2009

Abstract

Spatiotemporal cascades of exposed and hidden perturbations of the Couette flow are considered in the range of Reynolds numbers

�A cascade model of exposed perturbations, which are produced by the asymmetrically moving channel walls, is based on the non-orthogonal expansion into the spatial Taylor submodes and temporal harmonic submodes. Three alternative approaches to the cascade description of the exposed perturbations through a spatial cascade, a temporal cascade, and a spatiotemporal cascade are discussed and compared. At high Reynolds numbers, the exposed perturbations form two forced viscous sublayers on the moving walls, a free viscous sublayer at a symmetry center of the Couette flow, and two multiscale cascades of coherent structures, which spread as transversal waves in the core of the Couette flow towards the symmetry center and compensate there each other. A cascade model of hidden perturbations, which initially vanish with any prescribed tolerance, is constructed by hyperbolic temporal submodes and asymmetric spatial harmonic submodes. The temporal cascade of the hidden perturbations results in a dynamic model of the spatial cascade, which is reduced to an open-ended algebraic model for tensor coefficients of matrix amplitudes of the hidden perturbations and a quantization condition, connecting the excitation-relaxation parameters of the temporal cascade of coherent structures with the wave numbers of their spatial cascade. At high Reynolds numbers, the hidden perturbations model a uniform stochastic velocity profile. A spontaneous emergence of the hidden perturbations is also justified by an asymptotic Hamiltonian approach.

Keywords and phrases: multiscale cascades, the Couette flow, transition, coherent structures, exposed and hidden perturbations.

Communicated by Shahrdad G. Sajjadi

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ISSN: 0973-4686