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Visualizing the geometry of state space in plane Couette flow

Published online by Cambridge University Press:  25 September 2008

J. F. GIBSON ,
J. HALCROW  and
P. CVITANOVIĆ
J. F. GIBSON
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
J. HALCROW
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
P. CVITANOVIĆ
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
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Abstract

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 107 - 130
Copyright
Copyright © Cambridge University Press 2008

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