Visualizing the geometry of state space in plane Couette flow

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 plan...

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Bibliographic Details
Published in:Journal of fluid mechanics 2008-09, Vol.611, p.107-130
Main Authors: GIBSON, J. F., HALCROW, J., CVITANOVIĆ, P.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Fluid dynamics
Fluid mechanics
Fundamental areas of phenomenology (including applications)
Physics
Transition to turbulence
Turbulent flows, convection, and heat transfer
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/S002211200800267X