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Visualizing the geometry of state space in plane Couette flow. (English) Zbl 1151.76453
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 \(10^{5}\)-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.

MSC:
76E05 Parallel shear flows in hydrodynamic stability
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
channelflow; LAPACK
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