It has recently become possible to compute precise 3D, nonlinear
solutions of Navier-Stokes equations at Reynolds numbers above
the onset of turbulence, for simple geometries such as pipes and
channels. These solutions capture the form and dynamics of "coherent
structures" and provide a starting point for understanding low-Reynolds
turbulence as a dynamical system. In this talk I will present a number
of equilibrium, traveling wave, and periodic orbit solutions of plane
Couette flow, emphasizing visualizations of their physical structure
and state-space dynamics, and comparisons to turbulent flow. Certain
spatially localized solutions exhibit homoclinic snaking remarkably
similar to that observed in simpler 1D PDE systems. What emerges is a
picture of low-Reynolds turbulence as a walk among a set of weakly
unstable invariant solutions.
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