1. What these movies illustrate

There are two kinds of animations here:

• 3D movies of velocity fields, which show turbulent flows visiting unstable coherent structures
• State-space trajectories, which show
  1. that coherent structures result from close passes to unstable equilibrium solutions of Navier-Stokes, and
  2. that the equilibria and their unstable manifolds impart a rigid structure to state space that organizes the turbulent dynamics.

The dual views section shows some of these animations side-by-side. In these, you can see coherent structures appear in the turbulent velocity field as the state-space trajectory makes close passes to equilibrium solutions of Navier-Stokes.

In what follows “plane Couette” is used to illustrate key ideas. Similar phenomena have been observed in pipe flows.