Plane Couette - dual views of transient turbulence

Transient turbulence: a narrative

[keep the movie from the previous slide running in a separate window]

The state-space trajectory is plotted against three exact equilibria of plane Couette flow:

the lower-branch (LB, blue)

the upper-branch (UB, green)

the "newbie" (NB, red)

plus the laminar solution (LM, black) (see equilibria). The half-cell translations in x and z of the equilibria are points indicated with prefix "τ_x", "τ_z", and "τ_xz". The blue and green solid lines show the 1 and 2-dimensional unstable manifolds of the LB and UB equilibria. The moving red dot is the state of the fluid velocity field evolving under the Navier-Stokes equations.

Partial narrative:

t=0-150 : The initial condition is a small perturbation in the unstable manifold of the NB equilibrium. The perturbation is small enough that no visible changes occur until about t=150.
t=150-280 : An oscillation around the NB equilibrium, governed by the NB's unstable complex eigenvalue.
t=280-300 : A close pass to the half-cell z-translated τ_zLB equilibrium, escape back toward turbulence along LB's 1d unstable manifold.
t=300-1200 : Close passes to UB unstable manifold, equilibria still waiting to be discovered, τ_z LB, etc.
t=1200-1350 : Decay to laminar flow.