4c. Dual views of transient turbulence

1. Transient turbulence from the unstable manifold of the NB equilibrium.
   Other encodings:
   • (MPEG-4, h264 encoding, 15 MB)
   • (Quicktime .mov, h264 encoding, 18 MB)
   • (Microsoft .avi, wmv2 encoding, 22 MB)
   • (Xiph .ogg, theora encoding, 15 MB)

   Legend: The state-space trajectory is plotted against three exact equilibria of plane Couette flow: the lower-branch (LB, blue), the upper-branch (UB, green), and the "newbie" (NB, red) equilibria, 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.

2. Transient turbulence from the unstable manifold of the UB equilibrium. (12 MB)
   This movie shows several close passes the lower-branch equilibrium.