Energy norm

The kinetic energy of the fluid  E = (u,u)/2  endows the vectors in state space (velocity field across entire cell at a given instant in time) with a physically motivated bilinear L2 norm, a vector dot product:

Enorm

THE IDEA: pick a few fluid states frequently visited by the turbulent flow, and visualize the infinite-dimensional flow by projecting it onto coordinate frames constructed from such states.

For wall-bounded shear flows close to the onset of turbulence such state space portraits are surprisingly informative.

What are good choices of such states? We could start locally, close to an equilibrium.

or go directly to global visualization of the dynamics, next slide.