6a. 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 form (dot product):

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

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

What are good choices of such states? We start locally, close to an equilibrium. In order to describe its neighborhood we need to understand the linear stability of equilibria first.

 

energetics

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