Plane Couette -

Power-dissipation visualization

Power input I vs bulk dissipation rate D of various solutions:

[click on figure to enlarge]

Left frame:: (line) A typical turbulent trajectory. Green dots are equidistant in time.; (blue) Average value of I=D for the turbulent trajectory.; (red) I=D computed on the upper branch uUB - suprisingly close to the turbulent average.
Right frame:: (color) (I(t),D(t)) for the 3 periodic orbits embedded in the turbulent region of state space.; (dotted line) A typical turbulent trajectory.

These projections from 61,506 dimensions to 2 dimensions help us eliminate equilibria that lie outside regions frequented by the turbulent trajectory as physically unimportant, and suggest that the above periodic orbits are typical of turbulent motions.

The expression for the total kinetic energy of the fluid suggests, however, a much more versatile state space visualization.