5. Energy-dissipation visualization of Navier-Stokes solutions

3D movies give no hint of relative importance of, and interrelations between different exact invariant solutions. Some physical insights on this are gained by studying the energy E, power input I (shearing motion of the walls) and 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 UB - 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.

 

POsHKW

statespFrames

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 fluid contained within the cell suggests, however, a much more versatile visualization framework.