A new way to look at many time-dependent turbulent data sets at one glance?

Revolution will not be televised
(but it will be on video).

Within the last few years the dynamical (as opposed to statistical) theory of moderate Re turbulence in wall-bounded flows has made great strides; the geometry of the infinite-dimensional Navier-Stokes state space (not low-dimensional models) can now be triangulated by sets of exact solutions (equilibria, traveling waves, periodic orbits, ...). They, together with their stable/unstable manifolds and the heteroclinic connections between various exact solutions, form a rigid backbone which enables us to describe, control and predict the sinuous motions of a turbulent fluid.

Visualization of a high resolution 3D PVI snapshot of a turbulent pipe flow by iso-vorticity surfaces:

[click to enlarge]

pCf equilibria

Casimir W.H. van Doorne (PhD thesis, Delft 2004)

iso-vorticity surfaces
blue: rotating in one sense
red: in the other sense
Turbulent flow:
a small repertoire of unstable recurrent coherent structures

The following state space visualization shows that the time evolution of such turbulent flow is better visualized in terms of close passages to a set of unstable coherent structures observed in the turbulent regions of the state space.

[click to enlarge]

pCf equilibria

A family portrait of 25 exact equilibria and traveling waves of plane Couette flow.

The green dense region is the natural measure generated by typical long lived turbulent transients, with the set of invariant solutions providing a “cage” that organizes the turbulent flow.

The tutorial explains this picture.

The state space visualization is novel, and might become a useful tool for you, a fluid dynamics experimentalist. Instead of obtaining a set of exact invariant solutions from a theorist, pick a few typical coherent structures observed in turbulent flow, and use these to construct a low-dimensional state-space projection from your high-dimensional data set. Evolution of a typical turbulent flow (green dense region above) is then visualized in terms of close passages to unstable coherent structures observed in the turbulent regions of the state space.

The tutorial develops the theory behind this state space representation in a sequence of gentle steps.

tutorial.html