In the world of everyday, moderately turbulent fluids flowing across planes and down pipes, a velvet revolution is taking place. Experiments are as detailed as simulations, there is a zoo of exact numerical solutions that one dared not dream about a decade ago, and portraits of turbulent fluid's state space geometry are unexpectedly elegant.

We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the tutorial is aimed at anyone who had ever wondered how we know a cloud when we see one, if no cloud is ever seen twice? And how do we turn that into mathematics?

There are two kinds of animations here:

3D movies of velocity fields

turbulent flows visiting unstable coherent structures

State-space trajectories, which show

1. that coherent structures result from close passes to unstable equilibrium solutions of Navier-Stokes, and

2. that the equilibria and their unstable manifolds impart a rigid structure to state space that organizes the turbulent dynamics.

 

The dual views shows some of these animations side-by-side. In these, you will see recurrent coherent structures appear in the turbulent velocity field as the state-space trajectory makes close passes to equilibrium solutions of Navier-Stokes.

In what follows, the key ideas are illustrated in the context of plane Couette flow. Similar phenomena have been observed in pipe flows.