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.
|Author Institutional Affiliation|
John F. Gibson
In collaboration with Predrag Cvitanović, Jonathan Halcrow, Fabian Waleffe, and Divakar Viswanath
Center for Nonlinear Science
School of Physics
|Tutorial Level||Advanced Tutorial|