Geometry of Turbulence in Wall-bounded Shear Flows: A Stroll Through 61,506 Dimensions

By John F. Gibson
Geometry of Turbulence in Wall-bounded Shear Flows: A Stroll Through 61,506 Dimensions

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
Georgia Tech
Author Email
Tutorial LevelAdvanced Tutorial
Contest EntryNo

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