We are currently investigating the utility of “exact coherent structures” (ECS) in describing, forecasting, and even controlling turbulent fluid flows. ECS are special unstable solutions of the Navier-Stokes equations that have regular temporal behavior (such as equilibria, periodic orbits, etc) and are thought to provide the “road map” that governs turbulent evolution. We are exploring this new dynamical description of turbulence in 2D flows, which provide an ideal testbed because of their experimental accessibility and computational tractability.
Two-dimensional flows have long been studied as models for oceanic and atmospheric flows. In practice, however, 2D flows can only be approximated by quasi-2D (Q2D) experiments, which are always subject to three-dimensional effects. We generate Q2D turbulence by driving a thin layer of electrolyte with electromagnetic forces, approximating a canonical fluids problem known as “Kolmogorov flow.” Kolmogorov flow is known to undergo a sequence of transitions (“bifurcations”) before becoming turbulent.
We have derived a strictly 2D model of the Q2D experiment by depth-averaging the full 3D Navier-Stokes equations (our paper is here). This 2D model quantitatively captures what happens in the experiment, which we’ve validated by making comparisons with numerical simulations. Specifically, we find agreement in the first and second bifurcations to about 3% (details are available in this preprint).
With quantitative agreement established between the experiment and simulation, we are now searching for evidence of the dynamical role of ECS in the weakly turbulent regime. We are very excited about our most recent results, which we are in the process of submitting for publication. Stay tuned!
As a preview, below is one such ECS that we’ve computed and a snapshot from the turbulent experiment. They look similar, suggesting ECS are dynamically important!