Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

TL;DR

This paper demonstrates that unstable periodic orbits (UPOs) are dynamically relevant in turbulent flows, with flows transiently shadowing these orbits and sharing similar energy statistics, thus linking turbulence to specific unstable solutions.

Contribution

The study shows that unstable periodic orbits are statistically significant and dynamically relevant in turbulent flows, bridging experimental and numerical observations.

Findings

Turbulent flows transiently shadow UPOs.

Flows spend significant time near UPOs.

Energy input and dissipation rates are similar for flows and UPOs.

Abstract

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.