Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

TL;DR

This paper investigates extracting recurrent flows from 2D Kolmogorov turbulence to predict flow statistics, finding improved accuracy with more data at low forcing, but limited success at higher forcing due to increased chaos complexity.

Contribution

It demonstrates the effectiveness of recurrent flow extraction in 2D Kolmogorov flow and introduces algorithmic improvements for better prediction of flow statistics.

Findings

More recurrent flows found than previous studies.

Improved predictions of dissipation and energy PDFs at low forcing.

Limited success at high forcing due to increased chaos complexity.

Abstract

Motivated by recent success in the dynamical systems approach to transitional flow, we study the efficiency and effectiveness of extracting simple invariant sets (recurrent flows) directly from chaotic/turbulent flows and the potential of these sets for providing predictions of certain statistics of the flow. Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a rectangular torus extended in the forcing direction. In the former case, an order of magnitude more recurrent flows are found than previously (Chandler & Kerswell 2013) and shown to give improved predictions for the dissipation and energy pdfs of the chaos via periodic orbit theory. Over the extended torus at low forcing amplitudes, some extracted states mimick the statistics of the spatially-localised chaos present surprisingly well recalling the striking finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At higher forcing amplitudes, however, success is limited highlighting the increased dimensionality of the chaos and the need for larger data sets. Algorithmic developments to improve the extraction procedure are discussed.