Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

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, 2π]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 [G. J. Chandler and R. R. Kerswell, “Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow,” J. Fluid Mech. 722, 554–595 (2013)] and shown to give improved predictions for the dissipation and energy pdfs of the chaos via periodic orbit theory. Analysis of the recurrent flows shows that the energy is largely trapped in the smallest wavenumbers through a combination of the inverse cascade process and a feature of the advective nonlinearity in 2D. Over the extended torus at low forcing amplitudes, some extracted states mimic the statistics of the spatially localised chaos present surprisingly well recalling the findings of Kawahara and Kida [“Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst,” J. Fluid Mech. 449, 291 (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.