[Paper Review] Simple invariant solutions embedded in 2D Kolmogorov turbulence

Gary J. Chandler, Rich R. Kerswell|arXiv (Cornell University)|Jul 19, 2012

Quantum chaos and dynamical systems19 references3 citations

TL;DR

This paper proposes extracting simple invariant solutions—exact recurrent flows—from long direct numerical simulations (DNS) of 2D Kolmogorov turbulence to reconstruct turbulence statistics using principles inspired by Periodic Orbit Theory. At low Reynolds number (Re=40), 50 recurrent flows were successfully identified, primarily in the most populated phase-space region, enabling a reasonable reproduction of energy and dissipation probability density functions, though the method fails at higher Re due to insufficient data length.

ABSTRACT

We consider long simulations of 2D Kolmogorov turbulence body-forced by $\sin4y \ex$ on the torus $(x,y) \in [0,2π]^2$ with the purpose of extracting simple invariant sets or `exact recurrent flows' embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics in the spirit of Periodic Orbit Theory derived for certain types of low dimensional chaos. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation probability density functions. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.

Motivation & Objective

• To identify simple invariant solutions—exact recurrent flows—embedded in 2D Kolmogorov turbulence using long direct numerical simulations.
• To test whether these recurrent flows can reconstruct key turbulence statistics, inspired by Periodic Orbit Theory from low-dimensional chaos.
• To assess the feasibility of using recurrent flows as a basis for predicting turbulence behavior in higher-dimensional, fully turbulent systems.
• To evaluate the impact of simulation length on the number and diversity of extractable recurrent flows, especially in the asymptotic turbulent regime.
• To explore the potential for automation and extrapolation of recurrent flow sets across different Reynolds numbers.

Proposed method

• Conducting long direct numerical simulations (DNS) of 2D Kolmogorov turbulence on a torus with monochromatic forcing sin(4y) in the x-direction.
• Using near-recurrence detection to identify trajectories that return close to their initial state, indicating the presence of periodic or recurrent behavior.
• Applying arc-length continuation to refine approximate recurrent solutions into exact invariant solutions (i.e., closed orbits in phase space).
• Evaluating the stability of each extracted recurrent flow to assess its dynamical significance and potential role in the turbulent attractor.
• Using the set of extracted recurrent flows to reconstruct turbulence statistics such as energy and dissipation probability density functions via weighted sums.
• Assessing the success of reconstruction by comparing the weighted sum of recurrent flow properties to the full DNS statistics.

Experimental results

Research questions

• RQ1 Can exact recurrent flows be reliably extracted from long DNS of 2D Kolmogorov turbulence?
• RQ2 To what extent can these recurrent flows reproduce the statistical properties of the turbulence, such as energy and dissipation distributions?
• RQ3 How does the number and quality of extracted recurrent flows depend on the Reynolds number and simulation length?
• RQ4 Is the approach based on Periodic Orbit Theory viable for reconstructing turbulence statistics in higher-dimensional systems like 2D turbulence?
• RQ5 Can the extracted recurrent flows be used to predict statistics at different Reynolds numbers through continuation and reweighting?

Key findings

• At Re=40, 50 recurrent flows were successfully extracted, with the majority located in the most frequently visited region of phase space, enabling a reasonable reconstruction of energy and dissipation probability density functions.
• The reconstruction performance at Re=40 was more reliant on the density of recurrent flows in the dominant phase-space region than on sophisticated weighting schemes.
• At Re=60, 80, and 100, the number of extractable recurrent flows was insufficient to make viable statistical predictions, primarily due to the limited length of the DNS data.
• A simulation length of 10^5 time units was barely adequate for Re=40 but was estimated to be two orders of magnitude too short for Re=60 and above.
• Despite limited success at higher Re, each extracted recurrent flow is a dynamically meaningful, exact invariant solution that represents a sustained sequence of coherent dynamical processes underlying the turbulence.
• The method is promising for automation and could, in principle, be extended to predict statistics at different Reynolds numbers if sufficiently large recurrent flow sets are available.