Coarse graining the state space of a turbulent flow using periodic orbits

TL;DR

This paper demonstrates that turbulent flow dynamics in a 3D Navier-Stokes simulation can be effectively modeled as transitions between neighborhoods of unstable periodic orbits, reducing complexity to a 17-node Markov chain.

Contribution

It introduces a novel coarse-grained model of turbulence dynamics by representing the system as a Markov chain of periodic orbit neighborhoods, simplifying high-dimensional flow data.

Findings

The Markov chain accurately reproduces long-term averages of flow observables.

Turbulent dynamics can be captured by transitions among a small set of unstable periodic orbits.

The approach reduces the complexity of turbulent flow analysis from over 10^5 degrees of freedom to 17 states.

Abstract

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than $10^5$ degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.