Mathematically established chaos and forecast of statistics with recurrent patterns in Taylor-Couette flow

TL;DR

This paper mathematically demonstrates chaos in Taylor-Couette flow, showing how unstable periodic orbits underpin turbulence and enabling statistical predictions of chaotic fluid behavior.

Contribution

It establishes a rigorous mathematical link between chaos, unstable periodic orbits, and turbulence in Taylor-Couette flow, with a novel approach to reconstruct flow statistics from UPOs.

Findings

Chaotic dynamics are well approximated by a simple discrete map.

Unstable periodic orbits correspond to the chaotic set in the flow.

Statistical properties of turbulence can be reconstructed from UPOs.

Abstract

The transition to chaos in the subcritical regime of counter-rotating Taylor-Couette flow is investigated using a minimal periodic domain capable of sustaining coherent structures. Following a Feigenbaum cascade, the dynamics are found to be remarkably well approximated by a simple discrete map that admits rigorous proof of its chaotic nature. The chaotic set that arises for the map features densely distributed periodic points that are in one-to-one correspondence with unstable periodic orbits (UPOs) of the Navier-Stokes system. This supports the increasingly accepted view that UPOs may serve as the backbone of turbulence and, indeed, we demonstrate that it is possible to reconstruct every statistical property of chaotic fluid flow from UPOs.