Direct path from turbulence to time-periodic solutions

TL;DR

This paper uncovers the geometric pathways linking laminar flow to turbulence in pipe and channel flows, revealing how turbulence emerges at lower Reynolds numbers and transitions to stochastic behavior.

Contribution

It identifies the complete reversible path from turbulent to invariant solutions in Navier-Stokes flows, advancing understanding of turbulence onset at lower Re.

Findings

Turbulence can be traced back to lower Reynolds numbers than previously known.

A reversible path linking turbulent and invariant solutions is identified.

Transition to stochastic dynamics coincides with attractor dimension explosion.

Abstract

Viscous flows through pipes and channels are steady and ordered until, with increasing velocity, the laminar motion catastrophically breaks down and gives way to turbulence. How this apparently discontinuous change from low- to high-dimensional motion can be rationalized within the framework of the Navier--Stokes equations is not well understood. Exploiting geometrical properties of transitional channel flow we trace turbulence to far lower Reynolds numbers (Re) than previously possible and identify the complete path that reversibly links fully turbulent motion to an invariant solution. This precursor of turbulence destabilizes rapidly with Re, and the accompanying explosive increase in attractor dimension effectively marks the transition between deterministic and de facto stochastic dynamics.