Revealing the state space of turbulent pipe flow by symmetry reduction

TL;DR

This paper applies symmetry reduction to turbulent pipe flow, transforming complex flow states into simpler forms, revealing the organization of turbulence and identifying key solutions like equilibria and periodic orbits.

Contribution

The study introduces symmetry reduction via the method of slices to pipe flow, enabling visualization and analysis of turbulent dynamics and solution structures in a reduced state space.

Findings

All travelling waves reduce to equilibria in the symmetry-reduced space.

Identification of relative periodic orbits embedded within the chaotic attractor.

Visualization of unstable manifolds and connections between solutions.

Abstract

Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear flows. Visualisations of the flow within the slice and its linearisation at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions, and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic attractor, which capture turbulent dynamics at transitional Reynolds numbers.