Symmetry reduction in high dimensions, illustrated in a turbulent pipe

TL;DR

This paper extends a symmetry reduction method to 3D fluid flows, enabling visualization of unstable manifolds and revealing the structure of turbulence through relative periodic orbits in pipe flow.

Contribution

It introduces an improved fixed Fourier mode slice approach for symmetry reduction in 3D fluid dynamics, facilitating the analysis of turbulent state space.

Findings

Discovery of many relative periodic orbits in pipe flow.

Enhanced visualization of turbulent attractor structure.

Demonstration of the method's effectiveness in complex fluid systems.

Abstract

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied 1-dimensional PDEs, to a spatially 3-dimensional fluid flow, and show that is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.