# Relative periodic orbits form the backbone of turbulent pipe flow

## TL;DR

This paper investigates the role of relative periodic orbits in turbulent pipe flow, demonstrating their embedding in the chaotic saddle and their guiding influence on turbulence dynamics in high-dimensional Navier-Stokes systems.

## Contribution

It provides a detailed analysis of relative periodic orbits in pipe flow, showing their significance in turbulent dynamics and extending the understanding from low-dimensional systems.

## Key findings

- Relative periodic orbits are embedded in the turbulent saddle.
- These orbits guide the chaotic turbulent dynamics.
- A library of invariant solutions was identified using symmetry reduction.

## Abstract

Chaotic dynamics of low-dimensional systems, such as Lorenz or R\"ossler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in lower-dimensional systems---is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.