Relative periodic orbits in transitional pipe flow

TL;DR

This paper introduces a numerical method to identify relative periodic orbits in pipe flow, revealing their role in the transition to turbulence and their structural features near a Hopf bifurcation.

Contribution

It presents a new numerical approach to find RPOs in pipe flow and demonstrates their significance in the transition process near a Hopf bifurcation.

Findings

RPOs are found near a Hopf bifurcation from a traveling wave.

RPOs feature weakly modulated streaks as a dominant structure.

These RPOs lie on the laminar-turbulent boundary, influencing transition dynamics.

Abstract

A dynamical system description of the transition process in shear flows with no linear instability starts with a knowledge of exact coherent solutions, among them travelling waves (TWs) and relative periodic orbits (RPOs). We describe a numerical method to find such solutions in pipe flow and apply it in the vicinity of a Hopf bifurcation from a TW which looks to be especially relevant for transition. The dominant structural feature of the RPO solution is the presence of weakly modulated streaks. This RPO, like the TW from which it bifurcates, sits on the laminar-turbulent boundary separating initial conditions which lead to turbulence from those which immediately relaminarise.