The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow

TL;DR

This paper explores how localized solutions in pipe flow originate from global traveling waves through symmetry-breaking bifurcations, offering a new pathway to understand transition to turbulence.

Contribution

It demonstrates the emergence of localized solutions from global solutions via symmetry-breaking bifurcations in pipe flow, providing a constructive method to generate localized invariant sets.

Findings

Localized solutions originate from global waves via bifurcations

Symmetry-breaking bifurcations lead to localized orbits

Global solutions can be systematically transformed into localized solutions

Abstract

The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can co-exist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate the origin of the first localized solution in a modulational symmetry-breaking Hopf bifurcation from a known global travelling wave that has 2-fold rotational symmetry about the pipe axis. Similar behaviour is found for a global wave of 3-fold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that all global solutions should be expected to lead to more realistic localised counterparts through such bifurcations, which provides a constructive route for their generation.