From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

TL;DR

This paper investigates a sequence of bifurcations in pipe flow leading from simple travelling waves to chaotic states, revealing the underlying dynamical mechanisms and symmetry properties involved in transition to turbulence.

Contribution

It identifies the bifurcation cascade and heteroclinic tangles responsible for mild chaos in pipe flow, linking invariant manifold interactions to turbulent dynamics.

Findings

Successive transitions produce diverse time-dependent solutions.

Chaotic states are confined within specific Reynolds number ranges.

Symmetry considerations facilitate bifurcation analysis.

Abstract

We study numerically a succession of transitions in pipe Poiseuille flow that leads from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade possess a shift-reflect symmetry and are both axially and azimuthally periodic with wave numbers {\kappa} = 1.63 and n = 2, respectively. As the Reynolds number is increased, successive transitions result in a wide range of time dependent solutions that includes spiralling, modulated-travelling, modulated-spiralling, doubly-modulated-spiralling and mildly chaotic waves. We show that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift-reflect-symmetric modulated-travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The states studied here belong to a subspace of discrete symmetry which makes many of the bifurcation and path-following investigations presented technically feasible. However, we expect that most of the phenomenology carries over to the full state-space, thus suggesting a mechanism for the formation and break-up of invariant states that can sustain turbulent dynamics.