Complexity of the laminar-turbulent boundary in pipe flow

TL;DR

This paper investigates the laminar-turbulent boundary in pipe flow, revealing a new traveling wave solution and analyzing the complex structure of the boundary using symmetry reduction and dynamical systems techniques.

Contribution

It introduces a novel symmetry-reduction method for analyzing invariant solutions and uncovers a previously unknown traveling wave on the laminar-turbulent boundary.

Findings

Discovery of a new traveling wave solution on the boundary

Visualization of dynamical paths connecting solutions to turbulence

Identification of different regions on the boundary influenced by invariant solutions

Abstract

Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry-reduction, which we formulate here for the first time for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.