Transition in pipe flow: the saddle structure on the boundary of turbulence

TL;DR

This paper investigates the laminar-turbulent boundary in pipe flow, revealing that trajectories are organized around a few traveling waves and their connections, with new solutions found under symmetry constraints.

Contribution

Develops a numerical technique to identify coherent structures on the laminar-turbulent boundary and discovers new traveling wave solutions in pipe flow.

Findings

Trajectories on the boundary are organized around specific traveling waves.

New types of traveling waves were identified under symmetry constraints.

Flow dynamics on the boundary involve heteroclinic connections between solutions.

Abstract

The laminar-turbulent boundary S is the set separating initial conditions which relaminarise uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow look to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognising approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on S are organised around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally-attracting travelling waves embedded within S. Four new types of travelling waves were found using this approach.