Stable Manifolds and the Transition to Turbulence in Pipe Flow

TL;DR

This paper develops a numerical method to identify specific disturbances in pipe flow that lead to transition or relaminarization, connecting laminar flow to turbulent states via stable manifolds.

Contribution

It introduces a new numerical approach to find disturbances that steer flow toward lower-branch solutions, linking laminar and turbulent states in shear flows.

Findings

Identified disturbances that approach lower-branch solutions

Demonstrated how small changes in disturbances lead to relaminarization or turbulence

Connected laminar flow, lower-branch solutions, and turbulence through stable manifolds

Abstract

Lower-branch traveling waves and equilibria computed in pipe flow and other shear flows appear intermediate between turbulent and laminar motions. We take a step towards connecting these lower-branch solutions to transition by deriving a numerical method for finding certain special disturbances of the laminar flow in a short pipe. These special disturbances cause the disturbed velocity field to approach the lower-branch solution by evolving along its stable manifold. If the disturbance were slightly smaller, the flow would relaminarize, and if slightly larger, it would transition to a turbulent state.