The critical point of the transition to turbulence in pipe flow

• Published in: Journal of fluid mechanics 2018-03, Vol.839, p.76-94
• Main Authors: Mukund, Vasudevan, Hof, Björn
• Format: Article
• Language: English
• Subjects: Boundary conditions; Clustering; Computational fluid dynamics; Critical point; Critical velocity; Decay rate; Flow pattern; Flow stability; Fluid flow; Fluids; Friction; Initial conditions; Interactions; JFM Papers; Laminar flow; Percolation; Pipe flow; Pipes; Reynolds number; Statistical methods; Statistics; Steady state; Studies; Turbulence; Turbulent flow; Velocity
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2017.923

In pipes, turbulence sets in despite the linear stability of the laminar Hagen–Poiseuille flow. The Reynolds number ( $Re$ ) for which turbulence first appears in a given experiment – the ‘natural transition point’ – depends on imperfections of the set-up, or, more precisely, on the magnitude of finite amplitude perturbations. At onset, turbulence typically only occupies a certain fraction of the flow, and this fraction equally is found to differ from experiment to experiment. Despite these findings, Reynolds proposed that after sufficiently long times, flows may settle to steady conditions: below a critical velocity, flows should (regardless of initial conditions) always return to laminar, while above this velocity, eddying motion should persist. As will be shown, even in pipes several thousand diameters long, the spatio-temporal intermittent flow patterns observed at the end of the pipe strongly depend on the initial conditions, and there is no indication that different flow patterns would eventually settle to a (statistical) steady state. Exploiting the fact that turbulent puffs do not age (i.e. they are memoryless), we continuously recreate the puff sequence exiting the pipe at the pipe entrance, and in doing so introduce periodic boundary conditions for the puff pattern. This procedure allows us to study the evolution of the flow patterns for arbitrary long times, and we find that after times in excess of $10^{7}$ advective time units, indeed a statistical steady state is reached. Although the resulting flows remain spatio-temporally intermittent, puff splitting and decay rates eventually reach a balance, so that the turbulent fraction fluctuates around a well-defined level which only depends on $Re$ . In accordance with Reynolds’ proposition, we find that at lower $Re$ (here 2020), flows eventually always resume to laminar, while for higher $Re$ ( ${\geqslant}2060$ ), turbulence persists. The critical point for pipe flow hence falls in the interval of $2020