Pipe flow is a prominent example among the shear flows that undergo transition to turbulence without mediation by a linear instability of the laminar profile. Experiments on pipe flow, as well as plane Couette and plane Poiseuille flow, show that triggering turbulence depends sensitively on initial conditions, that between the laminar and the turbulent states there exists no intermediate state with simple spatial or temporal characteristics, and that turbulence is not persistent, i.e., it can decay again, if the observation time is long enough. All these features can consistently be explained on the assumption that the turbulent state corresponds to a chaotic saddle in state space. The goal of this review is to explain this concept, summarize the numerical and experimental evidence for pipe flow, and outline the consequences for related flows.

The linear stability of pipe flow implies that only perturbations of sufficient strength will trigger the transition to turbulence. In order to determine this threshold in perturbation amplitude we study the edge of chaos which separates perturbations that decay towards the laminar profile and perturbations that trigger turbulence. Using the lifetime as an indicator and methods developed in Skufca et al., Phys. Rev. Lett. 96, 174101 (2006), we show that superimposed on an overall $1/\mathrm{Re}$ scaling predicted and studied previously there are small, nonmonotonic variations reflecting folds in the edge of chaos. By tracing the motion in the edge we find that it is formed by the stable manifold of a unique flow field that is dominated by a pair of downstream vortices, asymmetrically placed towards the wall. The flow field that generates the edge of chaos shows intrinsic chaotic dynamics.

We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in one spatial dimension. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.

Taming Turbulence: When fluid flows through a pipe, if the inertial forces are increased or the viscosity is decreased, the flow will become increasing noisy and will shift from being laminar to turbulent. Turbulence can be triggered by roughness in the pipe or other irregularities, which cause local eddies that grow into full-scale disruption of the otherwise smooth flow. Hof et al. (p. 1491 ; see the Perspective by McKeon ) show that a continuous turbulent eddy, downstream, eliminates the growth of upstream disturbances and can prevent the overall flow from becoming turbulent. Unlike many other control methods, the energy cost for implementing this strategy is less than the benefit gained by maintaining a laminar flow.