Sensitive dependence on initial conditions in transition to turbulence in pipe flow

The experiments by Darbyshire and Mullin (J. Fluid Mech. 289, 83 (1995)) on the transition to turbulence in pipe flow show that there is no sharp border between initial conditions that trigger turbulence and those that do not. We here relate this behaviour to the possibility that the transition to turbulence is connected with the formation of a chaotic saddle in the phase space of the system. We quantify a sensitive dependence on initial conditions and find in a statistical analysis that in the transition region the distribution of turbulent lifetimes follows an exponential law. The characteristic mean lifetime of the distribution increases rapidly with Reynolds number and becomes inaccessibly large for Reynolds numbers exceeding about 2200. Suitable experiments to further probe this concept are proposed.Comment: 10 pages, 12 figures; submitted to J. Fluid Mec

Transition from the Couette-Taylor system to the plane Couette system

We discuss the flow between concentric rotating cylinders in the limit of large radii where the system approaches plane Couette flow. We discuss how in this limit the linear instability that leads to the formation of Taylor vortices is lost and how the character of the transition approaches that of planar shear flows. In particular, a parameter regime is identified where fractal distributions of life times and spatiotemporal intermittency occur. Experiments in this regime should allow to study the characteristics of shear flow turbulence in a closed flow geometry.Comment: 5 pages, 5 figure

Travelling waves in pipe flow

A family of three-dimensional travelling waves for flow through a pipe of circular cross section is identified. The travelling waves are dominated by pairs of downstream vortices and streaks. They originate in saddle-node bifurcations at Reynolds numbers as low as 1250. All states are immediately unstable. Their dynamical significance is that they provide a skeleton for the formation of a chaotic saddle that can explain the intermittent transition to turbulence and the sensitive dependence on initial conditions in this shear flow.Comment: 4 pages, 5 figure

Dynamical systems and the transition to turbulence in linearly stable shear flows

Plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures, which transiently show up in the temporal evolution of the turbulent flow. Here we summarize the evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow and on the coherent structures in pipe flow