We study the dynamics of localised perturbations in plane Couette flow with periodic lateral boundary conditions. For small Reynolds number and small amplitude of the initial state the perturbation decays on a viscous time scale $t \propto Re$. For Reynolds number larger than about 200, chaotic transients appear with life times longer than the viscous one. Depending on the type of the perturbation isolated initial conditions with infinite life time appear for Reynolds numbers larger than about 270--320. In this third regime, the life time as a function of Reynolds number and amplitude is fractal. These results suggest that in the transition region the turbulent dynamics is characterised by a chaotic repeller rather than an attractor.Comment: 4 pages, Latex, 4 eps-figures, submitted to Phys. Rev. Le

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.

We study in numerical experiments in plane Couette flow the possibility to follow a turbulent state created at higher Reynolds numbers down to lower ones. For annealing rates faster than the escape time but slower than the relaxation rate to the turbulent state turbulence persisted for some time down to Reynolds numbers around 240. This is less than the accepted value for the transition to turbulence (about 320) but still larger than the lowest bifurcation point for stationary states (about 125).

Pipe flow and many other shear flows show a transition to turbulence at flow rates for which the laminar profile is stable against infinitesimal perturbations. In this brief review the recent progress in the understanding of this transition will be summarized, with a focus on the linear and nonlinear states that drive the transitions, the extended and localized patterns that appear, and on the spatio-temporal dynamics and their relation to directed percolation.