Turbulence Transition in Shear Flows: Chaos in High-Dimensional Spaces

The study of the transition to turbulence in parallel shear flows without linear instability of the laminar profile has profited immensely from the application of dynamical systems ideas. Studies of the transition in plane Couette flow and pipe flow, in particular, have shown that the transition is connected with the appearance of 3-d coherent structures that form a chaotic saddle which shows up in a transient turbulent dynamics. It is remarkable that these concepts, initially developed for low-dimensional systems, also work in such a high-dimensional setting. The present note contains a brief summary of key features and a short list of references for further reading.