The onset of shear flow turbulence

Shear flow turbulence is a complex and interesting phenomenon which has been well known for a long time. The story of the physical understanding of its onset is a marvellous and recent tale of a unique nonlinearity. It is the convection term $\left(\mathbf{u}\cdot \mathbf{\nabla }\right)\mathbf{u}$ in the equation of motion which rules the transition to turbulence by introducing a subtle interplay between the nonnormal bunching of flow perturbations and their nonlinear interaction. By nonnormal coupling to the basic laminar flow, those disturbances which are misfit to the eigendirections can transiently draw energy from the laminar shear flow and grow before they ultimately fade away. If their intermediate growth is strong enough, they interact nonlinearly. This interaction recreates new misfit flow amplitudes, which again start drawing energy, grow transiently, interact, etc. This feedback loop is able to sustain a flow which, as a consequence of the nonlinearity, is spatiotemporally deterministic and irregular at the same time. We call this flow turbulent. The feedback mechanism has to be contrasted to that of a common instability. The laminar flow stays linearly stable, all eigenvalues have negative, damping real parts. That is, laminar shear flow passes to turbulence without (linear) instability. The onset of turbulence is the nonnormal-nonlinear performance of many degrees of freedom and not merely of a single, unstable one. Both features are mediated by the convective nonlinearity.