Transition Threshold and the Self-Sustaining Process

Abstract

The self-sustaining process is a fundamental and generic three-dimensional nonlinear process in shear flows. It is responsible for the existence of non-trivial traveling wave and time-periodic states. These states come in pairs, an upper branch and a lower branch. The limited data available to date suggest that the upper branch states provide a good first approximation to the statistics of turbulent flows. The upper branches may thus be understood as the “backbone” of the turbulent attractor while the lower branches might form the backbone of the boundary separating the basin of attraction of the laminar state from that of the turbulent state. Evidence is presented that the lower branch states tend to purely streaky flows, in which the streamwise velocity has an essential spanwise modulation, as the Reynolds number R tends to infinity. The streamwise rolls sustaining the streaks and the streamwise undulation sustaining the rolls, both scale like R ⁻¹ in amplitude, just enough to overcome viscous dissipation. It is argued that this scaling is directly related to the observed R ⁻¹ transition threshold. These results also indicate that the exact coherent structures never bifurcate from the laminar flow, not even at infinity. The scale of the key elements, streaks, rolls and streamwise undulation, remain of the order of the channel size. However, the higher x-harmonics show a slower decay with R than naively expected. The results indicate the presence of a warped critical layer.