On stability of streamwise streaks and transition thresholds in plane channel flows

Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by (SV) streamwise vortices double right arrow streamwise streaks double right arrow streak breakdown double right arrow transition. The oblique wave scenario is described by (OW) oblique waves double right arrow streamwise vortices double right arrow streamwise streaks double right arrow streak breakdown double right arrow transition. The purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers. We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers alpha(0) satisfying 0 < alpha(min) < \alpha(0)\ < alpha(max). In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations. In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien-Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios(SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.