On Stability of Streamwise Streaks and Transition Thresholds in Plane Channel Flows

Streak breakdown caused by a spanwise in ectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette ow. The streamwise vortex scenario is described by (SV) streamwise vortices =) streamwise streaks =) streak breakdown =) transition: The oblique wave scenario is described by (OW) oblique waves =) streamwise vortices =) streamwise streaks =) streak breakdown =) 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, two-dimensional optimals, and random noise at subcritical Reynolds numbers. We nd that if instability occurs, it is con ned to disturbances with streamwise wavenumbers 0 satisfying 0 < min < j 0j < max. In these unstable cases, the least stable mode is located near the center of the channel with a phase velocity approximately equal to the centerline velocity of the mean ow. Growth rates for instability increase with streak amplitude. For Couette ow 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 nal part of the paper we show that the threshold energies for transition in Poiseuille ow 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 two-dimensional optimals. 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) scenario and two-dimensional optimals at Reynolds number 1500. 2