Streamwise-constant large-scale structures in Couette and Poiseuille flows

The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr--Sommerfeld and Squire operators are each considered separately. Physically this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. In this way we exploit the fact that, for streamwise-constant fluctuations, the dynamics governing the wall-normal velocity are independent of the mean velocity profile (and therefore the mean shear). The analysis is performed for both plane Couette flow and plane Poiseuille flow; and for each we consider linear amplification mechanisms about both the laminar and turbulent mean velocity profiles. The analysis reveals two things. First, that the most amplified structures (with a spanwise spacing of approximately $4h$, where $h$ is the channel half height) are to a great extent encoded in the Orr-Sommerfeld operator alone, thus helping to explain their prevalence. Second---and consistent with numerical and experimental observations---that Couette flow is significantly more efficient than Poiseuille flow in leveraging wall-normal velocity fluctuations to produce large-scale streamwise streaks.