A ``resonant'' spanwise perturbation frequency in streamwise-constant Couette flow

Turbulence in plane Couette flow is dominated by streamwise elongated structures that are approximately spanwise periodic with a preferred spatial frequency. It has been postulated that these approximately streamwise-constant coherent structures develop due to streamwise vortices in the flow. We investigate this idea by considering a streamwise-constant (2D/3C) model of plane Couette flow. We introduce streamwise vortices by imposing spanwise periodic cross-stream perturbations on the flow field and study it's energy amplification under stochastic disturbances. The periodic nature of the resulting equations allows us to cast the system into a convenient, so-called ``lifted,'' form that retains the periodic coefficients in the analysis. We can then efficiently solve for the energy amplification using a perturbation approach on the associated Lyapunov equation. Our results show the existence of a peak or ``resonant'' spanwise frequency that maximizes the disturbance amplification, suggesting that the 2D/3C equations capture the type of (spanwise frequency) selective mechanism that leads to spanwise periodic structures common in fully developed flows.