Energy amplification in channel flows with stochastic excitation

We investigate energy amplification in parallel channel flows, where background noise is modeled as stochastic excitation of the linearized Navier–Stokes equations. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves ${\mathrm{O(R}}^3)$ amplification. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity fields. The dependence of these quantities on both the Reynolds number and the spanwise wave number are explicitly computed. We show how the amplification mechanism is due to a coupling between wall-normal velocity and vorticity disturbances, which in turn is due to nonzero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the non-normality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.