Disturbance energy amplification in three-dimensional channel flows

We investigate energy amplification in parallel channel flows, where background noise is modeled as stochastic excitation of the linearized Navier-Stokes equations. This amplification is quantified as an H² norm of an infinite dimensional system. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves O(R³) amplification. Our basic technical tools are 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 computed. We show how the amplification mechanism is due to a coupling between normal velocity and vorticity disturbances, which in turn is due to non-zero mean shear and disturbance spanwise variation. The implications of very large H² norms, and the corresponding frequency response shapes are interpreted in the context of predicting transition to turbulence and the accompanying coherent structures