Frequency domain analysis of the linearized Navier-Stokes equations

We study the linearized Navier-Stokes (LNS) equations from an input-output point of view by analyzing their spatio-temporal frequency responses. We show how the relative roles of Tollmien-Schlichting (TS) waves and streamwise vortices and streaks can be explained as input-output resonances of the spatio-temporal frequency responses. Furthermore, we derive important conclusions about the effectiveness of input field components, and the contributions of the streamwise, wall-normal, and spanwise velocity perturbations to the kinetic energy density. In particular, we show that the wall-normal and spanwise forces have much stronger influence on the velocity field than the streamwise force. On the other hand, the velocity perturbations in the direction of a nominal flow achieve much bigger magnitudes than the perturbations in the other two spatial directions.