The Spatio-Temporal Impulse Response of the Linearized Navier-Stokes Equations

In this paper, we perform an input-output analysis of the linearized Navier-Stokes equations by investigating their spatio-temporal ’impulse response’. Our first motivation for this problem comes from experimental work, where transition to turbulence is observed to occur either naturally, with so-called ’turbulent spots’, or when these spots are intentionally induced in the flow by deliberate external excitation, which can be modeled as a spatio-temporal impulse body force in the Navier-Stokes equations. We compare this computed spatio-temporal impulse response to observed characteristics of turbulent spots and find significant qualitative agreement. Our other motivation is to fully probe the dynamics of the linearized equations, and we find that they exhibit rich and complex structures hitherto unseen by other analysis methods.