Nonmodal amplitude equations

We propose a general method to analytically derive a weakly nonlinear amplitude equation for the nonmodal response of a fluid flow to a harmonic forcing, a stochastic forcing, and an initial perturbation, respectively. Although ultimately recovering our previous results [1], the present approach is simpler, for neither the operator perturbation nor the ensuing compatibility condition were formally necessary. Additionally, it provides an explicit, rigorous treatment of the sub-optimal responses. The present method is also exploited to derive an amplitude equation for the stochastic response which is substantially easier to solve and interpret than the one we proposed previously. We also emphasize that despite being concerned with the response to external disturbances of three different natures, the derivations of the amplitude equations all proceed by the same general principle. This height of view was missing in our previous publications, whereas it is useful both for physical understanding and also to facilitate the adaptation of the method to some other types of external disturbances.

Eventually, the three amplitude equations that are derived are tested in parallel and non-parallel, two and three-dimensional flows. At extremely low numerical cost as compared to fully nonlinear techniques, they can predict the weakly nonlinear modification of the gains as the flow departs from the linear regime by increasing the amplitude of the external excitation. Owing to their simplicity, amplitude equations also furnish physical interpretations of the weakly nonlinear mechanisms at work.

[1] Ducimetière Y-M, Boujo E, Gallaire F. Weak nonlinearity for strong non-normality. Journal of Fluid Mechanics. 2022;947:A43. doi:10.1017/jfm.2022.664