Nonmodal amplitude equations

We consider fluid flows for which the linearized Navier-Stokes operator is strongly nonnormal. The responses of such flows to external perturbations are spanned by a generically very large number of nonorthogonal eigenmodes. They are therefore qualified as "nonmodal" responses, to insist on the inefficiency of the eigenbasis to describe them. In the aim of the article to reduce the system to a lower-dimensional one free of spatial degrees of freedom, (eigen)modal reduction techniques, such as the center manifold, are thus inappropriate precisely because the leading-order dynamics cannot be restricted to a low-dimensional eigensubspace. However, it is often true that only a small number (we assume only one) of singular modes is sufficient to reconstruct the nonmodal responses at the leading order. By adopting the latter paradigm, we propose a general method to analytically derive a weakly nonlinear amplitude equation for the nonmodal response of a fluid flow to a small harmonic forcing, stochastic forcing, and initial perturbation, respectively. In these last two problems, we assumed a parallel base flow with a spatially monochromatic external excitation. The present approach for deriving nonmodal amplitude equations is simpler than the one we previously proposed, for neither the operator perturbation nor the ensuing compatibility condition is formally necessary. Additionally, it provides an explicit, rigorous treatment of the suboptimal responses. When applied to the stochastic response, the present method makes it possible to derive an amplitude equation that is substantially easier to solve and interpret than the one we proposed previously. Eventually, the three derived amplitude equations are tested in three distinct, two- and three-dimensional flows. For sufficiently small excitation amplitudes, yet up to values large enough for the flow to depart from the linear regime, they can systematically predict the weakly nonlinear modification of the gains as the amplitude of the external excitation is increased. This, at an extremely low numerical cost as compared to fully nonlinear techniques. However, the proposed weakly nonlinear approach, precisely because it condemns the flow spatial structure to be mostly along the leading singular mode, is often too simplistic to predict the occurrence of subcritical transitions as the excitation amplitude is increased to too large values.