Weak nonlinearity for strong non-normality

Abstract

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and four application cases highlight common non-normal mechanisms in hydrodynamics: the streamwise convective non-normal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.

Weakly nonlinear evolution of stochastically driven non-normal systems

“…By generalising the method proposed in Ducimetière et al. (2022), we have derived an amplitude equation describing the weakly nonlinear, statistically steady response to stochastic forcing in arbitrarily stable yet non-normal systems. The amplitude depends solely on the frequency, and is therefore independent of the number of spatial degrees of freedom of the discretised system.…”

“…In our previous work (Ducimetière, Boujo & Gallaire 2022), we derived an amplitude equation to prolong the linear harmonic gain (and transient growth) curve in a weakly nonlinear regime by increasing the forcing amplitude. The method does not rely on modal quantities, in contrast to classical techniques, and the present paper aims at generalising this method to the response to stochastic forcing.…”

A weakly nonlinear amplitude equation approach to the bypass transition in the two-dimensional Lamb–Oseen vortex

“…We have derived an amplitude equation for non-normal systems, describing the transient response to an initial condition, in a weakly nonlinear regime. Unlike in Ducimetière et al (2022), the reference state of these systems can now depend arbitrarily on times, owing to the propagator formalism, without the need for this latter to take its particular operator exponential shape. This offers numerous possibilities of applications, and weak nonlinearities could be modelled, for instance, in pulsating pipe flows, which play a key role in the hemodynamic system of many species (Pier & Schmid 2017; it could also be applied to time-dependent stratified shear flows, which have revealed to support strong transient growth, for instance in Arratia, Caulfield & Chomaz (2013) and Parker et al (2021).…”

“…In reality, may be sufficiently large for the nonlinear corrections to the response not to be negligible anymore, thus for the transient gain to depart from its linear prediction. Building on our previous work (Ducimetière, Boujo & Gallaire 2022), we propose thereafter a method for capturing weakly nonlinear effects on the transient gain over a time-varying base flow.…”

Capturing the edge of chaos as a spectral submanifold in pipe flows

“…Starting with the discovery of the upper-and lower-branch fixed points of plane Couette flow by Nagata (1990), an extensive library of exact coherent states (ECS) has linear models. For example, Ducimetière, Boujo & Gallaire (2022) used the spectrum of the resolvent operator and multiple-scale expansion to derive Stuart-Landau-type amplitude equations (Landau 1959) for flows exhibiting non-normality. Among other examples, the amplitude equations were then used to predict the energy of the response observed in a plane Poiseuille flow subjected to harmonic forcing.…”