Weak nonlinearity for strong nonnormality

TL;DR

This paper develops a theoretical framework to derive amplitude equations for weakly nonlinear, nonnormal dynamical systems, capturing transient growth phenomena in fluid flows through a multiple-scale asymptotic expansion.

Contribution

It introduces a novel approach that combines nonmodal analysis with center manifold theory for weakly nonlinear nonnormal systems, applicable to fluid dynamics.

Findings

Unified treatment of transient growth and harmonic response

Application to flow past backward-facing step, plane Poiseuille flow

Identification of common nonnormal mechanisms in hydrodynamics

Abstract

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of nonnormal dynamical systems when they experience transient growth or respond to harmonic forcing. This approach reconciles the nonmodal nature of these growth mechanisms and the need for a center manifold to project the leading-order dynamics. Under the hypothesis of strong nonnormality, 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 nonnormal mechanisms in hydrodynamics: the streamwise convective nonnormal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.