Weak nonlinearity for strong nonnormality
In this work with Y.-M. Ducimetière and E. Boujo, 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 several application cases which 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.
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Bicentenaire des équations de Navier-Stokes, Cité des Sciences et de l’industrie, Paris, 5 Juillet 2023
Summer School on Wave Turbulence, MIT, Cambridge, USA, 24 Juillet 2023
200 Years of Navier-Stokes Equation and Turbulences, Les Houches, France, 31 Juillet 2023