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We propose a framework that elucidates the input-output characteristics offlows with complex dynamics arising from nonlinear interactions betweendifferent time scales. More specifically, we consider a periodicallytime-varying base flow, and perform a frequency-domain analysis of periodicperturbations about this base flow; the response of these perturbations isgoverned by the harmonic resolvent, which is a linear operator similar to theharmonic transfer function introduced by Wereley (1991). This approach makes itpossible to explicitly capture the triadic interactions that are responsiblefor the energy transfer between different time scales in the flow. Forinstance, perturbations at frequency $\alpha$ are coupled with perturbations atfrequency $\omega$ through the base flow at frequency $\omega-\alpha$. We drawa connection with resolvent analsyis, which is a special case of the harmonicresolvent when evaluated about a steady base flow. We show that the left andright singular vectors of the harmonic resolvent are the optimal response andforcing modes, which can be understood as full spatio-temporal signals thatreveal space-time amplification characteristics of the flow. We illustrate themethod on examples, including a three-dimensional system of ordinarydifferential equations and the flow over an airfoil at near-stall angle ofattack.

Analysis of amplification mechanisms and cross-frequency interactions in nonlinear flows via the harmonic resolvent