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

Published online by Cambridge University Press:  12 August 2020

Alberto Padovan Open the ORCID record for Alberto Padovan [Opens in a new window] ,
Samuel E. Otto  and
Clarence W. Rowley Open the ORCID record for Clarence W. Rowley [Opens in a new window]
Alberto Padovan*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08540, USA
Samuel E. Otto
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08540, USA
Clarence W. Rowley
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08540, USA
*
†Email address for correspondence: apadovan@princeton.edu
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Abstract

We propose a framework that elucidates the input–output characteristics of flows with complex dynamics arising from nonlinear interactions between different time scales. More specifically, we consider a periodically time-varying base flow, and perform a frequency-domain analysis of periodic perturbations about this base flow. The response of these perturbations is governed by the harmonic resolvent, which is a linear operator similar to the harmonic transfer function introduced by Wereley (1991 Analysis and control of linear periodically time-varying systems, PhD thesis, Massachusetts Institute of Technology). This approach makes it possible to explicitly capture the triadic interactions that are responsible for the energy transfer between different time scales in the flow. For instance, perturbations at frequency $\omega$ are coupled with perturbations at frequency $\alpha$ through the base flow at frequency $\omega -\alpha$. We draw a connection with resolvent analysis, which is a special case of the harmonic resolvent when evaluated about a steady base flow. We show that the left and right singular vectors of the harmonic resolvent are the optimal response and forcing modes, which can be understood as full spatio-temporal signals that reveal space–time amplification characteristics of the flow. Finally, we illustrate the method on examples, including a three-dimensional system of ordinary differential equations and the flow over an airfoil at near-stall angle of attack.

JFM classification

Flow Control: Control theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A14
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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