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A framework for input–output analysis of wall-bounded shear flows

Published online by Cambridge University Press:  28 June 2019

Mohamadreza Ahmadi Open the ORCID record for Mohamadreza Ahmadi [Opens in a new window] ,
Giorgio Valmorbida ,
Dennice Gayme Open the ORCID record for Dennice Gayme [Opens in a new window]  and
Antonis Papachristodoulou
Mohamadreza Ahmadi*
Affiliation:
Department of Mechanical Engineering, California Institute of Technology, 1200 East California Boulevard, MC 104-44, Pasadena, CA 91125, USA
Giorgio Valmorbida
Affiliation:
Laboratoire des Signaux et Systèmes, CentraleSupélec, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91192 Gif-sur-Yvette, France
Dennice Gayme
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Lathrobe Hall 223, 3400 North Charles Street, Baltimore, MD 21218, USA
Antonis Papachristodoulou
Affiliation:
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
*
†Email address for correspondence: mrahmadi@caltech.edu
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Abstract

We propose a new framework to evaluate input–output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise-constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach for examining flow properties such as energy growth, worst-case disturbance amplification and stability to persistent excitations (i.e. input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation rather than of a discretized form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input–output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.

JFM classification

Flow Control: Control theory Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 742 - 785
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Copyright
© 2019 Cambridge University Press 

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