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Ambiguity in mean-flow-based linear analysis

Published online by Cambridge University Press:  06 August 2020

U. Karban*
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France
B. Bugeat
Affiliation:
Department of Engineering, University of Cambridge, Cambridge CB2 3AP, UK
E. Martini
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
A. Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI48109, USA
A. V. G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
L. Lesshafft
Affiliation:
Laboratoire d'Hydrodynamique, CNRS – École Polytechnique – Institut Polytechnique de Paris, Palaiseau, France
A. Agarwal
Affiliation:
Department of Engineering, University of Cambridge, Cambridge CB2 3AP, UK
P. Jordan
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France
T. Colonius
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA91125, USA
*
Email address for correspondence: ugur.karban@univ-poitiers.fr

Abstract

Linearisation of the Navier–Stokes equations about the mean of a turbulent flow forms the foundation of popular models for energy amplification and coherent structures, including resolvent analysis. While the Navier–Stokes equations can be equivalently written using many different sets of dependent variables, we show that the properties of the linear operator obtained via linearisation about the mean depend on the variables in which the equations are written prior to linearisation, and can be modified under nonlinear transformation of variables. For example, we show that using primitive and conservative variables leads to differences in the singular values and modes of the resolvent operator for turbulent jets, and that the differences become more severe as variable-density effects increase. This lack of uniqueness of mean-flow-based linear analysis provides new opportunities for optimising models by specific choice of variables while also highlighting the importance of carefully accounting for the nonlinear terms that act as a forcing on the resolvent operator.

JFM classification

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JFM Rapids
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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