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Conditions for validity of mean flow stability analysis

Published online by Cambridge University Press:  06 June 2016

Samir Beneddine ,
Denis Sipp ,
Anthony Arnault ,
Julien Dandois  and
Lutz Lesshafft
Samir Beneddine*
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Denis Sipp
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Anthony Arnault
Affiliation:
ONERA-DAAP, 8 rue des Vertugadins, 92190 Meudon, France
Julien Dandois
Affiliation:
ONERA-DAAP, 8 rue des Vertugadins, 92190 Meudon, France
Lutz Lesshafft
Affiliation:
LadHyX, CNRS-Ecole Polytechnique, 91128 Palaiseau CEDEX, France
*
†Email address for correspondence: samir.beneddine@onera.fr
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Abstract

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

JFM classification

Instability: Instability Wakes/Jets: Separated flows Turbulent Flows: Turbulent Flows

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Papers
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Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 485 - 504
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© 2016 Cambridge University Press 

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