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

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References

Adams, E. W. & Johnston, J. P. 1988 Effects of the separating shear layer on the reattachment flow structure. Part 2: reattachment length and wall shear stress. Exp. Fluids 6 (7), 493–499.CrossRefGoogle Scholar
Amestoy, P. R., Duff, I. S., L’Excellent, J.-Y. & Koster, J. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Applics. 23 (1), 15–41.CrossRefGoogle Scholar
Andersson, P., Henningson, D. S. & Hanifi, A. 1998 On a stabilization procedure for the parabolic stability equations. J. Engng Maths 33 (3), 311–332.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Boujo, E. & Gallaire, F. 2015 Sensitivity and open-loop control of stochastic response in a noise amplifier flow: the backward-facing step. J. Fluid Mech. 762, 361–392.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5 (3), 774–777.Google Scholar
Cambier, L., Heib, S. & Plot, S. 2013 The onera elsa cfd software: input from research and feedback from industry. Mech. Ind. 14 (03), 159–174.CrossRefGoogle Scholar
Chernyshenko, S. I. & Baig, M. F. 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99–131.Google Scholar
Chun, K.-B. & Sung, H. J. 1996 Control of turbulent separated flow over a backward-facing step by local forcing. Exp. Fluids 21 (6), 417–426.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Dandois, J., Garnier, E. & Sagaut, P. 2007 Numerical simulation of active separation control by a synthetic jet. J. Fluid Mech. 574, 25–58.CrossRefGoogle Scholar
Deck, S. 2005 Zonal-detached-eddy simulation of the flow around a high-lift configuration. AIAA J. 43 (11), 2372–2384.CrossRefGoogle Scholar
Del Álamo, J. C. & Jimenez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Dergham, G., Sipp, D. & Robinet, J.-Ch. 2013 Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow. J. Fluid Mech. 719, 406–430.CrossRefGoogle Scholar
Driver, D. M., Seegmiller, H. L. & Marvin, J. 1987 Time-dependent behavior of a reattaching shear layer. AIAA J. 25 (7), 914–919.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.Google Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2013 Localization of flow structures using-norm optimization. J. Fluid Mech. 729, 672–701.CrossRefGoogle Scholar
Gallas, Q., Arnault, A., Monnier, J. C., Delva, J., Dandois, J. & Mialon, B. 2015 Rapport annuel d’activit du pr hybexcit. ONERA Tech. Rep. RT 1/20871.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.CrossRefGoogle Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128.CrossRefGoogle Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.CrossRefGoogle Scholar
Le, H., Moin, P. & Kim, J. 1997 Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech. 330, 349–374.CrossRefGoogle Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, vol. 6. SIAM.CrossRefGoogle Scholar
Mantič-Lugo, V.2015 Too big to grow: self-consistent model for nonlinear saturation in open shear flows. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.CrossRefGoogle ScholarPubMed
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27 (7), 074103.CrossRefGoogle Scholar
Marquet, O., Lombardi, M., Chomaz, J., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51 (02), 233–272.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Mettot, C., Sipp, D. & Bézard, H. 2014 Quasi-laminar stability and sensitivity analyses for turbulent flows: prediction of low-frequency unsteadiness and passive control. Phys. Fluids 26 (4), 045112.CrossRefGoogle Scholar
Mittal, S. 2008 Global linear stability analysis of time-averaged flows. Intl J. Numer. Meth. Fluids 58 (1), 111–118.CrossRefGoogle Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.Google Scholar
Oberleithner, K., Rukes, L. & Soria, J. 2014 Mean flow stability analysis of oscillating jet experiments. J. Fluid Mech. 757, 1–32.CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.CrossRefGoogle Scholar
Sartor, F., Mettot, C., Bur, R. & Sipp, D. 2015 Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis. J. Fluid Mech. 781, 550–577.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar
Spazzini, P. G., Iuso, G., Onorato, M., Zurlo, N. & Di Cicca, G. M. 2001 Unsteady behavior of back-facing step flow. Exp. Fluids 30 (5), 551–561.CrossRefGoogle Scholar
Trefethen, L., Trefethen, A., Reddy, S., Driscoll, T. & others, 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91 (4), 043009.Google Scholar