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A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

Published online by Cambridge University Press:  07 July 2016

P. Meliga ,
E. Boujo  and
F. Gallaire
P. Meliga*
Affiliation:
Aix-Marseille Université, CNRS, Ecole Centrale Marseille, Laboratoire M2P2, Marseille, France
E. Boujo
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
F. Gallaire
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Email address for correspondence: philippe.meliga@l3m.univ-mrs.fr
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Abstract

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

JFM classification

Flow Control: Instability control Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Wakes/Jets: Wakes

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Papers
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Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 327 - 357
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© 2016 Cambridge University Press 

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References

del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Alizard, F., Robinet, J.-C. & Rist, U. 2010 Sensitivity analysis of a streamwise corner flow. Phys. Fluids 22, 014103.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Boujo, E., Ehrenstein, U. & Gallaire, F. 2013 Open-loop control of noise amplification in a separated boundary layer flow. Phys. Fluids 25 (12), 124106.Google Scholar
Boujo, E. & Gallaire, F. 2014 Controlled reattachment in separated flows: a variational approach to recirculation length reduction. J. Fluid Mech. 742, 618–635.CrossRefGoogle Scholar
Brandt, L., Sipp, D., Pralits, J. O. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.Google Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5, 774–777.Google Scholar
Camarri, S., Fallenius, B. E. G. & Fransson, J. H. M. 2013 Stability analysis of experimental flow fields behind a porous cylinder for the investigation of the large-scale wake vortices. J. Fluid Mech. 715, 499–536.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Dušek, J., Le Gal, P. & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.Google 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
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Hammond, D. A. & Redekopp, L. G. 1997 Global dynamics of symmetric and asymmetric wakes. J. Fluid Mech. 331, 231–260.Google Scholar
Hill, D. C.1992 A theoretical approach for analyzing the restabilization of wakes. AIAA paper 92-0067.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Iungo, G. V., Viola, F., Camarri, S., Porté-Agel, F. & Gallaire, F. 2013 Linear stability analysis of wind turbine wakes performed on wind tunnel measurements. J. Fluid Mech. 737, 499–526.CrossRefGoogle Scholar
Luchini, P., Giannetti, F. & Pralits, J. 2009 Structural sensitivity of the finite-amplitude vortex shedding behind a circular cylinder. In IUTAM Symposium on Unsteady Separated Flows and their Control (ed. Braza, M. & Hourigan, K.), IUTAM Bookseries, vol. 14, pp. 151–160. Springer.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.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, 084501.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008a Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Marquet, O., Sipp, D., Jacquin, L. & Chomaz, J.-M.2008b Multiple scale and sensitivity analysis for the passive control of the cylinder flow. AIAA paper 2008-4228.CrossRefGoogle Scholar
Marquillie, M., Ehrenstein, U. & Laval, J.-P. 2011 Instability of streaks in wall turbulence with adverse pressure gradient. J. Fluid Mech. 681, 205–240.CrossRefGoogle Scholar
Maurel, A., Pagneux, V. & Wesfreid, J. E. 1995 Mean-flow correction as non-linear saturation mechanism. Europhys. Lett. 32 (3), 217–222.CrossRefGoogle Scholar
Meliga, P., Boujo, E., Pujals, G. & Gallaire, F. 2014 Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder. Phys. Fluids 26 (10), 104101.Google Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009a Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J. Fluid Mech. 633, 159–189.Google Scholar
Meliga, P., Gallaire, F. & Chomaz, J.-M. 2012a A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262.Google Scholar
Meliga, P., Pujals, G. & Serre, E. 2012b Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Phys. Fluids 24 (6), 061701.CrossRefGoogle Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2009b Elephant modes and low frequency unsteadiness in a high Reynolds number, transonic afterbody wake. Phys. Fluids 21 (5), 054105.CrossRefGoogle Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2010 Open-loop control of compressible afterbody flows using adjoint methods. Phys. Fluids 22 (5), 054109.Google 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
Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.Google Scholar
Piot, E., Casalis, G., Muller, F. & Bailly, C. 2006 Investigation of the PSE approach for subsonic and supersonic hot jets. Detailed comparisons with LES and Linearized Euler Equations results. Intl J. Aeroacoust. 5 (4), 361–393.Google Scholar
Pralits, J. O., Bottaro, A. & Cherubini, S. 2015 Weakly nonlinear optimal perturbations. J. Fluid Mech. 785, 135–151.CrossRefGoogle Scholar
Pralits, J. O., Brandt, L. & Giannetti, F. 2010 Instability and sensitivity of the flow around a rotating circular cylinder. J. Fluid Mech. 650, 513–536.Google 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
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54, 263–288.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.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex ‘shedding’ at low Reynolds numbers. J. Fluid Mech. 218, 71–107.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.Google Scholar
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91, 043009.Google ScholarPubMed
Viola, F., Iungo, G. V., Camarri, S., Porté-Agel, F. & Gallaire, F. 2014 Prediction of the hub vortex instability in a wind turbine wake: stability analysis with eddy-viscosity models calibrated on wind tunnel data. J. Fluid Mech. 750, R1.Google Scholar
Williamson, C. H. K. 1988 Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids 31 (10), 2742–2744.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.Google Scholar
Zielinska, B. J. A., Goujon-Durand, S., Dušek, J. & Wesfreid, J. E. 1997 Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79, 3893–3896.Google Scholar