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Flow control of weakly non-parallel flows: application to trailing vortices

Published online by Cambridge University Press:  01 June 2017

F. Viola ,
E. Pezzica ,
G. V. Iungo ,
F. Gallaire  and
S. Camarri Open the ORCID record for S. Camarri [Opens in a new window]
F. Viola
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
E. Pezzica
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
G. V. Iungo
Affiliation:
Wind Fluids and Experiments (WindFluX) Laboratory, University of Texas at Dallas, Richardson, TX 75080, USA
F. Gallaire*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
S. Camarri
Affiliation:
Department of Civil and Industrial Engineering, University of Pisa, Pisa 56122, Italy
*
†Email address for correspondence: [email protected]
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Abstract

A general formulation is proposed to control the integral amplification factor of harmonic disturbances in weakly non-parallel amplifier flows. The sensitivity of the local spatial stability spectrum to a base-flow modification is first determined, generalizing the results of Bottaro et al. (J. Fluid Mech., vol. 476, 2003, pp. 293–302). This result is then used to evaluate the sensitivity of the overall spatial growth to a modification of the inlet flow condition. This formalism is applied to a non-parallel Batchelor vortex, which is a well-known model for trailing vortices generated by a lifting wing. The resulting sensitivity map indicates the optimal modification of the inlet flow condition enabling the stabilization of the helical modes. It is shown that the control, formulated using a single linearization of the flow dynamics carried out on the uncontrolled configuration, successfully reduces the total spatial amplification of all convectively unstable disturbances.

JFM classification

Flow Control: Flow Control Flow Control: Instability control Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 342 - 363
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Copyright
© 2017 Cambridge University Press 

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References

Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. (B/Fluids) 27 (5), 501–513.CrossRefGoogle Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20 (4), 645–658.Google Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google 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.Google 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
Camarri, S. 2015 Flow control design inspired by linear stability analysis. Acta Mechanica 226 (4), 979–1010.CrossRefGoogle Scholar
Camarri, S. & Iollo, A. 2010 Feedback control of the vortex-shedding instability based on sensitivity analysis. Phys. Fluids 22 (9), 094102.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
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.Google Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.Google Scholar
Duck, P. W. & Foster, M. R. 1980 The inviscid stability of a trailing line vortex. Z. Angew. Math. Phys. 31 (4), 524–532.Google Scholar
Eckhoff, K. S. 1984 A note on the instability of columnar vortices. J. Fluid Mech. 145, 417–421.Google Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 500, 239–262.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274.Google Scholar
Gallaire, F. & Chomaz, J.-M. 2003 Mode selection in swirling jet experiments: a linear stability analysis. J. Fluid Mech. 494, 223–253.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Heaton, C. J. 2007 Optimal growth of the Batchelor vortex viscous modes. J. Fluid Mech. 592, 495–505.Google Scholar
Heaton, C. J., Nichols, J. W. & Schmid, P. J. 2009 Global linear stability of the non-parallel Batchelor vortex. J. Fluid Mech. 629, 139–160.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic Instabilities in Open Flows, chap. 2. Cambridge University Press.CrossRefGoogle Scholar
Iungo, G. V., Viola, F., Camarri, S. & Gallaire, F. 2013 Linear stability analysis of wind turbine wakes performed on wind tunnel measurements. J. Fluid Mech. 737, 499–526.Google Scholar
Khorrami, M. 1991 A Chebyshev spectral collocation method using a staggered grid for the stability of cylindrical flows. Intl J. Numer. Meth. Fluids 12, 825–833.Google Scholar
Leibovich, S. & Stewartson, K. 1983 Sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63 (04), 753–763.Google Scholar
Ludwieg, H. 1962 Zur erklärung der instabilität der über angestellten deltaflügeln auftretenden freien wirbelkerne. Z. Flugwiss. 10 (6), 242–249.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.CrossRefGoogle Scholar
Olendraru, C. & Sellier, A. 2002 Viscous effects in the absolute/convective instability of the Batchelor vortex. J. Fluid Mech. 459, 371–396.Google Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: absolute-convective transition and spatial branches. Phys. Fluids 11, 1805–1820.Google Scholar
Qadri, U. A., Mistry, D. & Juniper, M. P. 2013 Structural sensitivity of spiral vortex breakdown. J. Fluid Mech. 720, 558–581.Google Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworthy, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331–378.Google Scholar
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer.Google Scholar
Spalart, P. R. 1998 Airplane trailing vortices. Annu. Rev. Fluid Mech. 30 (1), 107–138.Google Scholar
Viola, F., Arratia, C. & Gallaire, F. 2016 Mode selection in trailing vortices: harmonic response of the non-parallel Batchelor vortex. J. Fluid Mech. 790, 523–552.Google Scholar
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
Zuccher, S., Bottaro, A. & Luchini, P. 2006 Algebraic growth in a Blasius boundary layer: nonlinear optimal disturbances. Eur. J. Mech. (B/Fluids) 25 (1), 1–17.Google Scholar