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Stabilization of absolute instability in spanwise wavy two-dimensional wakes

Published online by Cambridge University Press:  21 June 2013

Yongyun Hwang ,
Jinsung Kim  and
Haecheon Choi
Yongyun Hwang*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Jinsung Kim
Affiliation:
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Korea
Haecheon Choi
Affiliation:
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Korea
*
†Email address for correspondence: [email protected]
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Abstract

Controlling vortex shedding using spanwise-varying passive or active actuation (namely three-dimensional control) has recently been reported as a very efficient method for regulating two-dimensional bluff-body wakes. However, the mechanism of how the designed controller regulates vortex shedding is not clearly understood. To understand this mechanism, we perform a linear stability analysis of two-dimensional wakes, the base flow of which is modified with a given spanwise waviness. Absolute and convective instabilities of the spanwise wavy base flows are investigated using Floquet theory. Two types of base-flow modification are considered: varicose and sinuous. Both of these modifications attenuate absolute instability of two-dimensional wakes. In particular, the varicose modification is found to be much more effective in the attenuation than the sinuous one, and its spanwise lengths resulting in maximum attenuation show good agreement with those in three-dimensional controls. The physical mechanism of the stabilization is found to be associated with the formation of streamwise vortices from tilting of two-dimensional Kármán vortices and the subsequent tilting of these streamwise vortices by the spanwise shear in the base flow. Finally, the sensitivity of absolute instability to spanwise wavy base-flow modification is investigated. It is shown that absolute instability of two-dimensional wakes is much less sensitive to spanwise wavy base-flow modification than to two-dimensional modification. This suggests that the high efficiency observed in several three-dimensional controls is not due to the sensitive response of the wake instability to the spanwise waviness in the base flow.

JFM classification

Instability: Absolute/convective instability Flow Control: Instability control Wakes/Jets: Wakes/Jets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 727 , 25 July 2013 , pp. 346 - 378
Copyright
©2013 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
Antsaklis, P. J. & Michel, A. N. 1997 Linear Systems. McGraw-Hill.Google Scholar
Bearman, P. W. 1965 Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J. Fluid Mech. 21, 241–255.CrossRefGoogle Scholar
Bearman, P. W. 1967 The effect of base bleed on the flow behind a two-dimensional model with a blunt trailing edge. Aero. Q. 18, 207–224.Google Scholar
Bearman, P. W. & Owen, J. C. 1998 Reduction of bluff-body drag and suppression of vortex shedding by the introduction of wavy separation lines. J. Fluids Struct. 12, 123–130.CrossRefGoogle Scholar
Bers, A. 1983 Space-time evolution of plasma instabilities – absolute and convective. In Handbook of Plasma Physics (ed. Rosenbluth, M. N. & Sagdeev, R. Z.), vol. 1. pp. 451–517. North-Holland.Google Scholar
Bertolotti, F. P., Hein, S., Stolte, A. & Koch, W. 2004 Absolute/convective instability investigation of primary and secondary crossflow vortices. Notes Numer. Fluid Mech. Multidisc, Design 86, 51–62.Google Scholar
Brancher, P. & Chomaz, J. M. 1997 Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78, 658–661.CrossRefGoogle Scholar
Brandt, L., Cossu, C., Chomaz, J.-M., Huerre, P. & Henningson, D. S. 2003 On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485, 221–242.CrossRefGoogle Scholar
Brevdo, L. & Bridges, T. 1996 Absolute and convective instability of spatially periodic flows. Phil. Trans. R. Soc. Lond. A 354, 1027–1064.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle 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
Canuto, C., Hussaini, Y. M., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carriere, P. & Monkewitz, P. A. 1999 Convective versus absolute instability in mixed Rayleigh–Benard–Poiseuille convection. J. Fluid Mech. 384, 243–262.Google Scholar
Choi, H., Jeon, W.-P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60, 25–28.Google Scholar
Darekar, R. M. & Sherwin, S. J. 2001 Flow past a square-section cylinder with a wavy stagnation face. J. Fluid Mech. 426, 263–295.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Giannetti, F. & Luchini, F. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Hammond, D. A. & Redekopp, L. G. 1997 Global dynamics of symmetric and asymmetric wakes. J. Fluid Mech. 331, 231–260.CrossRefGoogle Scholar
Herbert, Th. 1988 Secondary instability of boundary-layers. Annu. Rev. Fluid Mech. 20, 487–526.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamic and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.Google Scholar
Hwang, Y. & Choi, H. 2006 Control of absolute instability by basic-flow modification in a parallel wake at low Reynolds number. J. Fluid Mech. 560, 465–475.Google Scholar
Hwang, Y. & Choi, H. 2008 Sensitivity of global instability of spatially developing flow in weakly and fully nonlinear regimes. Phys. Fluid 20, 071703.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010a Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.Google Scholar
Hwang, Y. & Cossu, C. 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.Google Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.Google Scholar
Kim, J. 2005 Distributed forcing of flow over a circular cylinder. PhD thesis, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea.Google Scholar
Kim, J. & Choi, H. 2005 Distributed forcing of flow over a circular cylinder. Phys. Fluids 17, 033133.Google Scholar
Kim, J., Hahn, S., Kim, J., Lee, D., Choi, J., Jeon, W.-P. & Choi, H. 2004 Active control of turbulent flow over a model vehicle for drag reduction. J. Turbul. 5, 019.CrossRefGoogle Scholar
Kwon, K. & Choi, H. 1996 Control of laminar vortex shedding behind a circular cylinder using splitter plates. Phys. Fluids 8, 478–496.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.Google Scholar
Leu, T.-S. & Ho, C.-M. 2000 Control of global instability in a non-parallel near wake. Astron. Astrophys. 404, 345–378.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
Mathis, C., Provansal, M. & Boyer, L. 1984 The Bénard-von Kármán instability: an experimental study near the threshold. J. Phys. Paris Lett. 45, 483–491.Google Scholar
Min, C. & Choi, H. 1999 Suboptimal feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 401, 123–156.Google Scholar
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers. Phys. Fluids 31, 999–1006.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.Google Scholar
Nishioka, M. & Sato, H. 1974 Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 765, 97–112.Google Scholar
Owen, J. C., Szewczyk, A. A. & Bearman, P. W. 2000 Suppression of Kármán vortex shedding. Phys. Fluids 12, S9.Google Scholar
Park, H., Jeon, W.-P., Choi, H. & Yoo, J. Y. 2006 Drag reduction in flow over a two-dimensional bluff body with a blunt trailing edge using a new passive device. J. Fluid Mech. 563, 389–414.Google Scholar
Park, J., Hwang, Y. & Cossu, C. 2011 On the stability of large-scale streaks in the turbulent Couette and Poiseulle flows. C. R. Mèc. 339 (1), 105103.Google 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
Pier, B 2007 Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Engng Maths 57, 237–251.Google 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
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.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, 015109.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.Google Scholar
Rodriguez, O. 1991 Base drag reduction by the control of three-dimensional unsteady vortical structures. Exp. Fluids 11, 218–226.CrossRefGoogle Scholar
Roshko, A. 1955 On the wake and drag of bluff bodies. J. Aeronaut. Sci. 22, 124–132.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.Google Scholar
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech. 271, 17–53.Google Scholar
Son, D., Jeon, S. & Choi, H. 1940 A proportional-integral-differential control of flow over a circular cylinder. Phil. Trans. R. Soc. Lond. A 369, 1540–1555.Google Scholar
Stansby, P. K. 1974 The effects of end plates on the base pressure coefficient of a circular cylinder. Aerosp. J. 78, 36–37.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex ‘shedding’ at low Reynolds number. J. Fluid Mech. 218, 71–107.Google Scholar
Tanner, M. 1972 A method of reducing the base drag of wings with blunt trailing edges. Aero. Q. 23, 15–23.Google Scholar
Tombazis, N. & Bearman, P. W. 1997 A study of three-dimensional aspects of vortex shedding from a bluff body with a mild geometric disturbance. J. Fluid Mech. 330, 85–112.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95, 319–343.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.Google Scholar
Wood, C. J. 1967 The effect of base bleed on a periodic wake. J. Aeronaut. Soc. 68, 477–482.CrossRefGoogle Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155–165.Google Scholar