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Sensitivity analysis and passive control of cylinder flow

  • OLIVIER MARQUET (a1), DENIS SIPP (a1) and LAURENT JACQUIN (a1)

Abstract

A general theoretical formalism is developed to assess how base-flow modifications may alter the stability properties of flows studied in a global approach of linear stability theory. It also comprises a systematic approach to the passive control of globally unstable flows by the use of small control devices. This formalism is based on a sensitivity analysis of any global eigenvalue to base-flow modifications. The base-flow modifications investigated are either arbitrary or specific ones induced by a steady force. This leads to a definition of the so-called sensitivity to base-flow modifications and sensitivity to a steady force. These sensitivity analyses are applied to the unstable global modes responsible for the onset of vortex shedding in the wake of a cylinder for Reynolds numbers in the range 47≤Re≤80. First, it is demonstrated how the sensitivity to arbitrary base-flow modifications may be used to identify regions and properties of the base flow that contribute to the onset of vortex shedding. Secondly, the sensitivity to a steady force determines the regions of the flow where a steady force acting on the base flow stabilizes the unstable global modes. Upon modelling the presence of a control device by a steady force acting on the base flow, these predictions are then extensively compared with the experimental results of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, p. 71). A physical interpretation of the suppression of vortex shedding by use of a control cylinder is proposed in the light of the sensitivity analysis.

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Airiau, C., Bottaro, A., Walther, S. & Legendre, D. 2003 A methodology for optimal laminar flow control: Application to the damping of Tollmien-Schlichting waves in a boundary layer. Phys. Fluids 15, 11311145.
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Eurphys. Lett. 75, 750756.
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293302.
Chomaz, J. M. 2005 Global instabilities in spatially developping flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
Chomaz, J.-M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60, 2528.
Chomaz, J.-M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. App. Maths 84, 119144.
Davis, T. A. 2004 A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Transactions on Methematical Software 30 (2), 165195.
Davis, T. A. & Duff, I. S. 1997 An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM J. Matrix Anal. Applics. 18, 140158.
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the cylinder wake's first instability. J. Fluid Mech. 581, 167197.
Gunzburger, M. D. 2007 Inverse design and optimization methods. Von Karman Institute for Fluid Dynamics Lecture Series 1997-05.
Hill, D. C. 1992 A theoretical approach for analysing the restabilization of wakes. AIAA 92-0067.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilites in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Hwang, Y. & Choi, H. 2006 Control of absolute instability by basic-flow modification in parallel wake at low Reynolds number. J. Fluid Mech. 560, 465475.
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 2345.
Kim, H. B. & Chang, K. S. 1995 Numerical study on vortex shedding from a circular cylinder influenced by a nearby control wire. Comput. Fluid Dyn. J. 4, 151164.
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users's Guide. SIAM, Philadelphia.
Mathis, C., Provansal, M. & Boyer, L. 1984 Benard-von Karman instability: an experimental study near the threshold. J. Phys. Lett. Paris 45, 483491.
Mittal, S. & Raghuvanshi, A. 2001 Control of vortex shedding behind circular cylinder for flows at low Reynolds number. Intl. J. Numer. Meth. Fluids 35, 421447.
Monkewitz, P. A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 120.
Morzynski, M., Afanasiev, K. & Thiele, F. 1999 Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Meth. Appl. Mech. Engng 169, 161176.
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297330.
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake J. Fluid Mech. 458, 407417.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bnart-von Karman instability: transient and forced regimes. J. Fluid Mech. 182, 122.
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, 333358.
Sorensen, D. C. 1992 Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357385.
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. In Proc. Forum on Unsteady Flow Separation. (ed. Ghia, K.), pp. 113. ASME
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex shedding at ‘low’ Reynolds numbers. J. Fluid Mech. 218, 71107.
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477539.
Zebib, A. 1987 Stability of viscous flow past a circular cylinder J. Engrg Maths 21, 155165.
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