J. Fluid Mech. (2008), vol. 615, pp. 221–252. c  2008 Cambridge University Press doi:10.1017/S0022112008003662 Printed in the United Kingdom 221 Sensitivity analysis and passive control of cylinder flow OLIVIER MARQUET, DENIS SIPP AND LAURENT JACQUIN ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France (Received 13 August 2007 and in revised form 23 July 2008) 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. 1. Introduction Many studies have been dedicated to understanding the dynamics of a cylinder flow for various values of the Reynolds number. In particular it is well known that at a critical Reynolds number Re c ≈ 47 the flow experiences a Hopf bifurcation from a steady symmetric state towards a time-periodic non-symmetric state (Provansal, Mathis & Boyer 1987; Sreenivasan, Strykowski & Olinger 1987; Noack & Eckelmann 1994). A global instability has clearly been identified as responsible for the onset of the vortex shedding process (Jackson 1987; Zebib 1987) but substantial work is still devoted to understanding the mechanism for selecting its frequency (Pier 2002; Barkley 2006; Sipp & Lebedev 2007). Control of vortex shedding has also received much attention and various passive and active control techniques have been tested on this flow both experimentally and numerically. Concerning passive control, Strykowski & Sreenivasan (1990) have experimentally investigated how a small control cylinder suitably placed in the wake of the main cylinder alters the vortex shedding. For various diameter ratios of the two cylinders they determined the regions of the flow where the placement of the control