Multiple timescale and sensitivity analysis for the passive control of the cylinder flow

A theoretical approach is developed to predict the effect of a passive control device on the global stability of a cylinder flow close to the critical Reynolds number Re_c = 46.8. The passive control device is here a small control cylinder that is modelled by a local force. This force is the sum of a steady component, acting on the base-flow, and of an unsteady component, acting at the perturbation level. To understand how these two components affect the flow stability, a multiple timescale analysis is first performed at the bifurcation. The leading global mode responsible for the onset of the Von Karman street is assumed to oscillate on a "fast" timescale and its amplitude to grow on a "slow" timescale. Then a sensitivity analysis of the leading global eigenvalue with respect to the steady and unsteady components of the force is developed. By combining these two analysis, one obtains maps of the growth rate and pulsation of the perturbation as a function of the position of the local force. The regions of the flow where the placement of a control cylinder suppresses the Von Karman instability are very well reproduced.