Model-based control of vortex shedding at low Reynolds numbers

Abstract

Model-based feedback control of vortex shedding at low Reynolds numbers is considered. The feedback signal is provided by velocity measurements in the wake, and actuation is achieved using blowing and suction on the cylinder's surface. Using two-dimensional direct numerical simulations and reduced-order modelling techniques, linear models of the wake are formed at Reynolds numbers between 45 and 110. These models are used to design feedback controllers using H ∞ loop-shaping. Complete suppression of shedding is demonstrated up to Re = 110-both for a single-sensor arrangement and for a three-sensor arrangement. The robustness of the feedback controllers is also investigated by applying them over a range of off-design Reynolds numbers, and good robustness properties are seen. It is also observed that it becomes increasingly difficult to achieve acceptable control performance-measured in a suitable way-as Reynolds number increases. which occurred naturally for the oval-shaped cylinder at Reynolds numbers of 77 and above, was suppressed by feedback at Reynolds numbers up to 80.The study of Berger [3] shares a number of features with its earliest successors [8,33,36,42]: all used a velocity measurement in the wake as the feedback signal; in all cases the control law was a simple proportional feedback gain chosen by trial and error (possibly with some phase shift); and all were conducted at low Reynolds numbers near the onset of vortex shedding. (A notable exception is the study by Ffowcs Williams and Zhou [8], performed at a Reynolds number of 400.) The actuation strategies employed in these early studies included transverse displacement of the cylinder [3,42], loudspeakers [8,36] and blowing and suction on the cylinder surface [33].All of these early studies successfully eliminated vortex shedding, but did so only slightly above the critical Reynolds number. Roussopoulos [36], for example, increased the critical Reynolds number by about 20 % in his experimental study, whilst Park et al. [33] increased it by about 27 % in simulations. Gunzburger and Lee [15] achieved complete suppression at slightly higher Reynolds numbers (of Re = 80, about 60 % above criticality) by using surface pressure measurements as the feedback signal and by adding an additional orifice for blowing and suction at the cylinder's trailing edge. Keles [23] used a similar experimental arrangement to Roussopoulos [36] at a higher Reynolds number of 170. The amplitude of vortex shedding was reduced along the entire span of the wake at a streamwise location two cylinder diameters downstream, but it is not clear whether a similar reduction occurred for all streamwise positions. Cohen et al.[7] also used a proportional feedback gain but, rather than use any sensor measurement directly, the controller acted on the first POD mode of a reduced-order model. The amplitude of the POD mode was estimated from several sensor measurements using linear stochastic estimation. Siegel et al. [38] extended this work by applying proportional-differential control to th...

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