Closed-loop control of vortex breakdown: a model study

Abstract

The objective of the study is to develop a closed-loop control scheme that is capable of preserving the columnar swirling flow state in the finite-length pipe model of Wang & Rusak (1996a). The base state consists of a solid body rotation superimposed on axial plug flow, with two dimensionless parameters: the swirl $\Omega$ and the pipe aspect ratio $L$. The linear stability properties of the columnar base state are documented and shown to give rise to unstable global modes above a critical swirl level $\Omega_1$, thereby triggering vortex breakdown. Our study focuses on the derivation of a control method in order to quench the linear development of the Wang & Rusak instability. An optimal control approach is then devised for a reduced-order system which is obtained by a suitable projection on a low-order subspace of the $N$ least-stable eigenmodes. The actuator consists of perturbations of the inlet circulation and its time history is selected so as to minimize a cost-functional incorporating both the state energy and the control energy. A Riccati-based formulation leads to the determination of the optimal gain matrix for the low-order system. When applied to the full linear system, the feedback law for $N=4$ succeeds in maintaining the columnar base state for swirl levels as high as 13% above global onset. The control scheme is found to be robust with respect to noise and to uncertainties in parameter settings. It remains effective even under partial-state information conditions.