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On the active feedback control of a swirling flow in a finite-length pipe

Published online by Cambridge University Press:  25 November 2013

Shixiao Wang ,
Zvi Rusak ,
Steve Taylor  and
Rui Gong
Shixiao Wang*
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
Zvi Rusak
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA
Steve Taylor
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
Rui Gong
Affiliation:
Department of Mathematics, University of Auckland, 38 Princes Street, Auckland, 1142, New Zealand
*
†Email address for correspondence: wang@math.auckland.ac.nz
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Abstract

The physical properties of a recently proposed feedback-stabilization method of a vortex flow in a finite-length straight pipe are studied for the case of a solid-body rotation flow. In the natural case, when the swirl ratio is beyond a certain critical level, linearly unstable modes appear in sequence as the swirl level is increased. Based on an asymptotic long-wave (long-pipe) approach, the global feedback control method is shown to enforce the decay in time of the perturbation’s kinetic energy and thereby quench all of the instability modes for a swirl range above the critical swirl level. The effectiveness of an extended version of this feedback flow control approach is further analysed through a detailed mode analysis of the full linear control problem for a solid-body rotation flow in a finite-length pipe that is not necessarily long. We first rigourously prove the asymptotic decay in time of all modes with real growth rates. We then compute the growth rate and shape of all modes according to the full linearized control problem for swirl levels up to 50 % above the critical level. We demonstrate that the flow is stabilized in the whole swirl range and can be even further stabilized for higher swirl levels. However, the control effectiveness is sensitive to the choice of the feedback control gain. A potentially best range of the gain is identified. An inadequate level of gain, either insufficient or excessive, could lead to a marginal control or failure of the control method at high swirl levels. The robustness of the proposed control law to stabilize both initial waves and continuous inlet flow perturbations and the elimination of the vortex breakdown process are demonstrated through numerical computations.

JFM classification

Flow Control: Control theory Vortex Flows: Vortex breakdown Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 280 - 307
Copyright
©2013 Cambridge University Press 

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