Feedback stabilization of the solid body rotation flow in a finite-length pipe

Abstract

Wang & Rusak developed a novel weakly nonlinear model to study the evolution of the per- turbations in an incompressible, inviscid, axisymmetric, near critical swirling flow in a long but finite, straight, circular pipe. Their study has deepened our understanding of the flow transition mechanism in vortex breakdown phenomenon which is necessary for us to develop an effective methodology to control the vortex breakdown. For the case of a solid body rotation flow, the physi- cal properties of a recently-proposed feedback-stabilization method of a vortex flow in a finite-length straight pipe are studied. In the natural case, linearly unstable modes appear in sequence as swirl level is increased when the swirl ratio is beyond a certain critical level. Based on an asymptotic long-wave approach, the global feedback control method is shown to enforce the decay in time of perturbation’s kinetic energy and thereby quench all the instability modes for a swirl range above the critical swirl level. The effectiveness of this feedback flow control approach is further analyzed through a detailed mode analysis of the full linear control problem for a flow in a finite-length pipe that is not necessarily long. We first rigorously 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 100% 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. We identified an optimal range of the gain. Either insufficient or excessive gain is considered as inadequate which could lead to a marginal control or failure of the control method at high swirl levels. The control robustness in terms of variations of the actual physical setting at the pipe outlet is also discussed.