Nonlinear Control of Axisymmetric Swirling Flows in a Long Finite Length Pipe

Abstract

Feedback stabilization of inviscid and high Reynolds number, axisymmetric, swirling flows in a long finitelength circular pipe using active variations of pipe geometry as a function of the evolving inlet radial velocity is studied. The complicated dynamics of the natural flow requires that any theoretical model that attempts to control vortex stability must include the essential nonlinear dynamics of the perturbation modes. In addition, the control methodology must establish a stable desired state with a wide basin of attraction. The present approach is built on a weakly nonlinear model problem for the analysis of perturbation dynamics on nearcritical swirling flows in a slightly areavarying, long, circular pipe with unsteady changes of wall geometry. In the natural case with no control, flows with incoming swirl ratio above a critical level are unstable and rapidly evolve to either vortex breakdown states or accelerated flow states. Following an integration of the model equation, a perturbation kineticenergy identity is derived, and an active feedback control methodology to suppress perturbations from a desired columnar state is proposed. The stabilization of both inviscid and highRe flows is demonstrated for a wide range of swirl ratios above the critical swirl for vortex breakdown and for largeamplitude initial perturbations. The control gain for the fastest decay of perturbations is found to be a function of the swirl level. Large gain values are required at nearcritical swirl ratios while lower gains provide a successful control at swirl levels away from critical. This feedback control technique cuts the feedforward mechanism between the inlet radial velocity and the growth of perturbation's kinetic energy in the bulk and thereby enforces the decay of perturbations and eliminates the natural explosive evolution of the vortex breakdown process. The application of this proposed robust active feedback control method establishes a branch of columnar states with a wide basin of attraction for swirl ratios up to at least 50% above the critical swirl. This study provides guidelines for future flow control simulations and experiments. However, the present methodology is limited to the control of highReynolds number (nearly inviscid), axisymmetric, weakly nonparallel flows in long pipes.