The interaction of near-critical swirling flows in a pipe with inlet azimuthal vorticity perturbations

Abstract

The interaction of a near-critical axisymmetric incompressible swirling flow in a straight pipe with small inlet azimuthal vorticity perturbations is studied. Certain flow conditions that may reflect the physical situation are prescribed along the pipe inlet and outlet. It is first demonstrated that under these conditions a regular-expansion solution in terms of the small azimuthal vorticity perturbations has a singular behavior around the critical swirl. This singularity infers that large-amplitude disturbances may be induced by the small perturbations when the incoming flow to the pipe has a swirl level around the critical swirl. In order to understand the nature of flows in this swirl range, a small-disturbance analysis is developed. It shows that under the prescribed inlet/outlet conditions, a small but finite inlet azimuthal vorticity perturbation breaks the transcritical bifurcation of solutions of the Euler equations at the critical swirl into two branches of perturbed solutions. When the azimuthal vorticity perturbations are positive these branches show a regular behavior. However, when they are negative, the perturbed branches fold at limit points near the critical swirl, with a finite gap between the two branches, and no near-columnar equilibrium state can exist for an incoming flow with swirl close to the critical level. The flow must develop large disturbances in this swirl range. Beyond this range, two equilibrium states may exist under the same inlet/outlet conditions. When the negative inlet vorticity perturbations become larger in their size, this special behavior uniformly changes into a branch of single equilibrium state for each incoming swirl. The relevance of the results to the appearance of the axisymmetric vortex breakdown in a pipe and the control of this phenomenon using inlet vorticity perturbations is also discussed. The results suggest that, in general, positive inlet azimuthal vorticity perturbations may be used to delay vortex breakdown to higher swirl levels whereas negative perturbations induce the appearance of vortex breakdown at levels below the critical swirl.