The stability of noncolumnar swirling flows in diverging streamtubes

Abstract

A linear stability analysis of a family of steady, noncolumnar and axisymmetric, swirling flows that may develop in a finite-length slightly diverging pipe is presented. These flow states are described by the asymptotic analysis of Rusak et al. (1998). There exists a limit level of the incoming flow swirl ratio ωcσ1 which is the corrected critical swirl as a result of the pipe divergence. When the swirl ratio is in a certain range below ωcσ1, two steady states can exist for the same inlet, outlet, and wall conditions: One which describes a near-columnar vortex state and another which describes a swirling flow with a large-amplitude disturbance. When the swirl level is above ωcσ1, no near-columnar, steady, and axisymmetric state exists. The stability of this family of flows is examined by studying the linearized dynamics of an unsteady and axially symmetric perturbation which also satisfies the boundary conditions. The stability analysis shows that ωcσ1 is a point of exchange of stability for the family of the noncolumnar vortex flows. The near-columnar states have a linearly stable mode of disturbance whereas the states with large disturbances are unstable. Also, the near-columnar states lose their stability characteristics as the swirl level approaches ωcσ1. Therefore, the analysis implies that when the swirl level of the incoming flow is above ωcσ1, the flow in the pipe must develop a transition process that involves large-amplitude perturbations and may lead to vortex breakdown states. The effect of the increase of pipe divergence on the flow dynamics and transition to breakdown states is also discussed.