The effect of small pipe divergence on near critical swirling flows

Abstract

The effect of small pipe divergence on an inviscid, incompressible, near critical axisymmetric swirling flow is investigated. The singular behavior of a regular expansion solution, in terms of the pipe divergence parameter, around the critical swirl of a flow in a straight pipe is demonstrated. This singularity infers that large-amplitude disturbances may be induced by the small pipe divergence when incoming flows have a swirl level near the critical swirl. In order to gain insight to the behavior of flows in this swirl range, a small-disturbance analysis is developed. It is found that a small but finite pipe divergence breaks the transcritical bifurcation of solutions of a flow in a straight pipe into two equilibrium solution branches. These branches fold at limit swirl levels near the critical swirl resulting in a finite gap of swirl that separates the two branches. This suggests that no near-columnar axisymmetric state can exist within this range of incoming swirl around the critical level; the flow must develop large disturbances in this swirl range. Beyond this range, two steady states may exist under the same inlet/outlet conditions. However, when the pipe divergence is increased, this special behavior uniformly changes into a branch of solutions with no fold. A weakly nonlinear approach to study the effect of slight pipe divergence on standing waves in a long pipe is also derived. The behavior of the asymptotic solutions match the bifurcation diagrams from previous theoretical and numerical studies and extends their results. The relevance of the results to axisymmetric vortex breakdown in a diverging pipe is discussed.