Role of Vortex Line Coiling in Vortex Breakdown

We propose a conceptual foundation for the vortex breakdown mechanism within a column with axial flow using the dynamics of vortex line coiling to develop a breakdown criterion. The stagnation point at the onset of breakdown’s recirculation bubble can be found by setting the axial velocity of the column equal to the axial velocity induced by the coiled vortex lines. Deriving the coiled vorticity and associated axial flow yields a model for the breakdown criterion q (the ratio of inlet peak azimuthal to peak axial velocities) in terms of the core radius variation (δr) and axial length (z_L) of the coiling. The model gives q²=[16πm(1+2µ+µ²)]/[7(2µ²+µ³)], where m=δr/z_L and µ=δr/r₁ (r₁ is the upstream core radius, i.e. at the start of the core radius variation), as the breakdown threshold. Prior models result in particular q values, while ours adapts q to a variety of vortex spreading rates (which also controls the coiling rate). For a given sinuously varying vortex core, the model predicts a stagnation point (hence breakdown criterion) at q=1.2, which occurs numerically between q=1.15 and q=1.25; prior data is also compared to our model. We have thus derived a simple, intuitive model for breakdown via the generation of axial flow using only the core radius variation of a column.