Dynamics of a polarized vortex ring

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Abstract
This paper builds on our claim that most vortical structures in transitional and turbulent flows are partially polarized. Polarization is inferred by the application of helical wave decomposition. We analyse initially polarized isolated viscous vortex rings through direct numerical simulation of the Navier-Stokes equations using divergence-free axisymmetric eigenfunctions of the curl operator. Integral measures of the degree of polarization, such as the fractions of energy, enstrophy, and helicity associated with right-handed (or left-handed) eigenfunctions, remain nearly constant during evolution, thereby suggesting that polarization is a persistent feature. However, for polarized rings an axial vortex (tail) develops near the axis, where the local ratio of right- to left-handed vorticities develops significant non-uniformities due to spatial separation of peaks of polarized components. Reconnection can occur in rings when polarized and is clearly discerned from the evolution of axisymmetric vortex surfaces; but interestingly, the location of reconnection cannot be inferred from the vorticity magnitude. The ring propagation velocity Up decreases monotonically as the degree of initial polarization increases. Unlike force-balance arguments, two explanations based on vortex dynamics provided here are not restricted to thin rings and predict reduction in Up correctly. These results reveal surprising differences among the evolutionary dynamics of polarized, partially polarized, and unpolarized rings.

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