Universal translational velocity of vortex rings behind conical objects

TL;DR

This study investigates the universal behavior of vortex rings generated behind conical objects, revealing that their non-dimensional energy and circulation converge to similar values regardless of cone geometry, with vortex velocity being a key governing factor.

Contribution

It demonstrates that vortex rings behind cones exhibit universal non-dimensional energy and circulation values, independent of cone shape and speed, with vortex velocity as a primary governing parameter.

Findings

Vortex rings behind cones reach universal non-dimensional energy and circulation values.

Vortex velocity is the most consistent parameter influencing vortex ring formation.

Vortex pinch-off does not cause the non-dimensional energy to reach a minimum.

Abstract

Ring vortices are efficient at transporting fluid. They are often produced by ejecting a volume of fluid through a circular orifice. The impulse given to the vortex rings results in a propulsive force on the generator. Propulsive vortex rings have been widely studied and characterised. After four convective times, the vortex moves faster than the shear layer it originates from, and separates. This separation corresponds to a maximum in circulation and a maximal spread of the vorticity, quantified by the non-dimensional energy. The simultaneity of these three events obfuscates the causality between them. To analyse the temporal evolution of the non-dimensional energy of ring vortices independent of their separation, we analyse the development of vortices generated in the wake of cones. Cones with different apertures and diameters were accelerated from rest to produce a wide variety of vortex rings and their energy, circulation, and velocity were extracted based on time-resolved velocity field measurements. The rings that form behind the cones have a self-induced velocity that causes them to follow the cone and they continue to grow as the cone travels well beyond the limiting vortex formation times scales observed for propulsive vortices. The non-dimensional circulation, based on the vortex diameter, and the non-dimensional energy of the drag vortex rings converge after three convective times to values comparable to their propulsive counterparts. This result proves that vortex pinch-off does not cause the non-dimensional energy to reach a minimum value. The limiting values of the non-dimensional circulation and energy are mostly independent of the cone geometry and translational velocity and fall within an interval of 10% around the mean value. With only 6% of variation, the velocity of the vortex is the most unifying quantity that governs the formation of vortex rings behind cones.