Scaling of the translational velocity of vortex rings behind conical objects

Ring vortices are efficient at transporting fluid across long distances. They can be found in nature in various ways: they propel squids, inject blood in the heart, and entertain dolphins. These vortices are generally produced by ejecting a volume of fluid through a circular orifice. The impulse given to the vortex rings moving away results in a propulsive force on the vortex generator. Propulsive vortex rings have been widely studied and characterized. After four convective times, the vortex moves faster than the shear layer it originates from and separates from it. When the vortex separates, the circulation of the vortex reaches a maximum value, and the nondimensional energy attains a minimum. The simultaneity of these three events obfuscates the causality between them. To analyze the temporal evolution of the nondimensional energy of ring vortices independent of their separation, we analyze the spatiotemporal 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. The energy, circulation, and velocity of these vortices were extracted based on time-resolved velocity field measurements. The vortex 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 timescales observed for propulsive vortices. The nondimensional circulation, based on the vortex diameter, and the nondimensional 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 nondimensional energy to reach a minimum value. The limiting values of the nondimensional circulation and energy are mostly independent of the cone geometry and translational velocity and fall within an interval of 10% around the mean value. The velocity of the vortex shows only 6% of variation and is the most unifying quantity that governs the formation of vortex rings behind cones.