Estimating the non-dimensional energy of vortex rings by modelling their roll-up

Abstract

The non-dimensional energy of starting vortex rings typically converges to values around 0.33 when they are created by a piston-cylinder or a bluff body translating at a constant speed. To explore the limits of the universality of this value and to analyse the variations that occur outside of those limits, we present an alternative approach to the slug-flow model to predict the non-dimensional energy of a vortex ring. Our approach is based on the self-similar vortex sheet roll-up described by Pullin (J. Fluid Mech., vol. 88, 1978, pp. 401–430). We derive the vorticity distribution for the vortex core resulting from a spiralling shear layer roll-up and compute the associated non-dimensional energy. To demonstrate the validity of our model for vortex rings generated through circular nozzles and in the wake of disks, we consider different velocity profiles of the vortex generator that follow a power law with a variable time exponent $m$ . Higher values of $m$ indicate a more uniform vorticity distribution. For a constant velocity ( $m=0$ ), our model yields a non-dimensional energy of ${{E}^{{*}}}=0.33$ . For a constant acceleration ( $m=1$ ), we find ${{E}^{{*}}}=0.19$ . The limiting value $m \rightarrow \infty$ corresponds to a uniform vorticity distribution and leads to ${{E}^{{*}}}=0.16$ , which is close to values found in the literature for Hill's spherical vortex. The radial diffusion of the vorticity within the vortex core results in the decrease of the non-dimensional energy. For a constant velocity, we obtain realistic vorticity distributions by radially diffusing the vorticity distribution of the Pullin spiral and predict a decrease of the non-dimensional energy from 0.33 to 0.28, in accordance with experimental results. Our proposed model offers a practical alternative to the existing slug flow model to predict the minimum non-dimensional energy of a vortex ring. The model is applicable to piston-generated and wake vortex rings and only requires the kinematics of the vortex generator as input.