Modelling vortex ring growth in the wake of a translating cone

Abstract

Vortex rings have the ability to transport fluid over long distances. They are usually produced by ejecting a volume of fluid through a circular orifice or nozzle. When the volume and velocity of the ejected fluid are known, the vortex' circulation, impulse, and energy can be estimated by the slug flow model. Vortex rings also form in the wake of accelerating axisymmetric bodies. In this configuration, the volume and velocity of the fluid that is injected into the vortex is not known a priori. Here, we present two models to predict the growth of the vortex behind disks or cones. The first model uses conformal mapping and assumes that all vorticity generated ends up in the vortex. The vortex circulation is determined by imposing the Kutta condition at the tip of the disk. The position of the vortex is integrated from an approximation of its velocity, given by Fraenkel. The model predicts well the maximum circulation of the vortex, but does not predict the tail shedding observed experimentally. A second model is based on an axisymmetric version of the discrete vortex method. The shear layer formed at the tip of the cone is discretised by point vortices, which roll-up into a coherent vortex ring. The model accurately captures the temporal evolution of the circulation and the non-dimensional energy. The detrainment of vorticity from the vortex, through the process of tail shedding, is qualitatively captured by the model but remains quantitatively sensitive to the numerical parameters.