Reynolds-number effect on vortex ring evolution

An analytical model describing a vortex ring for low Reynolds numbers (Re) proposed previously by Kaplanski and Rudi [Phys. Fluids,17, 087101 (2005)], is extended to a vortex rings for higher Reynolds numbers. The experimental results show that the vortex ring core takes the oblate ellipsoidal shape with increasing Re. In order to model this feature, we suggest an expression for the vorticity distribution, which corrects the linearized solution of the Navier-Stokes equation, with two disposable nondimensional parameters λ and β governing the shape of the vortex core, and derive the new expressions for the streamfuction, circulation, energy and translation velocity on the basis of it. The appropriate values of λ and β are calculated by equating the nondimensional energy E_d and circulation Γ_d of the theoretical vortex to the corresponding values obtained from the experimental or numerical vortex ring. To validate the model, the data adapted from the numerical study of a vortex ring at Re=1400 performed by Danaila and Helie [Phys. Fluids, 20, 073602 (2008)], is applied. It is shown that the predicted temporal evolution of the translation velocity at high Reynolds numbers matches very well with the experiments and numerical simulations.