Reynolds-number effect on vortex ring evolution in a viscous fluid

It is known that the cross section of the vortex ring core takes an approximately elliptical shape with increasing Reynolds number. In order to model this feature, the functional form of a vortex ring solution of the Stokes equations is modified so as to be able to model higher Reynolds number rings. The model introduces two nondimensional parameters that govern the shape of the vortex core:λ ≥ 1 and β ≥ 1. Based on this modification, new expressions for the translation velocity, energy, circulation, and streamfunction are derived for a wide range of section ellipticity that are specific to such vortices. To validate the model, the data adapted from the numerical study of vortex ring at Reynolds number Re = 1400 performed by Danaila and Helie [Phys. Fluids20, 073602 (2008)], is used. In this case, the appropriate values of λ and β are calculated by equating the normalized energy E_d and circulation Γ_d of the theoretical vortex to the corresponding values obtained from the numerical data. The model provides a good prediction of the ring velocity evolution at high Reynolds numbers.