3. Viscous Vortex Rings

We present a viscous vortex ring model based on a solution to the time-dependent Stokes-flow equation. The solution has a quasi-isotropic Gaussian vorticity distribution, which is more realistic compared to the one assumed in steady inviscid vortex rings models discussed in the previous chapter. We derive closed formulae for the stream function, translational velocity of the vorticity centroid and integral characteristics (impulse, circulation, kinetic energy) of the viscous vortex ring. Well-known results for the translational velocity are recovered for the limiting regimes of short-time evolution (Saffman) and long-time evolution (Rott and Cantwell). The model is then generalised to include a time-evolving eddy viscosity scale, offering a unified framework to describe both laminar and turbulent vortex rings. A Reynolds-number correction of the model is derived using the method of matched asymptotic expansions. A comparison with experimental results and direct numerical simulations shows that the model describes well the integral quantities and provides a more accurate description of the vortex ring geometry than inviscid models.