2. Steady Inviscid Vortex Rings

Most of the vortex ring models developed so far are based on the hypothesis of an inviscid and steady fluid flow. Deriving these classical and idealistic models is a useful first exercise for anyone interested in the mathematical modelling of vortex rings. We first derive a fundamental solution to the vorticity equation using the Green function. This solution is then used to describe the circular vortex filament, characterised by an infinitely small core and singular vorticity distribution. A closed-form solution for the vortex filament is also provided using the Fourier–Haenkel transform. We subsequently introduce the general solution for the steady inviscid vortex ring with finite core size. As a particular case, we present Hill’s spherical vortex ring, which is the only known closed analytical solution for this problem. For a general shape of the vortex core, the Norbury–Fraenkel family of inviscid vortex rings is obtained by numerical calculation. This very popular model is described in detail. The existence of solutions for inviscid vortex rings with fixed elliptical cross section is also addressed. Finally, we derive the famous Kelvin’s formula for the translational velocity of the vortex ring. As a practical application of this theory, we finally address the problem of the reconstruction of the velocity field of a vortex ring using theoretical models or optimisation approaches.