Abstract
We derive a model for an axisymmetric vortex ring confined in a tube. We start by assuming that the vorticity distribution in the vortex ring is described by models for unconfined viscous vortex rings presented in Chaps. 3 and 4. The Stokes stream function of the confined vortex ring is then presented as the difference between the stream function of the unconfined vortex ring and a wall-induced correction. Based on the asymptotic development of the vorticity in the vicinity of the tube wall, we generalise Brasseur’s approach (Brasseur 1979) to derive the wall-induced correction. The model takes into account vortex ring cores with quasi-circular or elliptical shapes. For the confined vortex ring, closed formulae for the stream function and vorticity distribution are derived. The predictions of the model are shown to be in agreement with direct numerical simulations of confined vortex rings generated by a piston–cylinder mechanism. A simplified procedure for fitting experimental and numerical data with the predictions of the model is described. This opens the way for applying the model to realistic confined vortex rings in various applications.
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