A model for confined vortex rings with elliptical-core vorticity distribution

We present a new model for an axisymmetric vortex ring confined in a tube. The model takes into account the elliptical (elongated) shape of the vortex ring core and thus extends our previous model [Danaila, Kaplanski and Sazhin, J. of Fluid Mechanics, 774, 2015] derived for vortex rings with quasi-circular cores. The new model offers a more accurate description of the deformation of the vortex ring core, induced by the lateral wall, and a better approximation of the translational velocity of the vortex ring, compared with the previous model. The main ingredients of the model are the following: the description of the vorticity distribution in the vortex ring is based on the previous model of unconfined elliptical-core vortex rings [Kaplanski, Fukumoto and Rudi, Physics of Fluids, 24, 2012]; Brasseur's approach [Brasseur, PhD Thesis, 1979] is then applied to derive a wall-induced correction for the Stokes stream function of the confined vortex ring flow. We derive closed formulae for the flow stream function and vorticity distribution. An asymptotic expression for the long time evolution of the drift velocity of the vortex ring as a function of the ellipticity parameter is also 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. The predictions of the model support the recently suggested heuristic relation [Krieg andMohseni, J. Fluid Engineering, 135, 2013] between the energy and circulation of vortex rings with converging radial velocity. A new 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 including those in internal combustion engines.