Simulation of a vortex ring of significant core thickness

Consider axisymmetric, solenoidal motion of an inviscid uniform density fluid in which the region of rotational motion is interior to a torus. Let the fluid at remote distances from the torus be at rest and let the motion be steady relative to an observer moving with the torus. The simulation assumes that the azimuthal vorticity is directly proportional to the distance from the axis of symmetry (an assumption consistent with the fact that material lines, and hence vortex lines, undergo stretching and compression as they convect from one position to another in the core). The simulation assumes unit value of the centroidal radius---i.e. the distance from the centroid of a typical meridional cross section to the axis of symmetry---as well as unit value of the total dipole strength of the motion. The minor radius---i.e. the radius of a circular disk whose area is that of a typical meridional section the core---is one half the centroidal radius. The results show that if the core cross section is circular there is a non-physical slip between the fluid inside and outside the core. One may eliminate the slip by optimizing the cross sectional shape. The radius of curvature of the optimum shape is largest at the inner equator and smallest at the leading and trailing extremities.