Quasi-steady linked vortices with chaotic streamlines
Abstract This paper describes the motion and the flow geometry of two or more linked ring vortices in an otherwise quiescent, ideal fluid. The vortices are thin tubes of near-circular shape which lie on the surface of an immaterial torus of small aspect ratio. Since the vortices are assumed to be identical and evenly distributed on any meridional section of the torus, the flow evolution depends only on the number of vortices ($n$) and the torus aspect ratio (${r}_{1} / {r}_{0} $, where ${r}_{0} $ is the centreline radius and ${r}_{1} $ is the cross-section radius). Numerical simulations based on the Biot–Savart law showed that a small number of vortices ($n= 2, 3$) coiled on a thin torus (${r}_{1} / {r}_{0} \leq 0. 16$) progressed along and rotated around the symmetry axis of the torus in an almost uniform manner while each vortex approximately preserved its shape. In the comoving frame the velocity field always possesses two stagnation points. The transverse intersection, along $2n$ streamlines, of the stream tube emanating from the front stagnation point and the stream tube ending at the rear stagnation point creates a three-dimensional chaotic tangle. It was found that the volume of the chaotic region increases with increasing torus aspect ratio and decreasing number of vortices.