Linked Toroidal Vortices

Abstract

This paper deals with the dynamics of two or more toroidal filamentary vortices—i.e. thin tubular vortices coiled on an immaterial torus—in an otherwise quiescent, ideal fluid. If the vortices are identical and equally spaced on a meridional section of the torus, the flow evolution depends on the torus aspect ratio ($r_1/r_0,$ where $r_0$ is the radius of the centreline and $r_1$ is the radius of the cross section), the number of vortices (N), and the vortex topology ($V_{p,q},$ denoting a vortex that winds p times round the torus symmetry axis and q times round the torus centreline). The evolution of sets of $NV_{1,2}$ vortices was computed using the Rosenhead–Moore approximation to the Biot–Savart law to evaluate the velocity field and a fourth-order Runge–Kutta scheme to advance in time. It was found that when a small number of vortices is coiled on a thin torus the system progressed along and rotated around the torus symmetry axis in an almost steady manner, with each vortex approximately preserving its shape.