Swirling flow in an axisymmetric cavity of arbitrary profile, driven by a rotating magnetic field

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Abstract
We investigate the swirling flow of liquid metal in an axisymmetric cavity of arbitrary profile, generated by a rotating magnetic field. In addition to the primary swirling motion, a recirculation is generated by the Bödewadt-like boundary layers on the inclined sides of the cavity. As in the classic problem of ‘spin-up’ in a cylinder, this secondary flow has a dominating effect over the distribution of angular momentum. It is shown that, in the inviscid core, the angular momentum is independent of z, the axial coordinate, and that the applied body force is balanced by the Coriolis force. The bulk of the streamlines pass through both the core and the boundary layer, picking up energy in one region and losing it in the other. By matching the angular momentum and recirculating mass flux in the core to that in the boundary layer, a single governing equation is established for the swirl distribution. This second-order ordinary differential equation is valid for any axisymmetric shape, but is solved here for two cases; those of flow in a truncated cylinder and in a hemisphere. The former of these is compared with previously published experimental data, and with a full numerical simulation. Finally, we extend some of these ideas to buoyancy-driven flow. Here we take advantage of the analogy between centrifugal and thermally stratified flows to model natural convection of liquid metal in a cavity.

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