Theory and observations of waves on hollow-core vortices

Abstract

A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.