Circulation and formation number of laminar vortex rings

The formation time scale of axisymmetric Vortex rings is studied numericall y for relatively long discharge times. Experimental findings on the existen ce and universality of a formation time scale, referred to as the 'formatio n number', are confirmed. The formation number is indicative of the time at which a vortex ring acquires its maximal circulation. For vortex rings gen erated by impulsive motion of a piston, the formation number was found to b e approximately four, in very good agreement with experimental results. Num erical extensions of the experimental study to other cases, including cases with thick sheer layers, show that the scaled circulation of the pinched-o ff Vortex is relatively insensitive to the details of the formation process , such as the velocity programme, velocity profile, vortex generator geomet ry and the Reynolds number. This finding might also indicate that the prope rly scaled circulation of steady vortex rings varies very little. The forma tion number does depend on the velocity profile. Non-impulsive velocity pro grammes slightly increase the formation number, while non-uniform velocity profiles may decrease it significantly. In the case of a parabolic velocity profile of the discharged flow, for example, the formation number decrease s by a factor as large as four. These findings indicate that a major source of the experimentally found small variations in the formation number is th e different evolution of the velocity profile of the discharged flow.