An over-pressure correction to the slug model for vortex ring circulation

The slug model provides a formula for the circulation of vortex rings formed by starting jets ejected from a nozzle or orifice in terms of the time history of the jet velocity. For rapidly initiated jets, however, the model under-predicts circulation by an amount that remains nearly constant as circulation increases. To remedy this shortcoming, the present study derives an equation for circulation directly from the governing equations for the flow, illustrating that the circulation can be expressed as the sum of a flux term, $\Gamma_U$, determined by the jet centreline velocity and a ‘source’ term, $\Gamma_p$, determined by over-pressure at the jet centreline during vortex ring formation. For impulsively or nearly impulsively initiated jets, $\Gamma_p$ is modelled using potential flow analysis since over-pressure is significant primarily during the rapid jet initiation in these cases. The effects of the jet initiation and boundary-layer growth on the jet centreline velocity are also considered to provide a realistic model for $\Gamma_U$. Models for nozzle and orifice configurations are developed. In the case of a nozzle-type generator, the need to model the jet centreline velocity leads to an approximate solution for starting flow near the exit of an infinitely long pipe. For the nozzle case, the model results agree with available circulation measurements to within ${\pm}$6.2% for a wide range of conditions, except for jet velocity programs that are nearly triangular. In the orifice case, the model agrees with available results to within ${-}$13%. Both cases show substantial improvement over the slug model, which is in error by ${-}$21% to ${-}$67% for the data considered.