The formation number of vortex rings formed in uniform background co-flow

The formation of vortex rings generated by an impulsively started jet in the presence of uniform background co-flow is studied experimentally to extend previous results. A piston–cylinder mechanism is used to generate the vortex rings and the co-flow is supplied through a transparent shroud surrounding the cylinder. Digital particle image velocimetry (DPIV) is used to measure the development of the ring vorticity and its eventual pinch off from the generating jet for ratios of the co-flow to jet velocity ($R_{v})$ in the range 0 – 0.85. The formation time scale for the ring to obtain maximal circulation and pinch off from the generating jet, called the formation number ($F$), is determined as a function of $R_{v}$ using DPIV measurements of circulation and a generalized definition of dimensionless discharge time or ‘formation time’. Both simultaneous initiation and delayed initiation of co-flow are considered. In all cases, a sharp drop in $F$ (taking place over a range of 0.1 in $R_{v}$) is centred around a critical velocity ratio ($R_{crit}$). As the initiation of co-flow was delayed, the magnitude of the drop in $F$ and the value of $R_{crit}$ decreased. A kinematic model based on the relative velocities of the forming ring and jet shear layer is formulated and correctly predicts vortex ring pinch off for $R_{v} \,{>}\, R_{crit}$. The results of the model indicate the reduction in $F$ at large $R_{v}$ is directly related to the increased convective velocity provided to the ring by the co-flow.