The significance of vortex ring formation to the impulse and thrust of a starting jet

The recent work of Gharib, Rambod, and Shariff [J. Fluid Mech. 360, 121 (1998)] studied vortex rings formed by starting jets generated using a piston-cylinder mechanism. Their results showed that vortex rings generated from starting jets stop forming and pinch off from the generating jet for sufficiently large values of the piston stroke to diameter ratio $\mathrm{(L/D),}$ suggesting a maximization principle may exist for propulsion utilizing starting jets. The importance of vortex ring formation and pinch off to propulsion, however, rests on the relative contribution of the leading vortex ring and the trailing jet (which appears after pinch off) to the impulse supplied to the flow. To resolve the relative importance of the vortex ring and trailing jet for propulsion, a piston-cylinder mechanism attached to a force balance is used to investigate the impulse and thrust generated by starting jets for $\mathrm{L/D}$ ratios in the range 2–8. Two different velocity programs are used, providing two different $\mathrm{L/D}$ values beyond which pinch off is observed, in order to determine the effect of vortex ring pinch off. Measurements of the impulse associated with vortex ring formation show it to be much larger than that expected from the jet velocity alone and proportionally larger than that associated with a trailing jet for $\mathrm{L/D}$ large enough to observe pinch off. The latter result leads to a local maximum in the average thrust during a pulse near $\mathrm{L/D}$ values associated with vortex rings whose circulation has been maximized. These results are shown to be related to the nozzle exit over-pressure generated during vortex ring formation. The over-pressure is in turn shown to be associated with the acceleration of ambient fluid by vortex ring formation in the form of added and entrained mass.