Formation process of a two-dimensional starting jet

The formation process of a vortex pair generated by a two-dimensional starting jet has been investigated numerically over a range of Reynolds numbers from 500 to 2000. The effects of stroke ratio and nozzle configuration are examined. Only a single vortex pair can be observed in the vorticity field generated by small stroke ratios less than 10 while the leading vortex pair formed by larger stroke ratios eventually disconnects from the trailing jet. The formation numbers (13.6 and 9.3) for a straight nozzle and an orifice nozzle have been identified by the circulation criterion and they are further analysed by four other criteria. Using the contraction coefficient, formation numbers can be transformed into a universal value at about 16.5 for both nozzles. The effect of Reynolds number on the formation number is found to be within 12 % for parallel flow cases but it will increase up to 27 % for non-parallel flow cases due to shear-layer instability. A modified contraction-based slug model is proposed, and it can accurately predict the total invariants (e.g. circulation, hydrodynamic impulse and kinetic energy) shedding from the nozzle edge. Analytical estimation of the formation number is further conducted by matching the predicted total invariants to the Pierrehumbert model of steady vortex pairs. By assuming that pinch-off starts when the vortex pair achieves the steady state, two analytical models are proposed in terms of vortex impulse and translational velocity. The latter appears to be more appropriate to predict the formation number for two-dimensional flows.