An extended model for orifice starting jets and its implications on vortex ring formation

Starting jets emanating from a straight nozzle and orifices of different orifice-to-tube diameter ratios are investigated using planar particle image velocimetry. The invariants of the motion are measured and compared to the classic slug-flow model. An extension to the slug-flow model is proposed to account for the contraction the fluid is experiencing when being pushed through orifice geometries. The contraction coefficient obtained for two-dimensional jets is applied to the axisymmetric problem. This modified slug-flow model is shown to better predict the invariants of the motion with discrepancies of the order of 10% compared to underpredictions of 130%, 50% and 120% for circulation, hydrodynamic impulse and kinetic energy, respectively, using the classic slug-flow model. Moreover, the new model suggests the existence of a maximum in the production of impulse and energy at an orifice-to-tube diameter ratio of about 0.9, which was also recorded experimentally for the kinetic energy. Finally, the implications of the modified model on vortex ring formation are discussed; the formation time is redefined in terms of the quantities in the vena contracta as T^*=U_*t/D_*, which ultimately unifies the formation number of orifices and straight nozzles with a value of approximately 4.