Vortex ring formation process in starting jets with uniform background co- and counter-flow

The formation process of the leading vortex ring in starting jets with uniform background
co- and counter-flow has been studied numerically for −0.5 ≤ Rv ≤ 0.5, where Rv is the
ratio of background velocity to jet velocity. For the cases with background counter-flow,
the normal formation process of the leading vortex ring would be destroyed when Rv <
−0.4, i.e. the trailing jet would overtake the leading vortex ring through the centre,
a phenomenon reminiscent of vortex leapfrogging. As the velocity ratio Rv increases,
the formation number Ft∗ decreases from 9.6 at Rv = −0.4 to 1.92 at Rv = 0.5. An
analytical model based on the kinematic criterion has been developed so as to describe the
relationship between the formation number Ft∗ and velocity ratio Rv. A linear relationship
between the vortex core parameter and stroke ratio of starting jet (ε ∼ k1L/D) for the
Norbury vortex ring has been established and used effectively to close the model. For
co-flow with 0 < Rv ≤ 0.5, the results from this model are consistent with the present
numerical simulation and the experiments by Krueger et al. (J. Fluid Mech., vol. 556,
2006, pp. 147–166). For counter-flow, two different equations are proposed for −0.4 ≤
Rv ≤ −0.2 and −0.2 < Rv < 0, respectively.