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 < Rv 0, respectively.