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

Abstract

The formation process of the leading vortex ring in starting jets with uniform background co- and counter-flow has been studied numerically for $-0.5\leq R_v\leq 0.5$ , where $R_v$ 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 $R_v<-0.4$ , i.e. the trailing jet would overtake the leading vortex ring through the centre, a phenomenon reminiscent of vortex leapfrogging. As the velocity ratio $R_v$ increases, the formation number $F_{t^*}$ decreases from $9.6$ at $R_v=-0.4$ to $1.92$ at $R_v=0.5$ . An analytical model based on the kinematic criterion has been developed so as to describe the relationship between the formation number $F_{t^*}$ and velocity ratio $R_v$ . A linear relationship between the vortex core parameter and stroke ratio of starting jet ( $\varepsilon \sim k_1L/D$ ) for the Norbury vortex ring has been established and used effectively to close the model. For co-flow with $0< R_v\leq 0.5$ , 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 $-0.4\leq R_v\leq -0.2$ and $-0.2< R_v<0$ , respectively.