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 <= R-v <= 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 (epsilon k(1)L/D) for the Norbury vortex ring has been established and used effectively to close the model. For co-flow with 0<= 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 <= R-v <=-0.2 and -0.2<0, respectively.

Automated summary

The formation process of the leading vortex ring in starting jets with uniform background co- and counter-flow has been studied numerically. For counter-flow, the normal formation process of the leading vortex ring would be destroyed at lower velocity ratios. An analytical model based on the kinematic criterion has been developed to describe the relationship between the formation number and velocity ratio. Different equations are proposed for the co-flow and counter-flow cases.