Formation number of confined vortex rings

This paper investigates the formation number of vortex rings generated by a piston-cylinder mechanism in a confined tube. We use Direct Numerical Simulations (DNS) of axisymmetric confined vortex rings to study the influence of different parameters on the separation (or pinch-off) of the vortex ring from the trailing jet. It is shown that the structure of the vortex ring at pinch-off depends on the type of injection program (pulse dominated either by positive or negative acceleration ramps) and the confinement ratio Dw=D, where Dw is the inner diameter of the tube and D the diameter of the cylinder). For low confinement ratios (Dw=D 2), a vortex of opposite sign generated at the lateral wall strongly interacts with the vortex ring and the pinch-off is not clearly observed. The pinch-o is observed and analysed for confinement ratios Dw=D 2:5. DNS
data is used to estimate the value of the formation time, which is the time necessary for the vortex generator to inject the same amount of circulation as carried by the detached vortex ring. The confined vortex ring at pinch-o is described by the model suggested by Danaila, Kaplanski and Sazhin [A model for confined vortex rings with elliptical core vorticity distribution, Journal of Fluid
Mechanics, 811:67-94, 2017]. This model allows us to take into account the influence of the lateral wall and the elliptical shape of the vortex core. The value of the formation time is predicted using this model and the slug-flow model.