A Model For the Limiting Time in Vortex Ring Formation
In another presentation, Gharib et at provide experimental results to show that when a vortex ring is created from a pipe by a piston, there is a limiting time or piston stroke length beyond which multiple rings appear. This time appeared to be insensitive to piston velocity history and Reynolds number. Nature might exploit such a. limit in different contexts to coherently deliver mass or momentum flux with the least number of strokes. Here, a simple hypothesis is considered: the limiting time occurs when the apparatus is no longer able to deliver energy at a rate compatible with the requirement, due to Kelvin, that a steady vortex ring have maximum energy given circulation and impulse. More specifically, the limit is expected to occur when the quantity alpha = E/square root of Gamma(sup 3)I delivered by the piston drops below the value, alpha(sub lim) for a limiting steady vortex ring solution. The resulting predictions agree very well with the experiments (after using alpha(sub lim) measured using the experimental flow fields). The insensitivity to piston history also emerges from the model. Finally, piston histories are designed that may extend the limiting time somewhat.