A model for universal time scale of vortex ring formation

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Abstract
An analytical model for predicting the universal time scale for formation of vortex rings generated through impulsively started jets is considered. The model is based on two assumptions, namely the validity of the slug model in simulating the discharge process of the fluid out of the cylinder and the approximation of the vortex at the pinch off moment by a vortex in the Norbury family. The nondimensional stroke length L/D (referred to as “formation number,” following Gharib et al. [J. Fluid Mech. 360, 121 (1998)]) predicted by the model satisfactorily matches the experimental observation of Gharib et al. The model introduces two nondimensional parameters that govern the limiting formation number: nondimensional energy E nd and circulation Γ nd . The predicted value of E nd matches very well with the experimental data. It is also predicted that there is a limiting value for the nondimensional circulation in the range 1.77≲Γ nd ≲2.07.

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