The formation of ''optimal'' vortex rings, and the efficiency of propulsion devices (Q2709167)

The authors examine the formation of an axisymmetric vortex ring by impulsively forcing the fluid flow through a pipe. An idealized model is developed for circulation, impulse and energy provided by the injected plug, and these quantities are compared with the corresponding quantities of rings with finite cores described by \textit{J. Norbury} [J. Fluid Mech. 57, 417-431 (1973; Zbl 0254.76018)]. It is shown that, as the length-to-diameter aspect ratio \(L/D\) of the plug increases, the size of the core increases in comparison with all the fluid carried by the ring, until the limiting case of Hill's spherical vortex is reached. For aspect ratios larger than a certain value, it is not possible to produce a single ring by conserving simultaneously the circulation impulse, volume and energy. This implies that the limiting vortex is `optimal' in the sense that it has maximal impulse, circulation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the \(L/D\) ratio can be more appropriately taken from direct experimental measurements of \textit{M. Gharib} et al. [J. Fluid Mech. 360, 121-140 (1998; Zbl 0922.76021)] who concluded that the limiting value of \(L/D\) is 4. This is close to the value found in the present paper.