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

The formation of an axisymmetric vortex ring by forcing fluid impulsively t hrough a pipe is examined. An idealized model of the circulation, impulse a nd energy provided by the injected plug is developed, and these quantities are equated to the corresponding properties of the class of rings with fini te cores described by Norbury (1973). It is shown that, as the length-to-di ameter aspect ratio LID of the plug increases, the size of the core increas es in comparison with all the fluid carried along with the ring, until the limiting case of Hill's spherical vortex is reached. For aspect ratios larg er than a certain value it is not possible to produce a single ring while c onserving circulation, impulse, volume and energy. This implies that the li miting vortex is 'optimal' in the sense that it has maximum impulse, circul ation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the LID ratio that can be achieved in p ractice is more appropriately taken from the direct experimental measuremen ts of Gharib et al. (1998) who concluded that the limiting value is L/D = 4 . This is close to the value found in our calculation.