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

Abstract

The formation of an axisymmetric vortex ring by forcing uid impulsively through a pipe is examined. An idealized model of the circulation, impulse and energy provided by the injected plug is developed, and these quantities are equated to the corresponding properties of the class of rings with finite cores described by Norbury (1973). 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 along with 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 while conserving circulation, impulse, volume and energy. This implies that the limiting vortex is ‘optimal’ in the sense that it has maximum impulse, circulation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the L/D ratio that can be achieved in practice is more appropriately taken from the direct experimental measurements 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.