Paradoxical predictions of swirling jets

This paper examines the shape of a steady jet with a swirling component, ejected from a circular orifice at an angle to the horizontal. Assuming the Froude number to be large, we derive a set of asymptotic equations for a slender jet. In the inviscid limit, the solutions of the set predict that, if the swirling velocity of the flow exceeds a certain threshold, the jet bends against gravity and rises until the initial supply of the liquid's kinetic energy is used up. This effect is due to the fact that the contributions of the swirl and streamwise velocities to the centrifugal force are of opposite signs, with their sum to be balanced by gravity. As a result, swirl- and streamwise-dominated jets bend in opposite directions. Downward-bending jets also exhibit counter-intuitive behaviour. If the swirling velocity is strong enough (but is still below the above threshold), the streamwise velocity on the jet's axis may decrease with the distance from the orifice, despite the acceleration due to gravity. Eventually, a stagnation point emerges due to this effect, arguably destabilising the jet. Also paradoxically, viscosity-dominated jets can reach higher (if they bend upwards) and farther (in all cases) than their inviscid counterparts, due to the fact that viscosity suppresses formation of stagnation points.