Abstract:
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.