Invariants and asymptotics of axisymmetric swirling submerged jets

An axisymmetric laminar swirling jet of a viscous incompressible fluid flowing from a rotating semi-infinite tube in space filled with the same fluid is explored. The inner surface of the tube rotates with a constant angular velocity, the outer surface is stationary or rotates with the same angular velocity. It is shown that in the first case, the flow field far from the tube orifice is described by the Loitsyansky asymptotic solution, and in the second case (with a weak coflow flow), it is described by the Long–Goldshtik–Zoubtsov self-similar solution. The Goldshtik hidden invariant is generalized to arbitrary axisymmetric swirling jets, and its influence on the jet asymptotics is studied. Strong swirling jets are calculated, and the dependence of the parameters of the recirculation zone (vortex breakdown in a swirling jet) on the swirl number and the Reynolds number is examined.