Analysis of inviscid vortex breakdown in a semi-infinite pipe

Axisymmetric vortex breakdown in a steady, inviscid, incompressible flow in a semi-infinite circular pipe is considered analytically. We suggest a new perception of vortex breakdown and compare ours with other approaches. In our view, vortex breakdown occurs due to solution nonuniqueness in some range of inflow parameters when the entire steady flow experiences a jump to another metastable steady state with the same boundary conditions. These co-existing solutions are smooth along the pipe length; they have the same mechanical energy but, in general, different flow forces. Vortex breakdown necessarily occurs by a continuous change in flow parameters (usually the swirl number) when the solution fails to exist (locally) because of fold or similar catastrophe, but spontaneous jumps (in some range of parameters) between different metastable solutions (not on a fold) can also be caused by large flow perturbations. The folds can appear due to transcritical bifurcation, which is destroyed (in the case considered here) by the injection of azimuthal vorticity into the vortex core at the pipe entrance. A high level of the entrance swirl leads to separation zones (even for solid-body inflow!) where the steady flow is undetermined. We find that the nonuniqueness interval in parameter space is connected with the flow pattern inside the separation zone. We consider models for dealing with two such flow patterns: the traditional analytic continuation (leading to a recirculation zone) and a new stagnant separation zone model. We reveal serious defects of the analytic continuation approach. The stagnation zone model is superior in that solutions always exist and, for large enough inflow swirl, exhibit nonuniqueness and folds, thus explaining the experimentally observed hysteretic jump transitions in vortex breakdown. We also predict some new phenomena, which deserve experimental investigation.