The influence of upstream boundary conditions on the onset of axisymmetric vortex breakdown in a duct

Past numerical studies on vortex breakdown in a duct have identified the choice of up- stream boundary conditions as a source of major concern. While the choice for axial and azimuthal velocity profiles is often dictated by the experiments, the choice for the third upstream condition does not appear unanimous and its influence on the scenario of transition to breakdown is still unclear. Our purpose in this paper is to address this question in a theoretical framework. We investigate the axisymmetric flow of an inviscid fluid in a duct of constant cross-section and finite axial length. Solutions bifurcating from the columnar solution at criticality are analyzed via a weakly nonlinear expansion and computed in the fully nonlinear regime using numerical continuation, until a centerline recirculation is found at the duct outlet. The linear stability of these flows is also investigated. We apply these techniques to two typical flows frequently used in the literature: the Burgers-Rott vortex, and plug axial flow with solid body rotation. For the latter case in particular, we observe that imposing a vanishing radial velocity instead of a fixed azimuthal vorticity as the third upstream boundary condition leads to a bifurcation diagram different from the traditional picture given by Wang and Rusak [JFM 340, 1997]. According to this new scenario, as the swirl number (the ratio of azimuthal to axial velocity) is increased, vortex breakdown may set in as a continuous process with no loss of stability and no hysteresis, the swirl number for the appearance of the recirculation being larger than that defining flow criticality.