The role of boundary conditions in a simple model of incipient vortex breakdown

We consider incipient vortex breakdown and describe how infinitesimal perturbations may destabilize a columnar swirling jet. The framework is axisymmetric and inviscid following Wang and Rusak's [J. Fluid Mech. 340, 177 (1997)] analysis. The goal of the present study is to relate the local properties of swirling flows in infinite pipes to their global stability properties in pipes of finite length. A spatial linear stability analysis is pursued which gives a complementary point of view to the subcritical/supercritical concept introduced by Benjamin [J. Fluid Mech. 14, 593 (1962)]. In contrast to supercritical flows which exhibit two neutral spatial branches traveling downstream and two counterpropagating evanescent spatial branches, subcritical flows exhibit a frequency range where all spatial branches are neutral, three of which travel downstream and one upstream. By using global energy budget arguments and monitoring how the upstream wave is reflected into the downstream waves and conversely, the inlet and outlet conditions are shown to drive the instability in the limit of long but finite pipes. Various inlet and outlet conditions are proposed that stabilize or destabilize the flow, depending on their ability to supply energy. The analysis demonstrates therefore that the global instability accounting for incipient vortex breakdown in Wang and Rusak's model may arise from the combination of a locally neutral flow and suitable inlet and outlet conditions. (C) 2004 American Institute of Physics.