An examination of a group-velocity criterion for the breakdown of an idealized vortex flow

The phenomenon of vortex breakdown is believed to be associated with a finite amplitude wave that has become trapped at the critical or breakdown location. The conditions at which the propagating waves become trapped at a certain axial location were examined by use of a group-velocity criterion implied by Landahl's general theory of wave trapping. An ideal vortex having constant vorticity and uniform axial velocity at the inlet of a slowly diverging duct was studied. The linear wave propagation analysis is applied to the base flow at several axial stations for several values of the ratio of swirl velocity to axial velocity at the inlet of the divergent duct, assuming a locally parallel flow. The dipsersion relations and hence the group velocities of both the symmetric (n = 0) and asymmetric modes (n = + or - 1) were investigated. The existence of a critical state in the flow (at which the group velocity vanishes), and its relationship to the stagnation point on the axis of the duct and to the occurrence of an irregular singularity in the equations governing wave propagation in the flow field are discussed.