Highly swirling flows are often prone to precessing instabilities, with an azimuthal wavenumber of m=−1. We carry out a weakly non-linear analysis to determine the response behaviour of this instability to harmonic forcing. An incompressible flow is considered, where an annular inlet provides a swirling flow into a cylindrical region. For high swirl a vortex breakdown is induced, which is found to support an m=−1 instability. By expanding about the Reynolds number where this instability first occurs, a Stuart–Landau equation for the critical mode amplitude can be found and the effect of forcing can be assessed. Two types of forcing are considered. Firstly, a Gaussian forcing confined to the inlet nozzle is used to study m=0 and m=−1 forcings. Secondly, optimal forcings (measured by the two-norm) with azimuthal wavenumbers in the range −3≤m≤3 are considered. It is found that modal stabilization is highly dependent on the azimuthal wavenumber m, which governs whether the forcing is counter or co-rotating with the direction of swirl. Counter-rotating forcings are able to stabilize the mode for a wide range of forcing frequencies, while co-rotating forcings fail to yield a stable flow. In all cases, it is the base-flow modification induced by the forced response that is the dominant underlying feature responsible for the observed stabilization. This base-flow modification seeks to reduce axial momentum near the recirculation region for co-rotating forcings, and increase it for counter-rotating forcings, thus changing the size of the recirculation bubble and producing the two distinct response behaviours.Leverhulme Trust, Isaac Newton Trus
