Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow

Abstract

Previous studies of the flow created inside a cylindrical cavity (radius R, height H) of fluid by a single rotating end wall have shown that over a range of cylinder aspect ratios 1.6≲H/R≲2.8, the first unsteady flows to bifurcate with increasing Reynolds number retain axisymmetry, and subsequent bifurcations break axisymmetry to give solutions with modulated rotating wave (MRW) states. The underlying axisymmetric components of these MRW flows are nearly indistinguishable from corresponding axisymmetric flows at the same Reynolds numbers. Of the three solution branches so far identified for this flow at H/R=2.5, only one both supports MRWs and has a simple limit-cycle underlying axisymmetric flow. Here, we carry out three-dimensional Floquet stability analysis of this branch of axisymmetric solutions and demonstrate that only a subset of the linearly unstable MRWs are observed asymptotically at large times for full Navier–Stokes solutions. Stability analysis of the time-average axisymmetric flow shows rotating wave (RW) instabilities that are in many ways similar to the MRW Floquet modes. The orientation of the vorticity of the RW and MRW structures implies that they are unlikely to originate as centrifugal instabilities, while simplified inviscid shear flow stability analysis of the time-average velocity profiles suggests instead that they arise as a result of inflectional instability of the swirling wall-jet flow contained by the cylindrical walls of the cavity.