Non-axisymmetric flows in a differential-disk rotating system

Abstract

The non-axisymmetric structure of an unstable Stewartson shear layer generated via a differential rotation between flush disks and a cylindrical enclosure is investigated numerically using both three-dimensional direct numerical simulation and a quasi-two-dimensional model. Previous literature has only considered the depth-independent quasi-two-dimensional model due to its low computational cost. The three-dimensional model implemented here highlights the supercritical instability responsible for the polygonal deformation of the shear layer in the linear and nonlinear growth regimes and reveals that linear stability analysis is capable of accurately determining the preferred azimuthal wavenumber for flow conditions near the onset of instability. This agreement is lost for sufficiently forced flows where nonlinear effects encourage the coalescence of vortices towards lower-wavenumber structures. Time-dependent flows are found for large Reynolds numbers defined based on the Stewartson layer thickness and azimuthal velocity differential. However, this temporal behaviour is not solely characterized by Reynolds number but is rather a function of both the Rossby and Ekman numbers. At high Ekman and Rossby numbers, unsteady flow emerges through a small-scale azimuthal destabilization of the axial jets within the Stewartson layers; at low Ekman numbers, unsteady flow emerges through a modulation in the strength of one of the axial vortices rolled up by non-axisymmetric instability of the Stewartson layer.