Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially corotating lid

Abstract

The flow in an enclosed completely filled rapidly rotating cylinder that is driven by the differential corotation of the top lid is studied numerically. Although the flow is in a very simple geometry, the fast background rotation and large differential rotation of the lid lead to very thin boundary layers with a variety of instability modes with very fine spatial scales as well as inertial waves that are sustained in the fast rotating interior flow and that interact with the viscous modes in the sidewall boundary layer, leading to complex spatiotemporal dynamics. The numerical simulations are compared and contrasted to experimental visualizations of the sidewall boundary layer instabilities reported by Hart and Kittelman [“Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder,” Phys. Fluids 8, 692 (1996)]. The experiments report observing axisymmetric rolls propagating down the sidewall layer for small differential corotation of the top lid. As the differential rotation was increased, backward tilted diagonal rolls that precess slightly retrograde with respect to the rotating sidewall and forward tilted rolls with prograde precession significantly faster than the sidewall rotation were observed. For still larger differential rotation, a wavy turbulent state that has backward tilted structures erupting from deep within the sidewall layer into the interior and is riding on the forward tilted diagonal rolls in the deep layer was observed. Our simulations capture all of these states and strongly suggest that the observed axisymmetric rolls are unstable and were only transiently observed due to the slow and continuous increase in the differential rotation employed in the experiments. The influence of inertial waves driven by the sidewall instabilities on the three-dimensional wavy turbulent state is discussed.