Three-dimensional instabilities and inertial waves in a rapidly rotating split-cylinder flow

Abstract

The nonlinear dynamics of the flow in a differentially rotating split cylinder is investigated numerically. The differential rotation, with the top half of the cylinder rotating faster than the bottom half, establishes a basic state consisting of a bulk flow that is essentially in solid-body rotation at the mean rotation rate of the cylinder and boundary layers where the bulk flow adjusts to the differential rotation of the cylinder halves, which drives a strong meridional flow. There are Ekman-like layers on the top and bottom end walls, and a Stewartson-like side wall layer with a strong downward axial flow component. The complicated bottom corner region, where the downward flow in the side wall layer decelerates and negotiates the corner, is the epicentre of a variety of instabilities associated with the local shear and curvature of the flow, both of which are very non-uniform. Families of both high and low azimuthal wavenumber rotating waves bifurcate from the basic state in Eckhaus bands, but the most prominent states found near onset are quasiperiodic states corresponding to mixed modes of the high and low azimuthal wavenumber rotating waves. The frequencies associated with most of these unsteady three-dimensional states are such that spiral inertial wave beams are emitted from the bottom corner region into the bulk, along cones at angles that are well predicted by the inertial wave dispersion relation, driving the bulk flow away from solid-body rotation.

Experimental lateral wall boundary layer behavior of a differentially rotating split-cylinder flow

“…In this second case, the boundary layer of the bases (Ekman type) and the boundary layer of the cylindrical wall (Stewartson type) [25] meet on the corners and an instability in the form of periodic or quasiperiodic states can appear and propagate from the faster corner to the slower corner depending on the Ro according to Refs. [23,28]. Hence, the zone near the cylindrical wall has to adapt its angular velocity from the solid-body rotation ( ) to the wall rotation ( ± ω).…”

“…Gutierrez-Castillo and Lopez [28] reproduced this analysis numerically in two dimensions (axisymmetric flows) and extended it for large but finite rotation velocities (nonlinear viscous problem) and larger differential rotations studying the stability of the main flow. Later this numerical analysis was extended in three dimensions [23] finding nonaxisymmetric instabilities. For this corotating split cylinder, the flow developed inside is in almost solid-body rotation at the average rotation rate, as in the Stewartson's problem, and a meridional flow appears driving fluid from one end wall to the other with the sandwich structure found in Ref.…”

“…A differential rotation in a cylindrical geometry can trigger different instabilities where the boundary conditions play a decisive role [7,20]. But, when the background rotation is fast enough, the Coriolis restoring force tends to restrict secondary flows to the boundary layers [21][22][23]. Then the flow developed inside the container is basically in solid-body rotation at the mean rotation of the system.…”

Inertial waves in rapidly rotating flows: a dynamical systems perspective

“…As in [23], when the flow is not restricted to being axisymmetric, [22] have shown that the primary instabilities are three-dimensional. Depending on ω 0 and δω, rotating waves localised in the junction region of the slower rotating half of the cylinder are found, either with low or high azimuthal wave numbers.…”

“…However, the associated wave beams are not evident in the nonlinear solutions. High wave number beam do not penetrate deep into the interior [44,22], and so even if there is turbulence in the boundary layers (broad-band spatial spectrum), only the low wave number part of the spectrum drives beams into the interior. So, the interior remains laminar and coherent with low wave number shear layers.…”