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# The flow stability of shear layers in a differentially rotating container

thesis

posted on 2017-02-23, 04:14 authored by Vo, Tony DaiA numerical study of flows in a differentially rotating container is presented. The container is comprised of a cylindrical rotating tank coupled with differentially rotating disks placed flush with the top and bottom horizontal boundaries of the tank. The tank and the disk share the same axis of rotation. The differential speed imposed by the disks produce shear layers that are susceptible to instabilities. Flow transitions in the axisymmetric flow including steady, unsteady and time-periodic states are investigated. In addition, non-axisymmetric flows are examined via quasi-two-dimensional and three-dimensional models.
The flow conditions in this configuration are characterised primarily by the Rossby and Ekman numbers, which are non-dimensional parameters. The Rossby number, Ro, describes the balance between inertial to Coriolis forces, while the Ekman number, E, describes the balance between viscous to Coriolis forces. A third non-dimensional parameter, the Reynolds number, which is the ratio of inertial to viscous forces, is also used to characterise the onset of several critical flow transitions. The aspect ratio of the tank, defined by the ratio of the disk radius to the tank height, is explored between 1/6 ≤ A ≤ 2. Additionally, Rossby numbers between −4 ≤ Ro ≤ 0.6 and Ekman numbers ranging between 5×10⁻⁵ ≤ E ≤ 3×10⁻³ are primarily investigated.
A spectral-element method is employed to compute axisymmetric flows on a semi-meridional mesh. The numerous meshes used throughout the study are validated through grid resolution studies under computationally demanding flow conditions. Achieving grid independence for such flow conditions ensures that the flow solutions obtained for a wide range of Ro and E are accurate.
The axisymmetric base flow is obtained for a range of Ro and E, and in various aspect ratio containers. The vertical structure of the flow reveals that small-|Ro| flows demonstrate strong axial independence in accordance with the Taylor–Proudman theorem. This theorem becomes invalid at sufficiently large |Ro| with distinct depth-dependent features displayed in the positive and negative-Ro regime. The transition from reflectively symmetric flow to symmetry-broken flow is determined and reveals an independence on the aspect ratio. Measurements of the Stewartson layer thickness across the explored parameter space have established trends away from Ro ≈ 0 for the
first time. Transition to unsteady and time-dependent flow from these steady-state flows has been achieved by either increasing the Rossby number or decreasing the Ekman number, both of which serve to increase an internal Reynolds number based on the shear layer thickness and velocity differential.
Interest in the developing non-axisymmetric three-dimensional structures on an underlying axisymmetric base flow motivates an application of a linear stability analysis technique. The eigenmodes extracted from the analysis describe the growth rates and the mode shapes of the wavenumber instability. The differentially rotating flow under investigation exhibits an instability typical of barotropic instability as its primarily linear instability mode, with typical dominant scaled azimuthal wavenumbers ranging between 1 ≤ kA ≤ 6. The instability deforms the base flow in such a way that a polygonal structure described by the most unstable wavenumber is seen precessing around the centre of the tank. Secondary instability modes are also present, and often display depth-dependent structures with higher azimuthal wavenumbers. Increases to the aspect ratio demonstrates a shift in preference to lower-wavenumber structures and to a more stable flow. Provided that the shear layer in the axisymmetric base flow is not affected by the confining walls, the growth rate data universally collapses when the azimuthal wavenumber is scaled by the aspect ratio.
Non-axisymmetric studies are conducted using a spectral-element-Fourier method. Nonlinear effects are seen to encourage the coalescence of vortices, resulting in the selection of smaller azimuthal wavenumbers with increased forcing. Thus, the resultant wavenumber generally illustrates a larger difference compared to the predicted linear mode when the described flow conditions move further away from the linear instability threshold. The vacillation process typically occurs through unit increments and reversing the forcing yields an increasing wavenumber configuration with observable hysteresis. The transition from axisymmetric to non-axisymmetric flow is determined to be supercritical from the application of a Stuart–Landau model. The simulations reveal the laborious growth of the instabilities in the flow and suggests that under typical experimental parameters, many months may be required for a stable flow state
to saturate! Thus, these results may have implications for the experimental results previously reported in the literature.
Computations of a quasi-two-dimensional model are used to compare the qualitative and quantitative results of the axisymmetric and three-dimensional flows. Due to the inability of the quasi-two-dimensional model to capture the vertical structure of the flow, the linear stability analysis expresses the same characteristics between positive and negative-Ro flows. The predicted dominant azimuthal wavenumber demonstrates strong agreement with those obtained by the axisymmetric model. Contrasts between the quasi-two-dimensional model and the three-dimensional model are performed, with the same trends being established from both models such that increasing the forcing conditions produces a lower-wavenumber structure.