Linear stability analysis of a shear layer induced by differential coaxial rotation within a cylindrical enclosure

Abstract

The generation of distinct polygonal configurations via the instability of a Stewartson shear layer is numerically investigated. The shear layer is induced using a rotating cylindrical tank with differentially forced disks located at the top and bottom boundaries. The incompressible Navier–Stokes equations are solved on a two-dimensional semi-meridional plane. Axisymmetric base flows are consistently found to reach a steady state for a wide range of flow conditions, and details of the vertical structure are revealed. An axially invariant two-dimensional flow is ascertained for small $\vert \mathit{Ro}\vert $, which substantiates the Taylor–Proudman theorem. Sufficient increases in $\vert \mathit{Ro}\vert $ forcing develops flow features that break this quasi-two-dimensionality. The onset of this breaking occurs earlier with increasing $\vert \mathit{Ro}\vert $ for $\mathit{Ro}\gt 0$ compared with $\mathit{Ro}\lt 0$. The thickness scaling of the vertical Stewartson layers are in agreement with previous analytical results. Growth rates of the most unstable azimuthal wavenumber from a global linear stability analysis are obtained. The threshold between axisymmetric and non-axisymmetric flow follows a power law, and both positive- and negative-$\mathit{Ro}$ regimes are found to adopt the same threshold for instability, namely $\vert \mathit{Ro}\vert \geq 18. 1{E}^{0. 767} $. This relationship corresponds to a constant critical internal Reynolds number of ${\mathit{Re}}_{i, c} \simeq 22. 5$. A review of reported critical internal Reynolds number and their characteristic length scales yields a consistent instability onset given by $\vert \mathit{Ro}\vert / {E}^{3/ 4} = 15. 4{\unicode{x2013}} 16. 6$; here we find $\vert \mathit{Ro}\vert / {E}^{3/ 4} = 15. 8$. At the onset of linear instability, the initially circular shear layer deforms, resulting in a polygonal structure consistent with barotropic instability. Dominant azimuthal wavenumbers range from $3$ to $7$ at the onset of instability for the parameter space explored. Empirical relationships for the preferential wavenumber have been obtained. Additional instability modes have been discovered that favour higher wavenumbers, and these exhibit structures localized to the disk–tank interfaces.

Linear stability and energetics of rotating radial horizontal convection

“…Nevertheless, the observed agreement confirms that the relation obtained from the potential energy equation for horizontal convection (Winters & Young 2009;Barkan et al 2013) at steady state extends to radial horizontal convection with rotation in cylindrical enclosures. (Stewartson 1957;Hide & Titman 1967;Niino & Misawa 1984;Vo et al 2014Vo et al , 2015. The next section describes the linear stability analysis performed to elucidate the instability mechanisms active in the rotating radial horizontal convection system.…”

“…Geophysical applications provoke interest in baroclinic instability in these flows, though in practice the finite enclosure of this system in laboratory set-ups inevitably render them susceptible to other instability mechanisms, including Stewartson layers arising from differential rotation between interior flow and sidewall shear layers (Stewartson 1957;Hide & Titman 1967;Früh & Read 1999;Vo, Montabone & Sheard 2014, 2015 and thermal instability in regions of strong adverse temperature gradient (Bodenschatz, Pesch & Ahlers 2000;King & Aurnou 2012). A key question for the correct interpretation of laboratory investigations of baroclinically active flows and horizontally driven convection flows with rotation is therefore its global stability to axisymmetric and non-axisymmetric disturbances.…”

Non-axisymmetric flows in a differential-disk rotating system

“…This configuration has been studied previously by numerous authors (e.g. Früh & Read 1999;Aguiar 2008;Vo et al 2014Vo et al , 2015 and a schematic of the system is shown in figure 1. The key dimensions adopted here match the physical proportions used by Früh & Read (1999) such that the ratio of the disk and tank radii is R d /R t = 1/2 and the aspect ratio (ratio of the tank height to disk radius) is A = H/R d = 2/3.…”

“…Vo, Montabone & Sheard (2014) were the first to characterize the vertical structure of the axisymmetric base flow and identify azimuthal linear instability modes that can cause polygonal deformations to the Stewartson layer. These shear layers were produced by the differential rotation of disks in a rotating cylindrical tank.…”

“…Comparisons between the two methodologies are performed and the validity of the quasi-two-dimensional model is discussed. The significance of the most unstable azimuthal linear instabilities predicted by linear stability analysis (Vo et al 2014) is explored by resolving the three-dimensional flow in regimes both near and well beyond the onset of instability. This investigation stems from the unexplained discrepancies observed in the azimuthal wavenumbers obtained experimentally by Früh & Read (1999) to those predicted by linear stability analysis (Vo et al 2014).…”