This study considers the linear stability of Poiseuille-Rayleigh-Bénard flows, subjected to a transverse magnetic field to understand the instabilities that arise from the complex interaction between the effects of shear, thermal stratification and magnetic damping. This fundamental study is motivated in part by the desire to enhance heat transfer in the blanket ducts of nuclear fusion reactors. In pure MHD flows, the imposed transverse magnetic field causes the flow to become quasi-two-dimensional and exhibit disturbances that are localised to the horizontal walls. However, the vertical temperature stratification in Rayleigh-Bénard flows feature convection cells that occupy the interior region and therefore the addition of this aspect provides an interesting point for investigation.The linearised governing equations are described by the quasi-two-dimensional model proposed by Sommeria and Moreau [1] which incorporates a Hartmann friction term, and the base flows are considered fully developed and one-dimensional. The neutral stability curves for critical Reynolds and Rayleigh numbers, Rec and Rac, respectively, as functions of Hartmann friction parameter H have been obtained over 10 −2 ≤ H ≤ 10 4 . Asymptotic trends are observed as H → ∞ following Rec ∝ H 1/2 and Rac ∝ H. The linear stability analysis reveals multiple instabilities which alter the flow both within the Shercliff boundary layers and the interior flow, with structures consistent with features from plane Poiseuille and Rayleigh-Bénard flows.
This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of 5000 and 15 000 and an extensive range of Hartmann numbers 0Ha 800 were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes corresponding to low, moderate and high magnetic field strengths are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness R H , and Reynolds number built upon the Shercliff layer thickness R S , respectively. Transition between regimes respectively occurs at Ha ≈ 2 and no lower than R H ≈ 33.3. Notably for the high Hartmann number regime, quasitwo-dimensional magnetohydrodynamic models are shown to be an excellent predictor of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-twodimensional turbulence at much higher Hartmann numbers than is currently achievable.
The structure and stability of Stewartson shear layers with different heights are investigated numerically via axisymmetric simulation and linear stability analysis, and a validation of the quasi-two-dimensional model is performed. The shear layers are generated in a rotating cylindrical tank with circular disks located at the lid and base imposing a differential rotation. The axisymmetric model captures both the thick and thin nested Stewartson layers, which are scaled by the Ekman number ($\mathit{E}\,$) as $\mathit{E}\,^{1/4}$ and $\mathit{E}\,^{1/3}$ respectively. In contrast, the quasi-two-dimensional model only captures the $\mathit{E}\,^{1/4}$ layer as the axial velocity required to invoke the $\mathit{E}\,^{1/3}$ layer is excluded. A direct comparison between the axisymmetric base flows and their linear stability in these two models is examined here for the first time. The base flows of the two models exhibit similar flow features at low Rossby numbers ($\mathit{Ro}$), with differences evident at larger $\mathit{Ro}$ where depth-dependent features are revealed by the axisymmetric model. Despite this, the quasi-two-dimensional model demonstrates excellent agreement with the axisymmetric model in terms of the shear-layer thickness and predicted stability. A study of various aspect ratios reveals that a Reynolds number based on the theoretical Ekman layer thickness is able to describe the transition of a base flow that is reflectively symmetric about the mid-plane to a symmetry-broken state. Additionally, the shear-layer thicknesses scale closely to the expected ${\it\delta}_{vel}\propto A\mathit{E}\,^{1/4}$ and ${\it\delta}_{vort}\propto A\mathit{E}\,^{1/3}$ for shear layers that are not affected by the confinement ($A\mathit{E}\,^{1/4}\lesssim 0.34$ in this system, the ratio of tank height to shear-layer radius). The linear stability analysis reveals that the ratio of Stewartson layer radius to thickness should be greater than $45$ for the stability of the flow to be independent of aspect ratio. Thus, for sufficiently small $A\mathit{E}\,^{1/4}$ and $A\mathit{E}\,^{1/3}$, the flow characteristics remain similar and the linear stability of the flow can be described universally when the azimuthal wavelength is scaled against $A$. The analysis also recovers an asymptotic scaling for the normalized azimuthal wavelength which suggests that ${\it\lambda}_{{\it\theta},c}^{\ast }\propto (|\mathit{Ro}|/\mathit{E}\,^{2})^{-1/5}$ for geometry-independent shear layers at marginal stability.
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.
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