In this paper, the instability of the buoyancy-driven boundary layer on a vertical cylinder is considered. We provide a mathematical description of a vertical cylinder immersed in a stably stratified fluid, and the analytical solutions of the basic flow are derived. Based on linear stability analysis, the effects of Prandtl number and transversal curvature have been investigated. It is found that the most unstable mode for Pr = 0.7 is three-dimensional for the cylindrical radii between 0.03 and 46, while the critical modes are axisymmetric for Pr = 7 and 100. All the results for different Pr are consistent with those of the vertical plate when the radius is large enough. Additionally, for absolute instability analysis, there exists absolute instability in the present model when the dimensionless radius is less than 0.31, which is quite different from the results obtained in the vertical plate. These encouraging results revealed in this paper should be helpful for understanding such a buoyancy-driven flow system.
The linear instability of the buoyancy-driven flow adjacent to an inclined heated wall immersed in a thermally stratified medium is studied theoretically and numerically. For the temporally unstable system, spatiotemporal stability analysis is carried out to delineate the parameter space (Grashof number, Prandtl number, and tile angle) for convective/absolute instability. We provide an example of an absolute instability of the buoyancy layer on an inclined buoyancy layer. It is shown that the tile angle and Prandtl number have a dramatic influence on the spatial-temporal properties of the flow. For fixed Pr = 6.7, increasing tile angle decreases the domain of absolute instability, and when tile angle is greater than [Formula: see text], the absolute instability disappears. The flow will change from convectively unstable to absolutely unstable with the increase of Pr. Results from the direct numerical simulation are in agreement with the predictions of the linear temporal and spatial-temporal instabilities. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.
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