Abstract
This paper is concerned with convective and absolute instabilities in the boundary-layer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature effects are included and the analysis is conducted between latitudes of 10° and 80° from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specific parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below θ = 66°. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at θ = 66°. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at θ = 30°; beyond this latitude the discrepancy increases but remains relatively small below θ = 70°. It is suggested that this absolute instability may cause the onset of transition below θ = 70°. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.
Highlights
When a sphere rotates in still fluid a flow is induced in which the fluid moves over the outer surface from the poles to the equator and is ejected radially from the equator
The results of Kohama & Kobayashi show that the onset of transition to turbulence occurs at a repeatable consistent local Reynolds number that is roughly the same at all latitudes below θ = 70◦, despite the instability waves first appearing across a wide parameter range with varying rotation rate and sphere radius
In this paper the rotating sphere was considered in a fixed frame of reference. This formulation has been used in all previous theoretical investigations on the rotating-sphere boundary layer, but is in contrast to the related work of Lingwood (1995) on the rotating disk
Summary
When a sphere rotates in still fluid a flow is induced in which the fluid moves over the outer surface from the poles to the equator and is ejected radially from the equator. The results of Kohama & Kobayashi show that the onset of transition to turbulence occurs at a repeatable consistent local Reynolds number that is roughly the same at all latitudes below θ = 70◦, despite the instability waves first appearing across a wide parameter range with varying rotation rate and sphere radius. This observation is highly suggestive that the boundary layer becomes absolutely unstable at, or just before, the transition location, leading to the temporal growth of the disturbances at that point and triggering the nonlinear behaviour characteristic of the onset of transition.
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