Instabilities of the Stewartson layer Part 1. The dependence on the sign of $Ro$
Instabilities of the Stewartson layer Part 1. The dependence on the sign of $Ro$
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DOI：
10.1017/s0022112003005676
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发表时间：
2003-09
影响因子：
3.7
通讯作者：
Rainer Hollerbach
中科院分区：
文献类型：
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作者：
Rainer Hollerbach
我们认为在快速的整体旋转的球壳中的流体流动，另外施加在内部球体上的差分旋转。基本状态由位于切向圆柱体上的轴对称Stewartson剪切层组成，圆柱体平行于旋转轴并且刚好接触内球。在这项工作中，我们考虑的非轴对称不稳定性时出现的微分旋转变得足够大。我们发现，差分旋转的符号，也就是说，无论是内球旋转略快或略慢于外球，是至关重要的，与积极的差分旋转产生的进展，以更高的波数$m$作为整体的旋转速率增加，但负差分旋转产生$m\，{=}\，1 $在几乎整个范围内的旋转速率。这种差异特别有趣，因为它以前在一个密切相关的实验研究中看到过，但在另一个实验中没有。先验渐近分析也表明应该没有差异。因此，我们试图了解流动结构和/或几何形状的哪些细微特征会导致结果的这种差异。我们表明，几何形状是关键的功能，与高度沿着旋转轴突然改变整个切圆柱体。我们不能确定为什么这会产生如此大的差异，为什么只对负微分转动有差异。我们建议，而不是额外的实验和渐近，以进一步澄清这一点。
We consider the fluid flow in a spherical shell in rapid overall rotation, with additionally a differential rotation imposed on the inner sphere. The basic state consists of the axisymmetric Stewartson shear layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and just touching the inner sphere. In this work we consider the non-axisymmetric instabilities that arise when the differential rotation becomes sufficiently large. We find that the sign of the differential rotation, that is, whether the inner sphere is rotating slightly faster or slightly slower than the outer sphere, is crucial, with positive differential rotations yielding a progression to higher wavenumbers $m$ as the overall rotation rate increases, but negative differential rotations yielding $m\,{=}\,1$ over almost the entire range of rotation rates. This difference is particularly intriguing, as it has been seen before in one closely related experimental study, but not in another. A prior asymptotic analysis also suggested there should be no difference. We therefore try to understand what subtle features of the flow structures and/or geometries should cause this difference in results. We show that the geometry is the critical feature, with the height along the axis of rotation changing abruptly across the tangent cylinder. We are not able to identify why this should make such a difference, and why only for negative differential rotations. We suggest instead additional experiments and asymptotics to further clarify this point.