共振进动圆柱中的纬向流
Zonal flow in a resonant precessing cylinder
AI总结：
本研究通过渐近分析揭示了共振进动圆柱中纬向流的三个来源，并推广至任意开尔文模态，理论预测经数值与实验验证，且发现纬向流在共振条件下总是逆行的。
摘要：
当圆柱绕其轴旋转且该轴本身绕另一方向旋转时，圆柱发生进动。在充满流体的进动圆柱中，通常存在一个稳定且轴对称的方位角流分量，该分量称为纬向流。尽管纬向流在实验和数值模拟中经常被观察到，但其起源迄今一直未能在理论上得到解释。在此，我们发展了一种渐近分析来计算在共振进动圆柱中被强迫的纬向流，即当谐波响应由单一开尔文模态主导时。我们发现纬向流来源于三个不同的机制：（1）无粘开尔文模态与其粘性修正的非线性相互作用；（2）开尔文模态与自身非线性相互作用的稳态轴对称响应；以及（3）端部边界层中的非线性相互作用。在进动圆柱中，由于赤道科里奥利力和强迫剪切流，出现了两个额外的来源，但它们恰好相互抵消。因此，该研究推广到任何由进动或其他机制强迫的开尔文模态。通过与数值模拟和实验结果比较，本研究的理论预测得到了证实。我们还通过数值方法表明，在共振进动圆柱（m=1）中，或当纬向流由方位角波数m=2、3以及可能更高的共振开尔文模态产生时，纬向流总是逆行的。
英文摘要：
A cylinder undergoes precession when it rotates around its axis and this axis itself rotates around another direction. In a precessing cylinder full of fluid, a steady and axisymmetric component of the azimuthal flow is generally present. This component is called a zonal flow. Although zonal flows have been often observed in experiments and numerical simulations, their origin has eluded theoretical approaches so far. Here, we develop an asymptotic analysis to calculate the zonal flow forced in a resonant precessing cylinder, that is when the harmonic response is dominated by a single Kelvin mode. We find that the zonal flow originates from three different sources: (1) the nonlinear interaction of the inviscid Kelvin mode with its viscous correction; (2) the steady and axisymmetric response to the nonlinear interaction of the Kelvin mode with itself; and (3) the nonlinear interactions in the end boundary layers. In a precessing cylinder, two additional sources arise due to the equatorial Coriolis force and the forced shear flow. However, they cancel exactly. The study thus generalises to any Kelvin mode, forced by precession or any other mechanism. The present theoretical predictions of the zonal flow are confirmed by comparison with numerical simulations and experimental results. We also show numerically that the zonal flow is always retrograde in a resonant precessing cylinder (m=1) or when it results from resonant Kelvin modes of azimuthal wavenumbers m=2, 3, and presumably higher.
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