We investigate experimentally the instability of steady flow of fluid confined in weakly precessing spheroids. It is known that the conical-shear instability (CSI) proposed by Lin, Marti, and Noir [Phys. Fluids 27, 046601 (2015)] grows in a precessing sphere, and the elliptical and shearing instabilities [Kerswell, Geophys. Astrophys. Fluid Dyn. 72, 107 (1993)] can grow in precessing spheroids. Previous theories predict that when $\text{Re}\lesssim {\eta }^{-5}$, where $\text{Re}$ is the Reynolds number defined by the spin angular velocity and the equatorial radius of the spheroid and $\eta$ is its ellipticity, CSI dominates the other two instabilities even in a spheroid, and that, in particular, when $\eta <0.17$ the critical Poincaré number (i.e., the critical precession rate) ${\text{Po}}^{\left(c\right)}$ is proportional to ${\text{Re}}^{-3/10}$ for $200{\eta }^{-2}\ll \text{Re}\lesssim {\eta }^{-5}$. To experimentally verify these predictions, we measure a long time-series of fluid velocity to accurately estimate the power spectrum. Then, we determine ${\text{Po}}^{\left(c\right)}$ as a function of $\text{Re}$ and show the qualitative change of its scaling depending on $\eta$. Our experimental results perfectly support the theoretical predictions, implying that CSI can grow in precessing spheroids.

中文翻译：

---

进动球面中圆锥形剪切驱动的稳定流参数不稳定性

我们实验研究了弱进动球体中流体的稳定流动的不稳定性。众所周知，Lin，Marti和Noir提出了锥形剪切不稳定性（CSI）[ Phys。流体 27，046601（2015）]生长在一个进动球体，椭圆和剪切不稳定[Kerswell，地球物理。天体。流体动力学 72，107（1993）]可以在进动球状体生长。先前的理论预测$\text{回覆}\lesssim {\eta }^{-5}$，在哪里 $\text{回覆}$ 是雷诺数，由旋转角速度和球体的赤道半径定义，并且 $\eta$ 是它的椭圆率，CSI甚至在球体中也占据了其他两个不稳定性，尤其是当 $\eta <0.17$ 临界庞加莱数（即临界进动率） ${\text{宝}}^{（C）}$ 与...成正比 ${\text{回覆}}^{-3/10}$ 对于 $200{\eta }^{-2}\ll \text{回覆}\lesssim {\eta }^{-5}$。为了通过实验验证这些预测，我们测量了长时间的流体速度序列，以准确估算功率谱。然后，我们确定${\text{宝}}^{（C）}$ 根据 $\text{回覆}$ 并显示其缩放比例的质变 $\eta$。我们的实验结果完全支持理论上的预测，这意味着CSI可以在进动球体中增长。