How anisotropic particles rotate and orient in a flow depends on the hydrodynamic torque they experience. Here we compute the torque acting on a small spheroid in a uniform flow by numerically solving the Navier-Stokes equations. Particle shape is varied from oblate (aspect ratio $\lambda =1/6$) to prolate $(\lambda =6)$, and we consider low and moderate particle Reynolds numbers $(\mathrm{Re}\le 50)$. We demonstrate that the angular dependence of the torque, predicted theoretically for small particle Reynolds numbers, remains qualitatively correct for Reynolds numbers up to $\mathrm{Re}\sim 10$. The amplitude of the torque, however, is smaller than the theoretical prediction, the more so as $\mathrm{Re}$ increases. For Re larger than 10, the flow past oblate spheroids acquires a more complicated structure, resulting in systematic deviations from the theoretical predictions. Overall, our numerical results provide a justification of recent theories for the orientation statistics of ice crystals settling in a turbulent flow.

中文翻译：

---

静止均匀流中小球体上的惯性转矩

各向异性粒子如何在流体中旋转和定向取决于它们所经历的流体动力转矩。在这里，我们通过数值求解Navier-Stokes方程来计算均匀流中作用于小球体上的转矩。颗粒形状从扁圆形（长宽比）变化$\lambda =\mathrm{1个}/6$） $（\lambda =6）$，我们考虑了中低粒子雷诺数 $（\mathrm{回覆}\le 50）$。我们证明，理论上针对小粒子雷诺数预测的转矩的角度依赖性在定性上对于雷诺数仍保持定性正确$\mathrm{回覆}〜10$。但是，扭矩的幅度小于理论预测值，$\mathrm{回覆}$增加。如果Re大于10，则流经扁球体的流量将获得更为复杂的结构，从而导致系统偏离理论预测值。总的来说，我们的数值结果为在湍流中沉降的冰晶的定向统计提供了最新的理论依据。