Inertial torque on a small spheroid in a stationary uniform flow

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.

Figure 1

(a) Prolate spheroid with symmetry axis $\mathit{n}$ in a uniform flow with velocity $\mathit{u}=-U{\widehat{\mathbf{e}}}_1$. The angle $\phi$ between $\mathit{n}$ and the ${\widehat{\mathbf{e}}}_1$ axis is called the tilt angle. (b) Shape factor $F\left(\lambda \right)$ determining the torque ${\tau }_3$, Eq. (9), red solid line. Also shown are the slender-body asymptote (5), black dotted line, as well as the near-spherical asymptote (6), black dashed line.

Figure 2

(a) Dimensionless torque ${\tau }_3^{\prime }$ [Eq. (8)] upon a prolate spheroid in a uniform flow, as a function of the angle of inclination. Results are shown for $\lambda =6$ and $\mathrm{Re}=0.3$ (red, $\square$), $\mathrm{Re}=3$ (green, $\circ$), and $\mathrm{Re}=30$ (blue, $▵$). Filled symbols correspond to data from Ref. [19]. Theory (9) is shown as a solid black line. Colored lines are fits of the angular dependence in Eq. (9), proportional to $sin2\phi$. (b) Same for an oblate spheroid with $\lambda =1/6$. (c) Maximal torque ${\tau }_3^{\prime }$ (evaluated at $\phi ={45}^{\circ }$) as a function of aspect ratio $\lambda$, for different particle Reynolds numbers. Symbols show simulation results; the black solid line is $\frac{1}{2}F\left(\lambda \right)$. (d) Dependence of ${\tau }_3^{\prime }$ upon Reynolds number for $\phi ={45}^{\circ }$, $\lambda =6$ (black, $\diamond$), compared with the small-Re theory (9), solid line, with the parametrization of Ouchene et al. [11], dashed line, and with the parametrization of Fröhlich et al. [16], dash-dotted line.

Figure 3

Torque on a disk as a function of Reynolds number. Results for spheroids of two different shapes are shown: $\lambda =1/2$ (circles) and $\lambda =1/3$ (squares). Empty symbols correspond to tilt angle $\phi =30$, and full symbols correspond to $\phi =60$. At $\mathrm{Re}=3$ and 5, empty and full symbols lie on top of each other.

Figure 4

Streamlines of the flow around a disk in the $x\text{−}y$ plane at $z=0$, for $\mathrm{Re}=3$ and $\lambda =1/3$. The tilt angle is (a) $\phi ={30}^{\circ }$ and (b) $\phi ={60}^{\circ }$; (c) and (d) show the same but for $\mathrm{Re}=30$.