Statistical Model for the Orientation of Nonspherical Particles Settling in Turbulence

The orientation of small anisotropic particles settling in a turbulent fluid determines some essential properties of the suspension. We show that the orientation distribution of small heavy spheroids settling through turbulence can be accurately predicted by a simple Gaussian statistical model that takes into account particle inertia and provides a quantitative understanding of the orientation distribution on the problem parameters when fluid inertia is negligible. Our results open the way to a parametrization of the distribution of ice crystals in clouds, and potentially lead to an improved understanding of radiation reflection or particle aggregation through collisions in clouds.

Figure 1

Orientational bias of spheroids settling in turbulence. Distribution $P(n_g)$ of $n_g\equiv |\mathit{n}·\widehat{\mathit{g}}|$, with particle symmetry vector $\mathit{n}$ and direction $\widehat{\mathit{g}}$ of gravity. (a) DNS results for $P(n_g)$ for oblate spheroids [aspect ratios $\lambda =0.01$ (inverted triangle), 0.02 (star), 0.05 (right-pointing triangle)]. Statistical-model simulations: open symbols. Dashed line: isotropic distribution $P(n_g)=1$. (b) Same, but for prolate spheroids: $\lambda =5$ (circle), 7.5 (left pointing triangle), 10 (star). Parameters: ${\mathrm{Re}}_{\lambda }\approx 95$, $F_{\mathrm{K}}\approx 70$, and ${\text{St}}_{\mathrm{K}}\approx 4\min (\lambda ,1/\lambda )$ (see text); these correspond to values relevant to cloud physics, see the Supplemental Material [33].

Figure 2

Moments of $n_g=|\mathit{n}·\widehat{\mathit{g}}|$, for $p=1$ (circle), $p=2$ (square), $p=3$ (diamond), and $p=4$ (triangle). Dashed lines: moments of isotropic orientation distribution. Thin solid lines: Eq. (6) to $O(G^2)$. Thick solid lines: order 8-by-8 Padé-Borel resummation of Eq. (6) to $O(G^{32})$. Parameters: $\mathrm{Ku}=0.1$, $\mathrm{St}=10$, with (a) ${\lambda }^2=0.1$ or (b) ${\lambda }^2=10$.

Figure 3

History effect causes orientation bias. (a) Distribution of $n_g$ based on full statistical-model simulations (symbols), and based on straight deterministic paths (solid lines). Parameters: $\lambda =1/\sqrt{10}$, $\mathrm{Ku}=0.1$, $\mathrm{St}=10$, $F=1$ (circle), $F=10$ (square). (b) Same for parameters corresponding to data in Fig. 1, $\lambda =0.02$ (star), $\lambda =0.05$ (right-pointing triangle). Preferential-sampling contribution is hatched. (c) Moments $⟨n_g^{2p}{⟩}_{\infty }$ from statistical-model simulations in the persistent limit ($\mathrm{Ku}=10$) against ${\mathrm{St}}_{\mathrm{K}}$ ($p=1$, circle; $p=2$, square; $p=3$, diamond). Parameters $\lambda =1/\sqrt{10}$, $F_{\mathrm{K}}\approx 2.5$. Also shown are simulations based on straight deterministic paths (solid lines).