Inertia Induces Strong Orientation Fluctuations of Nonspherical Atmospheric Particles

The orientation of nonspherical particles in the atmosphere, such as volcanic ash and ice crystals, influences their residence times and the radiative properties of the atmosphere. Here, we demonstrate experimentally that the orientation of heavy submillimeter spheroids settling in still air exhibits decaying oscillations, whereas it relaxes monotonically in liquids. Theoretical analysis shows that these oscillations are due to particle inertia, caused by the large particle-fluid mass-density ratio. This effect must be accounted for to model solid particles in the atmosphere.

Figure 1

Experimental setup. (a) Optical table with top cameras ($\mathrm{T}X$ and $\mathrm{T}Y$) and bottom cameras ($\mathrm{B}X$ and $\mathrm{B}Y$) named after the shown coordinate system ($Z$ is the direction of gravity $\mathit{g}$), the settling chamber (SC), the synchronized pulsed LED unit, controlled with a waveform generator (WG), and the oscilloscope (OSC). (b) Snapshots of a settling prolate spheroid recorded at $2932\mathrm{frames}/\mathrm{s}$. The tilt angle—the angle between the particle symmetry axis and gravity—is shown in 5.1 ms intervals in units of degrees. See the Supplemental Material [45] for details.

Figure 2

Comparison between experiments and theory. (a) Time evolution of tilt angle $\phi$ from experiment (blue) and its model prediction (red) for spheroids with aspect ratios $\lambda =0.2$ and 5 from group I (Table 1). (b) Steady-state settling speed $v_g^{*}$, frequency, and decay rate of the tilt-angle oscillations as functions of the aspect ratio $\lambda$. Markers show averages obtained for all experiments with error bars indicating 95% confidence bounds for groups I (red circles), II (green squares), and III (blue diamonds) (Table 1). Solid lines show large-time asymptotes from a linear-stability analysis of the model described in Appendix A. Shaded regions indicate how much the theoretical predictions change as the measured settling speed varies along the particle trajectory. See Appendixes A and B for details. Dashed lines show results of a linear-stability analysis of a harmonic-oscillator approximation, Eq. (d2) in Appendix D.

Figure 3

(a) Bifurcation diagram. The dashed horizontal line distinguishes decay without oscillations ($\mathrm{\Delta }>0$) from decay with oscillations ($\mathrm{\Delta }<0$). Particles from group I in Table 1 are shown as red circles; group II as green squares; group III as blue diamonds; particles from Ref. [32] as $+$; fibers from Ref. [57] as a gray region. We approximated cylindrical fibers as slender prolate spheroids and estimated $\mathrm{\Delta }$ by setting $C_T=C_F=1$, since the corresponding ${\mathrm{Re}}_{\mathrm{p}}$ are very small. (b) Standard deviation of tilt-angle fluctuations for spheroids settling in weakly turbulent air with dissipation rate $ϵ$, as a function of ${\mathrm{Re}}_{\mathrm{p}}$. Shown are simulation results of the model described in Appendix E in symbols, for spheroids with $\lambda =0.2$, but different volumes $V_{\mathrm{p}}$. Solid lines correspond to the overdamped approximation, neglecting particle inertia [Eq. (S6) in the Supplemental Material [45] ]. The harmonic-oscillator bifurcation ($\mathrm{\Delta }=0$) is shown as a dashed vertical line.

Figure 4

Empirical coefficients $C_F$ and $C_T$ in Eq. (a2) for the parameters in Fig. 2, obtained as described in Appendix A. (a) Force coefficient $C_F$ as a function of aspect ratio $\lambda$. The groups refer to Table 1. (b) Torque coefficient $C_T$.