Settling and collision between small ice crystals in turbulent flows

Ice crystals are present in a variety of clouds, at sufficiently low temperature. We consider here mixed-phase clouds which, at temperature $\gtrsim -20{}^{\circ }\text{C}$, contain ice crystals, shaped approximately as thin oblate ellipsoids. We investigate the motion of these particles transported by an isotropic turbulent flow and, in particular, the collision between these crystals, a key process in the formation of graupels. Using fully resolved direct numerical simulations, and neglecting the effects of fluid inertia on the particle motion, we determine the influence of the turbulence intensity and of gravitational settling, in a realistic range of parameters. At small turbulent energy dissipation rate, collisions are induced mainly by differential gravitational settling between particles with different orientations. The effect has a clear signature on the relative orientation of colliding ellipsoids. As the turbulent energy dissipation rate increases, however, the influence of the turbulent velocity fluctuations becomes the dominant effect determining the collision rate. Using simple estimates, we propose an elementary understanding of the relative importance of gravitational settling and turbulent fluctuations.

Figure 1

(a) Illustration of the definition of the vector $\mathbf{n}$ parallel to the small axis of the ellipsoid. The angle $\theta$ is the angle between $\mathbf{n}$ and the direction of gravity, $\mathbf{g}$. With our definition, $cos\left(\theta \right)=n_z$. We restrict ourselves to $0\le n_z\le 1$. (b) Distribution of $n_z$ for ellipsoids with aspect ratio $\beta =0.02$ in the presence of gravity in a turbulent flow at $\varepsilon \approx 1{\text{cm}}^2/{\text{s}}^3$ (full line), $\varepsilon \approx 16{\text{cm}}^2/{\text{s}}^3$ (dashed line), and $\varepsilon \approx 256{\text{cm}}^2/{\text{s}}^3$ (dashed-dotted line). (c) Distribution of $n_z$ in the presence of gravity, at ${\text{Re}}_{\lambda }=95$, for ellipsoids of aspect ratio $\beta =0.01$ (full line), $\beta =0.02$ (dashed line), and $\beta =0.05$ (dashed-dotted line).

Figure 2

Correlation function of the vector $\mathbf{n}$ in the flow in the presence (upper panel) and in the absence (lower panel) of gravitational settling for ellipsoids with an aspect ratio $\beta =0.02$. The correlation function $\langle \mathbf{n}\left(0\right)\mathbf{n}\left(\tau \right)\rangle$ is plotted as a function of $\tau /{\tau }_K$.

Figure 3

Averaged settling velocity of ellipsoids in turbulent flows: (a) settling velocity as a function of $n_z$ for ellipsoids with an aspect ratio $\beta =0.02$, at several values of $\varepsilon$, as indicated in the legend (for comparison, the settling velocity in still air is shown by the curve with crosses); (b) ratio between the settling velocity and the rms of the fluctuations. This ratio, which is very close to the gravity parameter $S_{v,L}$, defined by Eq. (10), is shown for $\beta =0.02$, as a function of $\varepsilon$. The dashed line indicates the ${\varepsilon }^{-1/3}$ dependence, which captures qualitatively the dependence obtained numerically.

Figure 4

Comparison between velocity fluctuations and settling velocity at different turbulence intensity. The figure shows the joint PDFs of the orientation, $n_z$ (horizontal) and of the vertical velocity, $v_z$ (vertical) for ellipsoids at $\beta =0.02$, at the three values of $\varepsilon$, as indicated. The probability is color coded, as indicated on the color bar. The conditional average $\langle v_z|n_z\rangle$ is indicated by the full line in the figure (the dashed line shows the settling velocity in still fluid).

Figure 5

Collision rate of a suspension of ellipsoids. The collision rates in the presence (“+” symbols) and in the absence (“$\times$” symbols) of gravitational settling are compared. (a) Collision rate as a function of $\beta$, at a fixed Reynolds number, ${\text{Re}}_{\lambda }=95\left(\varepsilon \approx 16{\text{cm}}^2/{\text{s}}^3\right)$. The collision rate increases slightly with $\beta$ in both cases and is much larger in the presence of gravity. (b) Collision rate as a function of $\varepsilon$, at a fixed value of $\beta =0.02$. Generally the collision rates in the absence of gravity (shown with a “$\times$” symbol) are much smaller than in the presence of gravity (shown with a “+” symbol). The difference however, diminishes when $\varepsilon$ increases.

Figure 6

Distribution of $n_z$ for the colliding ellipsoids during their collision: upper ellipsoid (dashed red line) and lower ellipsoid (dashed-dotted magenta line). As a comparison, the distribution of the angles for all ellipsoids is shown with a full line. The Reynolds number increases from ${\text{Re}}_{\lambda }=56$ (left), ${\text{Re}}_{\lambda }=95$ (center), and ${\text{Re}}_{\lambda }=151$ (right). The results are shown for ellipsoids with $\beta =0.02$. The upper (lower) ellipsoids have a larger probability of having a small (large) value of $n_z$, which implies that the upper ellipsoids settle faster than the lower ones.

Figure 7

Cumulative probability distribution function of the cosine of the relative angle between two colliding ellipsoids, ${\mathbf{n}}_1·{\mathbf{n}}_2$. The full, dashed, and dashed-dotted lines correspond to ${\text{Re}}_{\lambda }=56,{\text{Re}}_{\lambda }=95$, and ${\text{Re}}_{\lambda }=151$, respectively. The three lowest (respectively highest) curves correspond to a case without (respectively with) gravitational settling. The cumulative probability distribution function is defined by Eq. (13).

Figure 8

Probability distribution function of the relative velocity between the center of mass of two colliding ellipsoids in the absence of gravity. The full, dashed, and dashed-dotted lines correspond to $\varepsilon \approx 1{\text{cm}}^2/{\text{s}}^3,\varepsilon \approx 16{\text{cm}}^2/{\text{s}}^3$, and $\varepsilon \approx 256{\text{cm}}^2/{\text{s}}^3$, respectively. Although the distributions for $\varepsilon \approx 1{\text{cm}}^2/{\text{s}}^3$ and $\varepsilon \approx 16{\text{cm}}^2/{\text{s}}^3$ are symmetric and do not extend much beyond $\mathrm{\Delta }{v}_l/\left[a{(\varepsilon /\nu )}^{1/2}\right]$, the one obtained for $\varepsilon \approx 256{\text{cm}}^2/{\text{s}}^3$ displays a broad tail for negative values of the relative velocity, thereby revealing the presence of large inertial effects.