Understanding droplet collisions through a model flow: Insights from a Burgers vortex

We investigate the role of intense vortical structures, similar to those in a turbulent flow, in enhancing collisions (and coalescences) which lead to the formation of large aggregates in particle-laden flows. By using a Burgers vortex model, we show, in particular, that vortex stretching significantly enhances sharp inhomogeneities in spatial particle densities, related to the rapid ejection of particles from intense vortices. Furthermore our work shows how such spatial clustering leads to an enhancement of collision rates and extreme statistics of collisional velocities. We also study the role of polydisperse suspensions in this enhancement. Our work uncovers an important principle, which, if valid for realistic turbulent flows, may be a factor in how small nuclei water droplets in warm clouds can aggregate to sizes large enough to trigger rain.

Figure 1

Representative plots of the final radial distance $r$ as a function of the initial radial distance $r_0$ for particles with $\text{St}=0.03$ at a short time $t=0.6{\tau }_{\eta }$ for a (a) strong ($r_{\mathrm{core}}=0.2$) and (b) weak ($r_{\mathrm{core}}=0.4$) vortex. We show data for different flow geometries characterized by $\sigma$, and the unbroken (purple) diagonal line is a guide to the eye to differentiate particles which are centrifuged out ($r>r_0$) from those which are drawn in ($r<r_0$).

Figure 2

The critical Stokes (see text for definition) as a function of the nondimensional stretching rate. For small stretching rates, the critical Stokes falls linearly with increasing rate (the slope of the dashed line is $-1/10$). There also seems to be an asymptotic value at high stretching rates.

Figure 3

Representative plot of the particle number density $\mathrm{\Phi }$, for $\text{St}=0.03$, as a function of $r$ at time $t=0.6{\tau }_{\eta }$ for different flow geometries (see legend). The effect of particle inertia is illustrated in the inset which shows $\mathrm{\Phi }$ vs $r$, for a given flow geometry ($r_{\mathrm{core}}=0.2$, $\sigma =0.08$), for particles with $\text{St}=0.03$, $\text{St}=0.06$, and $\text{St}=0.1$. The horizontal, green unbroken line is the initial uniform density profile.

Figure 4

Plots of the collision density $\mathrm{\Theta }$ (collisions occurring per unit volume) as a function of the radial distance $r$ for different flow geometries.

Figure 5

Representative plots of the collision number density $\mathrm{\Theta }$, as a function of $r$, illustrating the efficiency of a polydisperse suspension over a monodisperse one in facilitating collisions and hence coalescences. We show (see text) that polydispersity can overcome the effect of a weak vortex and increase the value of $\mathrm{\Theta }$.

Figure 6

A scatter plot of the relative (collisional) velocities for a small core and large core (inset) vortex corresponding to ($r_{\mathrm{core}}=0.2,\sigma =0.08$) and ($r_{\mathrm{core}}=0.4,\sigma =0.02$), respectively. The red, filled circles denote collisions where at least one of the colliding particles has emerged from the heart (cutoff zone) of the vortex, denoted by the blue vertical line (see text), whereas the green open circles are events where both particles were initially outside this cutoff zone.

Figure 7

The critical Stokes as a function of the nondimensional stretching rate. See also Fig. 2.