Polydisperse collision kernels in droplet-laden turbulence with implications for rain formation

TL;DR

This study investigates how turbulence modulates collisions among polydisperse cloud droplets in the bottleneck size range using fully resolved DNS at $Re_\lambda=418$, mapping the bidisperse collision kernel across $St\in[0.02,2]$ and comparing to existing models. It reveals that polydispersity enhances collisions via differential sampling at small $St$ but reduces collisions at large $St$ due to rapid decorrelation of droplet clusters, and identifies deficiencies in the Onishi and Zhou formalisms for bidisperse RDF. The authors propose a compact, first-order bidisperse kernel parameterization using a dispersion measure $\theta_{ij}=(St_i-St_j)/(St_i+St_j)$ via a fitted $\rho(\theta)$ and a simple mixing form $\tilde{\Gamma}_{ij}=\theta_{ij}\Gamma_{0i}+(1-\theta_{ij})\Gamma_{ii}$, achieving ~10% accuracy. They also demonstrate turbulent broadening of the DSD through coalescence simulations and show that fluctuations of the local dissipation rate markedly accelerate droplet growth. These results help inform cloud microphysics parameterizations and support future LES implementations to better predict rain formation.

Abstract

The collision kernel of droplets in warm clouds is a crucially important quantity for the parameterization of precipitation in weather and climate models. Nevertheless, its accurate representation remains a challenge, specifically in the bottleneck range $15\,μ\text{m}<r<40\,μ\text{m}$, within which turbulence is believed to be a key contributor to droplet growth. In this work, we address this problem by performing direct numerical simulations of polydisperse inertial particles suspended in three-dimensional turbulence at Reynolds number $Re_λ=418$. Collision statistics are obtained for droplet pairs across the Stokes number range $St\in[0.02,2]$. Our analysis reveals that polydispersity enhances collisions between light droplets through differential sampling, but attenuates collisions at larger Stokes numbers by rapidly reducing the spatial overlap of droplet clusters. We quantify and discuss this decorrelating effect in bidisperse clustering and compare our DNS results with the model of Onishi & Seifert [Atmos. Chem. Phys. 16, 12441 (2016)]. A novel phenomenological parameterization for the collision kernel is proposed, showing significant improvements in the predictions for the bidisperse case. Finally, we study the broadening of droplet size distributions due to turbulence and demonstrate the influence of fluctuations of the dissipation rate on accelerated droplet growth.

Figures (6)

• Figure 1: Monodisperse (a) collision kernel, (b) RRV and (c) RDF at contact as a function of the Stokes number. The blue lines depict the DNS data, whereas the green lines show the results from the model expressions.
• Figure 2: Bidisperse (a,d) collision kernel, (b,e) RRV and (c,f) RDF at contact obtained from the DNS (top row) and from the Onishi/Wang model (bottom row).
• Figure 3: (a) Bidisperse correlation coefficient as a function of $St_i/St_j$ obtained from the DNS for a range of $St_j$ (colored symbols) and obtained from the Zhou model with the original ($Re_\lambda=45$, black dashed line) and re-fitted ($Re_\lambda=418$, black dashed-dotted line) parameters, respectively. (b) Correlation coefficient as a function of $\theta_{ij}$ obtained from the DNS (symbols) and the prediction from \ref{['eqn:rho_theta']} (black solid line).
• Figure 4: (a) Bidisperse collision kernel matrix computed from \ref{['eqn:parameterization']} with $\Gamma_{ii}$ and $\Gamma_{0i}$ obtained from the DNS ($St_0=0.02$) and (b) its relative error with respect to the DNS data. (c) The relative error with respect to the DNS data obtained with the Onishi model. Panels (b) and (c) share the same color scale shown on the right.
• Figure 5: $\Gamma_{0i}$ obtained from the DNS with $St_0=0.02$ (blue dots). The green and red lines depict the empirical fits $\Gamma_{0i}=aSt_i$ with $a=5.3$ and $\Gamma_{0i}=b_0+b_1\exp(-b_2/St_i)$ with $b_0=4.3,~b_1=19.6,~b_2=3.05$, respectively.