Theoretical Formulation of Collision Rate and Collision Efficiency of Hydrodynamically Interacting Cloud Droplets in Turbulent Atmosphere

Abstract

A methodology for conducting direct numerical simulations (DNSs) of hydrodynamically interacting droplets in the context of cloud microphysics has been developed and used to validate a new kinematic formulation capable of describing the collision rate and collision efficiency of cloud droplets in turbulent air. The theoretical formulation is formally the same as the formulation recently developed for geometrical collision rate of finite-inertia, nonsettling particles. It is shown that its application to hydrodynamically interacting droplets requires corrections because of a nonoverlap requirement. An approximate method for correcting the kinematic properties has been developed and validated against DNS data. The formulation presented here is more general and accurate than previously published formulations that, in most cases, are some extension to the description of hydrodynamic-gravitational collision. General dynamic and kinematic representations of the properly defined collision efficiency in a turbulent flow have been discussed. In addition to augmenting the geometric collision rate, air turbulence has been found to enhance the collision efficiency because, in a turbulent flow, hydrodynamic interactions become less effective in reducing the average relative radial velocity. The level of increase in the collision efficiency depends on the flow dissipation rate. For example, the collision efficiency between droplets of 20 and 25 m in radii is increased by 59% and 10% by air turbulence at dissipation rates of 400 and 100 cm 2 s Ϫ3 , respectively. It is also shown that hydrodynamic interactions lead to higher droplet concentration fluctuations. The formulation presented here separates the effect of turbulence on collision efficiency from the previously observed effect of turbulence on the geometric collision rate.

Droplet growth in warm turbulent clouds

“…Wang et al (2005a) formulated a revised superposition method which avoids this limitation by imposing proper boundary conditions on the surface of the droplets. The hybrid DNS results of Wang et al (2005b), with the revised superposition method, show that air turbulence increases both the geometric collision rate and the collision efficiency, particularly as a 2 /a 1 → 1. They found that the results depend sensitively on the size of the larger droplet.…”

“…The geometric collision kernel (9) can be modified to account for droplet--droplet interactions by defining a turbulent collision efficiency (Wang et al, 2005b(Wang et al, , 2006bPinsky et al, 2007): (14) where 'HI' and 'No HI' respectively indicate that 12 has been computed with or without droplet-droplet interactions (keeping all other pertinent quantities fixed). In terms of the radial relative velocity and the RDF (14) with the dependence on r c now implied rather than explicit.…”

“…This misinterpretation may be problematic for high R λ flow, even in the limit of small St number. Wang et al (2005b) proposed a hybrid DNS approach to modelling hydrodynamic interactions based on the superposition method. This method (Pruppacher and Klett, 1997, p. 571) assumes that each sphere moves in a flow field generated by the other sphere falling in isolation.…”

The role of air turbulence in warm rain initiation

“…Recent systematic studies of the collection kernel for cloud droplets have been undertaken through either direct numerical simulation (DNS) (Wang et al, 2005Franklin et al, 2007;Ayala et al, 2008b) or a kinematic/stochastic representation of turbulence (Pinsky et al, 1999(Pinsky et al, , 2006. These studies provide not only quantitative data on the turbulent collision kernel, but also reveal that turbulent collisions of cloud droplets are dynamic events in a complex multiphase flow system affected by a range of scales from those governing the background air turbulence to those characterizing droplet-droplet aerodynamic interactions.…”

“…Motivated by the issues explained above, we have developed a consistent and rigorous simulation approach to the problem of turbulent collisions of cloud droplets (Wang et al, 2005;Ayala et al, 2007). The basic idea of the approach is to combine DNS of the background air turbulence with an analytical representation of the disturbance flow introduced by droplets ( Figure 1).…”