Abstract
We study the dynamic and kinematic collision statistics of cloud droplets for a range of flow Taylor microscale Reynolds numbers (up to 500), using a highly scalable hybrid direct numerical simulation approach. Accurate results of radial relative velocity (RRV) and radial distribution function (RDF) at contact have been obtained by taking advantage of their power-law scaling at short separation distances. Three specific but inter-related questions have been addressed in a systematic manner for geometric collisions of same-size droplets (of radius from 10 to 60 μm) in a typical cloud turbulence (dissipation rate at 400 cm2 s−3). Firstly, both deterministic and stochastic forcing schemes were employed to test the sensitivity of the simulation results on the large-scale driving mechanism. We found that, in general, the results are quantitatively similar, with the deterministic forcing giving a slightly larger RDF and collision kernel. This difference, however, is negligible for droplets of radius less than 30 μm. Secondly, we have shown that the dependence of pair statistics on the flow Reynolds number Rλ or larger scale fluid motion is of secondary importance, with a tendency for this effect to saturate at high enough Rλ leading to Rλ-independent results. Both DNS results and theoretical arguments show that the saturation happens at a smaller Rλ for smaller droplets. Finally, since most previous studies of turbulent collision of inertial particles concerned non-sedimenting particles, we have specifically addressed the role of gravity on collision statistics, by simultaneously simulating collision statistics with and without gravity. It is shown that the collision statistics is not affected by gravity when a < ac, where the critical droplet radius ac is found to be around 30 μm for the RRV, and around 20 μm for the RDF. For larger droplets, gravity alters the particle–eddy interaction time and significantly reduces the RRV. The effect of gravity on the RDF is rather complex: gravity reduces the RDF for intermediate-sized droplets but enhances the RDF for larger droplets. In addition, we have studied the scaling exponents of both RDF and RRV, and found that gravity modifies the RDF scaling exponents for both intermediate and large particles, in a manner very similar to the effect of gravity on the RDF at contact. Gravity is shown to cause the scaling exponents for RDF and RRV to level off for large droplets, in contrast to diminishing exponents for non-sedimenting particles.
Highlights
Background turbulent flowTables 1 and 2 list the average values of key flow parameters obtained from the simulations performed with both stochastic and deterministic schemes
Since most previous studies of the turbulent collision of inertial particles concerned non-sedimenting particles, our third question concerns the role of gravity on collision statistics. We address this by simultaneously simulating collision statistics with and without gravity
We reported on our efforts in developing high-resolution simulations of the turbulent collision of cloud droplets
Summary
The basic ideas and algorithms for the hybrid DNS (HDNS) approach have been presented in [8]. The use of the pseudo-spectral method for flow simulation has several advantages, such as the simplicity of implementing the large-scale forcing and imposing periodic boundary conditions, and high computational accuracy. Our new implementation utilizes a 2D DD of the 3D data field and a sequence of 1D FFT from the FFTW library (www.fftw.org) in each spatial direction This new implementation represents the optimal balance between computation and communication on a scalable computer with O(100 000) processors, as explained in [10]. To initialize the velocity field we used a random phase algorithm with a prescribed Kolmogorov energy spectrum as E(k) ∼ |k|−5/3 [11], where k is the wave vector Starting with such an initial setting and integrating the N–S equation with continuous injection of the kinetic energy at large scales, we obtain statistically stationary turbulent flows. Eswaran and Pope [13] showed that the average rate of energy input (which is the average dissipation rate) could be expressed as
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