Orlando Ayala

Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulation

There have been relatively few studies of turbulent collision rate of sedimenting droplets in the context of cloud physics, for which both the gravitational settling and inertial effects must be simultaneously considered. In this study, direct numerical simulations (DNS) were used to study the geometric collision rates of cloud droplets. Both Stokes drag law and a nonlinear drag law were considered, but the droplet–droplet local aerodynamic interactions were not included. Typical droplet and turbulence parameters of convective clouds were used to determine the flow dissipation rate , characteristic Stokes numbers, and the nondimensional terminal velocities. DNS results from a large number of runs covering the range from 10 to 400 cm2 s− 3 and droplet sizes from 10 to 60 μm in radius are presented. These results show that air turbulence can increase the geometric collision kernel by up to 47%, relative to geometric collision by differential sedimentation. This is due to both a moderate enhancement of the radial relative velocity between droplets and a moderate level of pair nonuniform concentration due to local droplet clustering. The turbulence enhancements increase with the flow dissipation rate and flow Reynolds number. Comparisons with related DNS studies show that our results confirm and extend the previous findings. The mean settling velocity of droplets in a turbulent flow was also obtained, showing that a maximum increase relative to the terminal velocity occurs for 20 μm cloud droplets. This agrees with a previous theory based on simple vortex flows and confirms the importance of a new nondimensional parameter τp3g2/ν for sedimenting droplets, where τp is the droplet inertial response time, g is the gravitational acceleration and ν is the air kinematic viscosity. Limitations of DNS and future directions are also noted.

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

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 droplet so f 20 and 25m in radii is increased by 59% and 10% by air turbulence at dissipation rates of 400 and 100 cm 2 s 3 , 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.

Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization

The effect of air turbulence on the geometric collision kernel of cloud droplets can be predicted if the effects of air turbulence on two kinematic pair statistics can be modeled. The first is the average radial relative velocity and the second is the radial distribution function (RDF). A survey of the literature shows that no theory is available for predicting the radial relative velocity of finite-inertia sedimenting droplets in a turbulent flow. In this paper, a theory for the radial relative velocity is developed, using a statistical approach assuming that gravitational sedimentation dominates the relative motion of droplets before collision. In the weak-inertia limit, the theory reveals a new term making a positive contribution to the radial relative velocity resulting from a coupling between sedimentation and air turbulence on the motion of finite-inertia droplets. The theory is compared to the direct numerical simulations (DNS) results in part 1, showing a reasonable agreement with the DNS data for bidisperse cloud droplets. For droplets larger than 30??m in radius, a nonlinear drag (NLD) can also be included in the theory in terms of an effective inertial response time and an effective terminal velocity. In addition, an empirical model is developed to quantify the RDF. This, together with the theory for radial relative velocity, provides a parameterization for the turbulent geometric collision kernel. Using this integrated model, we find that turbulence could triple the geometric collision kernel, relative to the stagnant air case, for a droplet pair of 10 and 20??m sedimenting through a cumulus cloud at R?=2?104 and =600?cm2?s?3. For the self-collisions of 20??m droplets, the collision kernel depends sensitively on the flow dissipation rate.

Turbulent collision efficiency of heavy particles relevant to cloud droplets

The collision efficiency of sedimenting cloud droplets in a turbulent air flow is a key input parameter in predicting the growth of cloud droplets by collision-coalescence. In this study, turbulent collision efficiency was directly computed, using a hybrid direct numerical simulation (HDNS) approach (Ayala et al 2007 J. Comput. Phys. 225 51-73). The HDNS results show that air turbulence enhances the collision efficiency partly due to the fact that aerodynamic interactions (AIs) become less effective in reducing the relative motion of droplets in the presence of background air turbulence. The level of increase in the collision efficiency depends on the flow dissipation rate and the droplet size ratio. For example, the collision efficiency between droplets of 18 and 20µm in radii is increased by air turbulence (relative to the stagnant air case) by a factor of 4 and 1.6 at dissipation rates of 400 and 100cm 2 s 3 , respectively. The collision efficiency for self-collisions in a bidisperse turbulent suspension can be larger than one. Such an increase in self-collisions is related to the far- field many-body AI and depends on the volumetric concentration of droplets. The total turbulent enhancement agrees qualitatively with previous results, but differs on a quantitative level. In the case of cross-size collisions between 18 and 20µm droplets, the total turbulent enhancement can be a factor of 7 and 2 at

A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles

A hybrid direct numerical simulation (DNS) approach is proposed for simulating turbulent collisions of hydrodynamically-interacting particles, under the assumptions that the disturbance flows due to particles are very localized in space and there is a sufficient length-scale separation between the particle size and the Kolmogorov scale of the undisturbed turbulent flow. The approach consists of direct simulation of the undisturbed turbulent flow and an analytical representation of local small-scale disturbance flows induced by the particles. This hybrid DNS approach provides, for the first time, a quantitative research tool to study the combined effects of turbulence and hydrodynamic interactions on the motion and collisional interactions of small particles. Several numerical implementation issues are discussed, along with consistency and accuracy of the approach. Areas for further development of the approach are indicated.

Parallel implementation and scalability analysis of 3D Fast Fourier Transform using 2D domain decomposition

3D FFT is computationally intensive and at the same time requires global or collective communication patterns. The efficient implementation of FFT on extreme scale computers is one of the grand challenges in scientific computing. On parallel computers with a distributed memory, different domain decompositions are possible to scale 3D FFT computation. In this paper, we argue that 2D domain decomposition is likely the best approach in terms of using a very large number of processors with reasonable data communication overhead. Specifically, we extend the data communication approach of Dmitruk et al. (2001) [21] previously used for 1D domain decomposition, to 2D domain decomposition. A thorough quantitative analysis of the code performance is undertaken for different problem sizes and numbers of processors, including scalability, load balance, dependence on subdomain configuration (i.e., different numbers of subdomain in the two decomposed directions for a fixed total number of subdomains). We show that our proposed approach is faster than the existing attempts on 2D-decomposition of 3D FFTs by Pekurovsky (2007) [23] (p3dfft), Takahashi (2009) [24], and Li and Laizet (2010) [25] (2decomp.org) especially for the case of large problem size and large number of processors (our strategy is 28% faster than Pekurovskis scheme, its closest competitor). We also show theoretically that our scheme performs better than the approach by Nelson et al. (1993) [22] up to a certain number of processors beyond which latency becomes and issue. We demonstrate that the speedup scales with the number of processors almost linearly before it saturates. The execution time on different processors differ by less than 5%, showing an excellent load balance. We further partitioned the execution time into computation, communication, and data copying related to the transpose operation, to understand how the relative percentage of the communication time increases with the number of processors. Finally, a theoretical complexity analysis is carried out to predict the scalability and its saturation. The complexity analysis indicates that the 2D domain decomposition will make it feasible to run a large 3D FFT on scalable computers with several hundred thousands processors.

Improved Formulations of the Superposition Method

Two formulations of an improved superposition method are proposed for studying droplet–droplet hydrodynamic interactions. The formulations make explicit use of the boundary conditions on the surface of the two interacting droplets. The improved formulations are described through a consistent and rigorous consideration of the relationship between the drag force and representation of disturbance flows. It is demonstrated that the improved formulations are much more accurate than the original implementation of the superposition method. Specifically, for the case of Stokes disturbance flows, the relative errors on the drag force can be reduced by one order of magnitude using the improved formulations, when compared with the original formulation, for situations when the lubrication effect is not dominant. Using the improved superposition method, collision efficiencies of small cloud droplets falling in calm air are also computed and compared with previously published results.

Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations

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 60µm) in a typical cloud turbulence (dissipation rate at 400cm 2 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

Comments on “Droplets to Drops by Turbulent Coagulation”

Riemer and Wexler (2005, henceforth RW05) have utilized the turbulent collision kernel model of Zhou et al. (2001) to study the effect of turbulence on the initiation and development of raindrops from cloud droplets. They concluded that, for a cloud dissipation rate at 300 cm s , turbulent coagulation can move 96% of the droplet mass to sizes over 100 m in radius in 30 min, as compared to only 7% without turbulence. This result shows that turbulence is capable of rapidly transforming droplets to sizes for which the gravitational coagulation can operate effectively, thus overcoming the size-gap bottleneck for rain initiation. Under the assumption that the turbulent collision kernel of Zhou et al. (2001) can be extrapolated to atmospheric Reynolds numbers, RW05 found that the turbulent coagulation kernel is several orders of magnitude larger than the sedimentation kernel for droplets smaller than 100 m. We believe that the effects of turbulence have been grossly overestimated in RW05 for reasons to be discussed below. First, we would like to point out an error in RW05 that led to an overestimation of the rms velocity u by a factor of 3 and thus an overestimation of the Taylor microscale Reynolds number R by a factor of 3. RW05’s estimations were based on the rms velocity u and the average cloud dissipation rate on the in-cloud measurements by MacPherson and Isaac, shown in Table 1 of MacPherson and Isaac (1977). The cloud turbulence is anisotropic and a rough estimate of u for equivalent isotropic turbulence would be

Effects of aerodynamic interactions on the motion of heavy particles in a bidisperse suspension

The effect of local aerodynamic interactions on the motion of heavy particles in a bidisperse suspension in both quiescent and turbulent air is studied by a hybrid simulation and a theoretical treatment. The particles are assumed to be small in size compared to the Kolmogorov length of the carrier air turbulence, and Stokes disturbance flows are used to represent the effect of particles. We first consider the case of no background air turbulence to validate the numerical and theoretical approaches by comparing with previous results in suspension mechanics. In a bidisperse suspension with background air turbulence, in addition to the previously known increase due to preferential sweeping, aerodynamic interactions contribute to a second augmentation in mean settling rate which depends on the flow dissipation rate. This additional increase in settling rate due to local aerodynamic interactions is also coupled with preferential concentration, in agreement with the experimental observations of Aliseda et al. (...