The geometric collision frequency per unit volume between particles with radius r1 and those with radius r2, Ncr1,r2, is expressed by the geometric collision kernel Kcr1,r2 as Ncr1,r2=Kcr1,r2np,1np,2, where np,i is the number density of particles with radius ri. The coagulation kernel Kcoag can be expressed by the combination of the geometric collision kernel, collision efficiency Ec, and coalescence efficiency Ecoal as Kcoagr1,r2=Ecoalr1,r2Ecr1,r2Kcr1,r2.

The gravitational collision kernel describes the collisions due to the settling velocity difference in the form of Kc,gravr1,r2=πR122V∞1-V∞2, where R12 (=r1+r2) is the collision radius and V∞i is the gravitational particle settling velocity. Turbulence enlarges the geometric collision kernel, i.e., the turbulent geometric kernel Kc,turb is larger than Kc,grav. Turbulence also enhances the coagulation kernel through enlarging Ec. The turbulence enhancement on the collision efficiency, ηE, is defined as ηEr1,r2=Ecr1,r2[T]Ecr1,r2[NoT], where [T] and [NoT] indicate the turbulent flow case and the stagnant (non-turbulent) flow case, respectively.

It had been difficult to confidently discuss the collision efficiency in a turbulent flow until developed a reliable superposition method, which iteratively solves the Stokes disturbance flows for a many-particle system. That superposition method is, however, computationally expensive due to its iteration procedure. later developed a less costly method, named the binary-based superposition method (BiSM), which has been adopted in the LCS (). BiSM assumes that interactions via three or more particles are negligible. This dramatically reduces the computational cost but maintains reliability as long as the particle number concentration is small, as observed in atmospheric clouds.

showed, by means of a DNS, that the preferential concentration of inertial particles, the so-called clustering effect, increases the collision frequency. The clustering effect is expressed in the spherical formulation derived by as

Kc(r1,r2)=2πR122wr(x=R12)g12(x=R12), where … denotes an ensemble average, wr(x=R12) (wr hereafter) is the radial relative velocity at contact separation, and g12(x=R12) (g12 hereafter) is the radial distribution function at contact separation and represents the clustering effect.

Droplet deformation and coalescence efficiency, which this study ignores, affect the collision growth of droplets with r>100 µm, although such effects only become significant for droplets with r>500 µm. It would, therefore, lead to some errors if extending the present results to such large droplets.

provided a parameterization for the turbulent geometric collision kernel of finite-inertia sedimenting droplets by proposing an empirical model for g12 in addition to a theoretical model for wr.

By following the expression by , the clustering effect for a monodisperse suspension of sedimenting droplets is expressed as

g11=ηrC1, where η is the Kolmogorov length. C1 is a function of St, Reλ, and the non-dimensional parameter for gravity V∞/vη with the Kolmogorov velocity vη. This parameterization was extended for a bidisperse system in a manner similar to that in :

g12=η2+rd2rL2+rd2C1/2, where rL=max⁡r1,r2 and C1 follows the same expression for the monodisperse case at Stmax⁡=max⁡(St1,St2)=St(rL), and rd is a length scale of the acceleration diffusion experienced by the particles. When two particles in a pair are two different sizes, any fluid acceleration or gravity will induce a relative velocity. This effect yields a diffusion-like process in the system and tends to smooth out inhomogeneities in the particle pair concentration. Thus, rd is larger for larger St1-St2 for the bidisperse case and a monodisperse suspension form is recovered for the case rd≪rL. It should be noted for the discussion in Sect  that the g12 model was designed to show maximum clustering at St∼1 and a higher droplet clustering for larger Reλ ().

In addition to the empirical g12 model, developed a theory for wr that is applicable to inertial droplets sedimenting under gravity in a turbulent flow. The basic assumption was that the droplet relative trajectory is mostly determined by gravitational sedimentation. Following , they decomposed the radial relative velocity (between two particles falling under gravity in a homogeneous isotropic turbulent flow) into a random part ξ caused by turbulent fluctuations and a deterministic part h due to gravity:

wr(ϕ)=ξ(ϕ)+h(ϕ), where the angle of contact, ϕ, is measured from the gravity axis. The random variable ξ(ϕ) is assumed to be normally distributed with a standard deviation σ(ϕ).

Using σ(ϕ=90∘) to approximate σ(ϕ), they obtained

wr=2πσ2+π8τp,1-τp,22g21/2, where σ is expressed in terms of τp,i, V∞i and flow parameters urms (the rms of the velocity fluctuations) in terms of ϵ and Reλ.

We now solve the three-dimensional continuity and Navier–Stokes equations for incompressible flows: ∇⋅U=0,∂U∂t+(U⋅∇)U=-1ρ∇p+ν∇2U+F(x,t). The kinematic viscosity ν is set to 1.5×10-5 m2 s-3, which is the value for atmospheric air at 1 atm and 298 K. The last term on the right-hand side represents the external forcing needed to achieve a statistically steady state. This study employs reduced-communication forcing (), which is suitable for massively parallel finite-difference models, to maintain the kinetic energy with k<2.5, where k is a wavevector. Spatial derivatives are calculated using fourth-order central differences. The conservative scheme of is employed for the advection term, and the second-order Runge–Kutta scheme is employed for time integration. To solve the velocity–pressure coupling, we use the highly simplified marker and cell (HSMAC) scheme (), which iterates until the rms of the velocity divergence becomes smaller than δ/Δ, where Δ is the grid spacing and δ is chosen to be 10-3. The governing equations are discretized by using a cubic domain of length 2πL0, where L0 is the representative length. Periodic boundary conditions are applied in all three directions. The flow cube is discretized uniformly into N3 grid points, resulting in Δ=2πL0N.

Under the limit of a large ratio of the density of the particle material to that of the fluid (ρp/ρf≫1), the governing equation for water droplets is given by

dVdt=-fτpV-U(x,t)+u(x,t)+Fimpulse+g, where V is the particle velocity, U is the air velocity at the position of the droplet, u is the disturbance flow velocity due to the surrounding droplets, and τp is the particle relaxation time defined as τp=(2/9)(ρp/ρf)(r2/ν), in which r is the particle radius. Fimpulse denotes the impulsive force due to collisions and g is the gravity vector (=(-g,0,0), where g is the gravitational acceleration). The ratio of the density of the particle material to that of the fluid, ρp/ρf, is set to 8.43×102 at 1 atm and 298 K, and f is the drag coefficient defined as the ratio between the nonlinear drag and the linear drag (). It should be noted that Eq. (), which adopts the point-particle assumption, is inaccurate for large St particles whose radii are not small enough compared to the Kolmogorov scale.

The second-order Runge–Kutta method is used for the time integration. The flow velocity at the droplet position U is linearly interpolated from the adjacent grid values. This simple linear interpolation is justified through comparisons with the cubic Hermitian, cubic Lagrangian, and fifth-order Lagrangian interpolations from . The disturbance flow u, which denotes the hydrodynamic interaction, is calculated by using the BiSM (). The particle mass and volume fractions are so dilute that the flow modulation is ignored.

After the background airflow has reached a statistically stationary state, monodispersed water droplets are introduced into the flow. After a period exceeding 3 times the eddy-turnover time T0=L0/U0, collision detection is then started. Droplets are allowed to overlap (ghost-particle condition) and a collision is judged from the trajectories of a pair of droplets by assuming linear particle movement for the time interval Δt.

The detailed description of the procedures for calculating collision statistics can be found in , who conducted the DNS for Reλ up to 530. This study performed additional simulations to push the maximum Reλ forward, up to 1140. The computational settings for the present simulations are summarized in Table .

To obtain reference data regarding droplet collisional growth, we tracked the growth of droplets that initially had the following exponential size distribution (e.g., ):

f0(x)=n0xm0exp⁡(-x/xm0), where xm0 is the mass of a droplet with a radius of rm0 and n0 is the initial number density. We carried out two cases: one with rm0=15 µm and n0=1.42×108 m-3 and the other with rm0=10 µm and n0=4.79×108 m-3. The corresponding initial liquid water content was 2.0 g m-3 for both cases. It was assumed that colliding particles immediately united without breakups and conserved mass and momentum.

Table summarizes the computational parameters for the flow calculation as well as the obtained flow statistics for the collision growth simulations. In cases T100, T, and T1000, the same grid configuration with the same Reynolds number was calculated, but the energy dissipation rates, which are in the typical range observed in turbulent atmospheric clouds, were 100, 400, and 1000 cm3 s-2, respectively. Cases T, TR127, TR206, and TR333 obtained flows with the same energy dissipation rate (400 cm3 s-2) but with different Reλ values. have already presented these cases, except for TR333 with rm0=15 µm. The present study additionally performed the case TR333 with rm0=15 µm to obtain a clear Reynolds-number dependence, as well as cases T, TR127, and TR206 with rm0=10 µm.

observed that the clustering effect and consequently the collision kernel decreases as the Reynolds number increases for Reλ>100 and St=0.4. Later, clarified that the Reynolds-number dependence of g11 observed for 1/3<St<1 is due to internal intermittency of the three-dimensional turbulence.

To quantify the influence of intermittence on g11, we need to separate the local quantity from the global (average) quantity. introduced the local energy dissipation as ϵl(x,t)=34πl3∫y≤lϵ#(x+y,t)dy, where superscript # denotes the local quantity. It was supposed that the probability density function (PDF) of ϵl follows a log-normal distribution if l is much smaller than the flow integral scale. Assuming l∼η, we obtain PLN(ϵ*|μ,σ2)=12πσϵ*exp⁡-ln⁡ϵ*-μ22σ2, where ϵ*=ϵη. Parameters σ and μ appear in the first and second moments of ϵ* as ϵ*=ϵ=exp⁡μ+σ2/2 and ϵ*2=exp⁡2μ+2σ2, respectively.

The intermittency is measured by the flatness factor F, defined as F=∂u1/∂x14∂u1/∂x122. It is observed that F follows a power law relation with Reλ, for example F∼Reλ3/8 (). Given ∂u1/∂x1∼ϵη/ν1/2=ϵ*/ν1/2, we obtain F∼ϵ*2ϵ2∼Reλ3/8. Substitution of Eqs. () and () into Eq. () yields σ2=38ln⁡Reλ. Eq. () then yields μ=ln⁡ϵReλ-3/16. That is, PLN(ϵ*|μ,σ2) can be rewritten as PLN(ϵ*|Reλ). We can define a local St, St*, as St*=St×ϵ*ϵ1/2, the PDF of which follows P(St*|St,Reλ)=2ϵSt*St2PLNϵSt*St2Reλ. It should be emphasized that the shape of PLN (and consequently P) depends on Reλ. If we assume a universal radial distribution function at contact separation against St*-g11#univ(St*)-, the global clustering effect can be obtained as g11St,Reλ=∫0∞g11#univ(St*)P(St*|St,Reλ)dSt*. It should be noted that g11 depends on Reλ, whereas g11#univ does not (which is why it is called universal). For St*≪1, the universal clustering effect would have the form g11#univ=A1St*2+1 by following Eq. (). Substitution of this form into Eq. () yields g11St≪1,Reλ=A1St2+1, regardless of the value of Reλ. This explains why the g11 for St=0.1 does not show a significant Reynolds-number dependence. For a moderate St*, we simply formulate the universal function by following Eqs. () and (11) but without the smoothing operators, as follows: g11#univ(St*)=H(St*-Sta*)A1*St*2+H(Sta*-St*)A2*St*-2, where A1* and A2* are empirical parameters and Sta* is defined as (A2*/A1*)1/4. Based on the DNS data for St=0.1, 0.4, and 0.6 in the flow with Reλ=130, we found that A1*=110 and A2*=0.073 work reasonably well. Although we have no justification for this universal function, it can provide g11 for arbitrary St (< 1) and Reλ through Eq. (). As we cannot analytically calculate the integration in Eq. (38), we have to numerically calculate it to obtain g11 for a certain combination of St and Reλ. We calculated g11 for St=0.1, 0.4, and 0.6 with Reλ=100, 200, 400, 1000, 4000, and 10 000. We then obtained the following empirical formulations by applying the least-squares method to the calculated results. g11St=0.1,Reλ∼2.1,g11St=0.4,Reλ∼19.3-1.9log⁡10Reλ,g11St=0.6,Reλ∼34.3-3.9log⁡10Reλ. Figure  shows a comparison between g11 values from the above equations and those from the DNS. The figure shows that the empirical estimates can reproduce the Reynolds-number dependence of g11 correctly. The figure includes the data for St=0.6 recently reported in . The two DNS datasets for St=0.6 agree well. argue that the Reynolds-number dependence is weak for the Reynolds number range explored by DNS, but the argument should be carefully examined when extrapolating to atmospheric flows. Figure  shows indeed a weak decrease in g11 for Stokes numbers between 0.3 and 1.

Figure shows a comparison between direct numerical simulation results and model predictions for g11. The dashed lines are the prediction by the Onishi model (), and the solid lines are the predictions by the present updated model. The DNS data for St≤1 and for Reλ≤530 were obtained from the table in . The data for St=1.4, 2, 4, and 8 were newly obtained. The results for Reλ=874 and 1140 (these Reynolds numbers are the largest ever achieved for turbulent particle collision statistics) are included in the figure. The DNS data show a decreasing trend for St<1 for the moderate Reynolds number range of 100≲Reλ≲1000. This decreasing trend with respect to Reλ is attributed to the flow intermittency () as discussed in the previous subsection. The black solid line is the estimated g11 for St=0.4 and the black dashed line is for St=0.6 (Eqs.  and , respectively). The present Onishi model shows slightly better agreement with the DNS data in terms of the slopes in comparison with the original model. For St>1, the DNS data show increasing trends for the moderate Reλ range, and those trends are predicted by the present parameterization, although the rate for St=2 is overestimated. One significant feature of the Onishi g11 model is that maximum clustering occurs at a larger St for a larger Reλ. This shows a clear contrast with the Ayala–Wang model, which was designed to show maximum clustering at St∼1 regardless of Reλ.

The updated parameterization leads to improvement, particularly for the St≥1 regime. For example, in the case of Reλ=127, the rms values of the relative errors of the prediction with the original parameters for (i) St=0.1, 0.2, 0.4, and 0.6 and for (ii) St=1, 1.4, 2, 4, and 8 were (i) 0.081 and (ii) 0.239. The rms values with the present parameters were (i) 0.075 and (ii) 0.113.

Figure shows a comparison between model predictions and DNS results of the coagulation kernel Kcoag(r1,r2) for r1=30 µm, Reλ=127, and ϵ=400 cm2 s-3. The kernel is normalized by the collision radius R and the local velocity gradient λ=ϵ/ν1/2. The reference DNS considers the hydrodynamic interaction and the gravitational droplet sedimentations. We observe a large discrepancy for r2∼30 µm (= r1), where the turbulence enhancement on collision efficiency is difficult to define, because the collision efficiency for r1=r2 cannot be defined for stagnant flow. Otherwise, the model predictions (Ayala–Wang model and Onishi model) agree well with the DNS results. As an example, we also observe a slight improvement in the Onishi model by including the sedimentation effect on wr (Sect. ) on the data for r2=40 µm.

The Ayala–Wang model shows a local maximum around r2=r1. The DNS results also show a convex shape, but the value at r2=r1 is much smaller than the prediction by the Ayala–Wang model. In contrast, the Onishi model does not show such a local maximum at r2=r1 but does provide values much closer to DNS elsewhere. The convex shape is related to the diffusion effect denoted by rd in Eq. (). Equation () for g12, employed in the Onishi model, was formulated for non-sedimenting droplets and this equation therefore leads to weaker acceleration-driven diffusion, i.e., smaller rd (). This can explain why the Onishi model does not show the convex shape.

Figure shows the ratio of the turbulent coagulation kernel to the Hall kernel for the turbulent flow with Reλ=127 and ϵ=400 cm2 s-3. The level of the ratio is basically similar for both the Ayala–Wang and Onishi models, and the ratio is nearly unity when the droplets are above 100 µm.

Figure shows the ratio of the coagulation kernel for Reλ=104 to that for Reλ=103. It should be noted that the Ec and ηE models employed in the Ayala–Wang and Onishi kernels do not consider the Reynolds dependence. Therefore, the figure actually shows the ratio of the geometric collision kernels, i.e., the ratio of wrg12. The Ayala–Wang kernel increases for the autoconversion region (r1,r2<40 µm) and the accretion region (r1<40 and r2>40, and r1>40 and r2<40 µm). The Onishi kernel decreases for the corresponding autoconversion region, but increases for the rain–rain self-collection region (r1,r2>40 µm).

Figure shows the ratio of g12 for Reλ=104 to that for Reλ=103. It should be noted that the form of Eq. () violates the spherical form and we cannot rigorously define g12,total and wrtotal that formulate Kc,total=2πR122wrtotalg12,total. Here, we simply considered g12 expressed by Eq. (12) as the g12,total for the total kernel and obtained wrtotal=Kc,total/2πR122g12. As designed, the Ayala–Wang kernel shows the increase for increasing Reλ for both the autoconversion and the accretion regions. In contrast, the Onishi kernel shows a decrease for the autoconversion region, but a significant increase for the accretion region and the rain–rain self-collection region (i.e., r1,r2>40 µm). This is due to the shift of the maximum clustering toward larger St with increasing Reλ.

Figure shows the ratio of the radial relative velocity for Reλ=104 to that for Reλ=103. The Ayala–Wang kernel shows little Reynolds-number dependence. In contrast, the Onishi kernel shows significant Reynolds-number dependence, which tends to be opposite to the Reynolds-number dependence of g12 and thus weakens the Reynolds-number dependence of the collision kernel.

The Reynolds-number dependence of the clustering effect is larger than that of the radial relative velocity, and the contour shape of Fig.  is more similar to Fig.  than to Fig.  for both the Ayala–Wang and the Onishi kernels. That is, the Reynolds-number dependence of the two kernels can mostly be attributed to the g12 parameterizations.

Note that the Fortran 90 code used to calculate the present Onishi kernel is provided in the Supplement.

We investigated the turbulence enhancement on the autoconversion rate, which is the conversion rate from the cloud category (r<40 µm) to the rain category due to collisions between the small cloud droplets. The Ayala–Wang kernel and the present Onishi kernel were employed to calculate the coagulation growth of droplets modeled by the stochastic collision–coalescence equation (SCE): ∂nf(m,t)∂t=∫0m/2Kcoag(m-m′,m′)nf(m-m′,t)nf(m,t)dm′-∫0∞Kcoag(m,m′)nf(m,t)nf(m′,t)dm′, where m is the particle mass and nf is the number density function. The coagulation component of the spectral bin model in the Multi-Scale Simulator for the Geoenvironment (MSSG-Bin) cloud physics model () was used to solve the SCE. The mass coordinate m was discretized as mk=21/smk-1, where s was set to 16. The representative radius of the first bin was 2.7 µm and 528 classes were calculated, the largest class of which had a representative radius of 5.4 mm. The SCE solution is basically a mean-field approximation. In contrast, the LCS acts as a reference model as it includes all turbulence effects directly in its Lagrangian particle simulation. Due to the high computational cost, however, the LCS is restricted to moderate Reynolds number (here up to Reλ=333).

Following , used a quantitative measure of the turbulence enhancement focusing on the timescale of the autoconversion process. The time required for a cloud to convert 10 % of its cloud mass into rain category drops is expressed as t10%, which can be used as a measure of the autoconversion timescale. Then, we can define the turbulence enhancement factor, Eturb, as

Eturb=PautoTPautoNoT=t10%‾NoTt10%‾T, where the overbar indicates the mean value.

Figure a shows Eturb as a function of ϵ for Reλ=66 in the rm0=10 µm case. The LCS data show an almost linear increase with increasing ϵ. Both the SCE simulation with the Ayala–Wang kernel (SCE-Ayala hereafter) and that with the Onishi kernel (SCE-Onishi hereafter) show the same trend with the LCS data, although the SCE-Ayala slightly overestimates the enhancement. The maximum relative difference between the SCE-Ayala and SCE-Onishi kernels was as small as 22 % at ϵ=500 cm2 s-3. Both the SCE-Ayala and the SCE-Onishi kernels show a kink at ϵ=600 cm2 s-3, where the turbulence enhancement on collision efficiency levels off. Figure b shows Eturb as a function of Reλ for ϵ=400 cm2 s-3 in the case of rm0=10 µm. The SCE-Ayala and the SCE-Onishi kernels show different trends: the SCE-Ayala predicts an increasing enhancement with increasing Reλ, while the SCE-Onishi predicts almost constant or slightly decreasing enhancement. The difference between the two SCE predictions becomes larger for larger Reλ, with the LCS result closer to the SCE-Onishi prediction. The difference between the SCE-Ayala and the SCE-Onishi kernels can be explained by the Reynolds-number dependence of the two kernels, as discussed in Sect. . This Reynolds-number dependence is relevant, because the SCE prediction becomes very different at large Reλ. For example, at Reλ=2×104, the SCE-Ayala prediction is 2.5 times larger than the SCE-Onishi prediction. The LCS results for Reλ≤206 support the SCE-Onishi prediction.

Figure shows Eturb for the rm0=15 µm case, which was also discussed in . This study additionally performed the simulation for Reλ=333 to investigate the Reynolds-number dependence more clearly. Basically, the results are similar to those in the previous figure. In Fig. , the SCE-Ayala and the SCE-Onishi kernels show closer results for Reλ=66, and both SCE-Ayala and SCE-Onishi slightly overestimate the enhancement for ϵ>400 cm2 s-3. The difference between the two predictions at Reλ=2×104 is larger: the SCE-Ayala prediction is 3.0 times larger than the SCE-Onishi prediction. The LCS results for Reλ up to 333 clearly support the SCE-Onishi prediction.

In summary, both Figs.  and show that the SCE-Ayala and the SCE-Onishi kernels produce consistent results for low Reλ with about a 20 % difference at most, but the two show very different values at large Reλ: the SCE-Ayala prediction becomes larger than the SCE-Onishi by a factor of up to 3 in cloud turbulence. This clearly suggests a strong demand for collision growth data with larger Reλ to construct a more robust turbulent kernel.

As noted in and discussed in Appendix A in , the periodicity of the computational domain may lead to errors for the settling particles with large St. defined the critical St, Stcrit, as Stcrit=FrLlu′uη, where Fr is the Froude number (=aη/g, where aη is the Kolmogorov-scale acceleration), L(=2πL0 in this study) is the domain size, l is the integral scale, and uη is the Kolmogorov-scale velocity. For St larger than Stcrit, the periodicity problem may arise.

Figures , , and are for settling particles. For those figures, we have calculated Stcrit to check the periodicity problem. (i) For Fig. , Stcrit=3.7, which corresponds to rcrit=75 µm; rcrit is the radius of particle with St=Stcrit. The two plots from DNS, which correspond to r2=80 µm and 120 µm, exceed rcrit. However, since the two plots are more or less similar with the gravitational (Hall) kernel values, the turbulent contribution would be small compared to the gravitational settling contribution. That is the error due to the periodicity would not significantly affect the results. (ii) For Figs. a and a, rcrit are 50, 65, and 70 µm for ϵ=100, 400, and 1000 cm2 s-3, respectively. For Figs. b and b rcrit are 65, 75, 85, and 90 µm for Reλ=66.1, 127, 206, and 333, respectively. The enhancement factor Eturb, shown in Figs.  and , was evaluated by t10%, which is defined as the time required for a cloud to convert 10 % of its cloud mass into rain category drops. The threshold between cloud and rain categories was set at r=40 µm. That is, 10 % of particles, in mass and volume, are larger than 40 µm in radius at t=t10% by definition. For example, according to the DNS results, 3 % of particles are larger than 50 µm and only 0.9 % of particles are larger than 60 µm at t=t10%. The percentage of particles that are larger than 50 µm in radius may have some impact on t=t10% and consequently Eturb. In this sense, the plot for ϵ=100 cm2 s-3 in Figs. a and a, whose rcrit is 50 µm, may contain some error associated with the periodicity problem. However, since Eturb for that plot is nearly unity, indicating small turbulence enhancement, the periodicity problem does not change the present findings.