To theoretically account for the new findings, we make some derivations that are partially similar to the ones in but under a new constraint due to coagulations: at contact (r=d), the radial component of the particle relative velocities can not be positive

In other words, particles may approach each other (and collide), but they can not be created at contact and then separate.

Simple analytical solutions to Eq. () may be elusive due to its integral nature (a consequence of the non-local diffusive flux). However, one could gain insights into it and test its accuracy via numerical solutions. To that end, we begin with a simple ansatz for g(r), then we curve-fit the ansatz to the DNS-produced RDF gDNS(r). This enables us to, firstly, verify that the ansatz could accurately represent gDNS(r) and, secondly, to obtain a “calibrated” ansatz that is a numerically accurate representation of gDNS(r). We then show that Eq. (), supplied with the calibrated ansatz, could numerically predict 〈Wr〉(r) (i.e., MRV) that agrees well with the DNS-produced MRV. In short, we will show that given a “correct” g(r), Eq. () produces the “correct” 〈Wr〉(r).

The ansatz has the form ga(r)=g0(r)gs(r), with gs(r)=c0r-c1, i.e., the RDF form for non-colliding particles under the same conditions. As a first-order analysis, we let g0, which embodies the effects of collision, take the simplest form that could still capture the main features of the RDFs seen in Fig. . Specifically, we let g0(r)=c00(r-d1)c10, where c00(r) and c10(r) are each piecewise constant quantities that switch from their small r to large r values at a crossover-scale rc (of the order of d); i.e., g0 is a two-piece power law of r-d1. (Note that our earlier finding of g(r→d)>0 implies that d1<d.)

From a given DNS-produced RDF gDNS(r), we first obtain a calibrated gs by fitting c0r-c1 to gDNS(r) in the power-law regime d≪r≲10η (see the red line in Fig. b). Next, we compute the DNS estimate of g0 via g0DNS=gDNS(r)/gs(r), which is essentially a compensated RDF (see the ○ plot in Fig. b). To get a calibrated g0, we then fit the general form of g0 given above to g0DNS (see the cyan line in Fig. b; note that each time, two pieces of power laws are fitted to one g0DNS, and rc results naturally from the intersection of the two). Figure b shows the calibrated ansatz for the case of St=0.054 and verifies its accuracy (the red line is gs(r), the cyan line is g0(r) and the dashed black line (g0⋅gs) accurately reproduces gDNS(r)). The inset in Fig. b shows that g0DNS(r) (i.e., gDNS(r)/gs(r)) is roughly St-independent for St≪1.

Next, we numerically evaluate the integral in the first term of Eq. (). The St≪1 assumption allows us to approximate g(r,St) inside the integral by its zero-St cousin g(r,St→0) . In practice, we replace g(r,St) with the ansatz fitted to the DNS result of g(r,St=0.001). Next, we use the DNS data to estimate Aτ and compute Rc* and cst (for this case, DNS gives Rc*=9.69×10-10dm3s-1; cst=|c1| as mentioned earlier). Finally we use Eq. () to predict 〈Wr〉(r).

Comparison of the predicted 〈Wr〉(r) with the values obtained directly from the DNS is shown in Fig. . The prediction shown was made for the case of St=0.054, to be compared with its DNS counterpart (the ○ symbols). (We also show the DNS result for St=0.001 and 0.11 to highlight an observation that 〈Wr〉(r) is almost St-independent in this small St regime.) We showed earlier that for r∼d, Aτ is given by Eq. (). However, as stated earlier, as r increases, the (statistical) asymmetry induced by collision–coagulation gradually becomes subdominant to the isotropy of turbulent fluctuation. Statistical isotropy implies aik(1)=0 , a fact our DNS data confirm. Thus, for r≫d, Aτ equals the order St2 term in Eq. (), exactly the same as the results of for non-colliding particles. For this reason, we show two versions of the prediction: 〈Wr〉r∼dtheory and 〈Wr〉r≫dtheory, which are respectively obtained by setting Aτ to its small r and large r limits (-2.6×10-3 s-1, 7.1×10-3 s-1) respectively. The agreement between DNS and the predictions is noteworthy, especially for small r. At r≈2d, the DNS result shows a weak tendency to first follow the upward trend of 〈Wr〉r∼dtheory and then drops off significantly at r≳2.5d. The latter is consistent with the fact that 〈Wr〉r≫dtheory is below 〈Wr〉r∼dtheory, but the drop is sharper than predicted.

Alternatively, Eq. () may be solved for the correct g(r), if 〈Wr〉 is given. As we are assuming St≪1, particle velocity statistics may be approximated by their fluid counterparts ; i.e., we may replace 〈Wr〉 with 〈Wr〉St=0, the latter being the MRV of fluid particles. Hence, if 〈Wr〉St=0 is known, it may be used, together with Eq. (), to predict RDF of any finite but small St. Figure a shows that 〈Wr〉St>0 from the DNS do not change significantly for St∈[0.001,0.1], supporting this approach

This is true in the relatively idealized system simulated but may not apply to the general problem that includes other effects

Here we provide a simple, first-order model for 〈Wr〉St=0. We limit ourselves to the regime of small particles (d≪η) and anticipate that 〈Wr〉 is non-trivial (nonzero) only for r∼d, a fact observable in Fig. a. We also assume that the relative trajectories of particles are rectilinear at such small scales. The coagulation constraint then implies that: in the rest frame of a particle (call it P1), a second particle nearby must move in such a way that the angle (θ) between its relative velocity and relative position (seen by P1) must satisfy sin⁡-1(d/r)≤θ≤π, under the convention of sin⁡-1(x)∈[-π2,π2] (more in the Supplement). We can thus write (by treating negative and positive wr separately, applying the K41 theory and the bounds on θ, details in the Supplement), for St≪1, that 〈Wr〉≡〈wr〉∗=p–〈wr|wr<0〉∗+p+〈wr|wr≥0〉∗11≈-p–ξ–r+p+ξ+r1+∫θm0Pθ+(θ′)cos⁡(θ′)dθ′∫0π2Pθ+(θ′)cos⁡(θ′)dθ′, where 〈.〉∗ denotes averaging over particle pairs, p+ (p-) is the probability of a realization of wr being positive (negative), and Pθ+ is a conditional PDF such that Pθ+≡Pθ|wr≥0≡Pθ|θ∈[0,π2], θm is the lower bound of θ described above. For a first-order account, we neglect skewness in the distribution of particle relative velocities and set p±=0.5. Following , we have set 〈wr|wr<0〉∗=ξ–r, where ξ±=Csε/(15ν), (Cs is a Kolmogorov constant, we found Cs=0.76 by matching ξ–r to the first-order fluid velocity structure-function from the DNS).

A simple phenomenological model for P(θ) may be constructed using the (statistical) central-limit theorem by assuming that the angle of approach θ at any time is the sum of many random-incremental rotations in the past, thus we write 12P(θ)=Nexp⁡[Kcos⁡(θ-μθ)]sin⁡(θ), where Nexp⁡[…] is the circular normal distribution, i.e., analog of Gaussian distribution for angular data; sin⁡(θ) results from integration over azimuthal angles (ϕ). We set μθ=π2 (neglect skewness in fluid's relative velocity PDF) and obtain K by matching the transverse to longitudinal ratio of structure functions (TLR) of the particle relative velocities with the ones via the DNS data; N is determined via normalization of P(θ). Figure b shows the 〈wr〉∗ derived via Eqs. () and (), using K calibrated with TLR of second-, fourth-, and sixth-order structure functions respectively. The results have a correct qualitative trend of vanishing values at large r that increase sharply as r approaches d, with the fourth-order's result giving the best agreement with DNS. Currently we do not have a satisfactory rationale to single out the fourth order. The TLR of different orders giving differing results may imply that our first-order model may be incomplete, possibly due to over-simplification in Eq. () or to the inaccuracy of the rectilinear assumption (d/η in the DNS may be insufficiently small).

We now discuss an important but precarious theoretical issue. clearly showed that the non-local diffusion (qrD) may only be converted from its general integral form into a differential version when the underlying RDF is a simple power law. However, and , working in two very different scenarios, found that their predictions using the differential form of the theory agree well with experiments, even when the RDFs involved were clearly not power laws. We shall attempt to remedy this apparent paradox in future work. To examine how well this albeit unjustified method works here, we recast Eq. () into its differential form : 13-τη-1Bnlr4∂g∂r+g(r)[r2〈Wr〉-Aτr3]=-Rc*, where Bnl=0.0397 (this value is computed from our DNS), and Bnl is expected to depend on flow characteristics, e.g., Rλ and τη (more in the Supplement). Using Eq. (), the same gsg0 ansatz, we make another prediction for 〈Wr〉St=0.054, which is plotted in Fig. a (dashed green line). This prediction is far from the DNS at r∼d but performs as well as the integral version at r≫d (the jump in the curve is just an artifact from the kink in the ansatz).

One advantage of Eq. () is that it allows for a general solution, which we now give, assuming 〈Wr〉 is given by Eqs. () and (): 14g(r)=1β(r)∫β(r)q(r)dr+C, with q(r)=Rc*τη/(Bnlr4), β(r)=exp⁡∫p(r)dr and p(r)=Aτr-〈wr〉∗τη/(Bnlr2) (more in the Supplement).

Thus far, we have not considered the effects of gravity on the particles. Here we provide a glimpse into the role of gravity (a detailed analysis is beyond the scope of the present study). In keeping with the scope of current work, we restrict ourselves to the case of monodisperse particles only. For this, we rerun the DNS cases of St=0.054 and 0.54 with gravity (to be compared with the zero-gravity case). The new particle advection equation is dv/dt=(u-v)/τp+g (all other details of the DNS remain unchanged). We choose to have the particle settling parameter Sg≡τpg/uη (where uη is the Kolmogorov velocity scale) be in the range O(0.1)-O(1) (this is achieved by letting |g|=10 dms-2). As a result, the range of Sg and St explored here is well aligned with measured values in natural clouds . For the case of St=0.054 (Sg=0.49), we find no discernible difference for both RDF and MRV between the “with gravity” and zero-gravity results (corresponding figures in the Supplement). For the St=0.54 (Sg=4.9) case, Fig.  shows the effects of gravity on the RDF and MRV. We see that the slope (exponent of g(r) in the range d≪r<20η) of the RDF in the gravitational case is reduced by about 15 % compared to the zero-gravity case (Sg=0). However, the shape of the RDF in the collision regime (r∼d) is approximately preserved, suggesting that a construct of the form gcollision×ggravity may be a good first-order model for the full RDF (close examination of the compensated RDFs gives substantial support for this idea, details in the Supplement). These observations imply that as Sg increases from O(0.1) to O(1), the effects of gravity on RDF grow from negligible to significant but not dominant; the main effect is the reduction of the exponent while the collision-related “modulation” (gcollision) remains largely intact. The inset of Fig.  shows that the MRV is also weakened by gravity, though the statistical noise limits the strength of this conclusion. Lastly, it is worth noting that in the complementary DNS by that included gravity but not actual collisions, a much stronger gravitational effect was found on the statistics of bidisperse particles relative to the monodisperse case.

As mentioned, the fundamental focus of our work precludes the DNS and theory from considering a number of complexities relevant to some applications. As a result, this limits the direct quantitative applicability of our results to some realistic problems (e.g., in clouds). Besides gravity, another neglected factor is the hydrodynamic inter-particle force (HDI). Recent works, e.g., found that the HDI also has a strong impact on RDF for r∼d. For monodisperse particles with small to moderate St, the HDI is expected to be more important than gravity. While we expect that the HDI should not alter the qualitative trend that g(r) should fall towards a small value at r→d (the same applies to the observed trend of MRV), it is likely that the HDI and collision would affect RDF and MRV in a coupled manner.

Also neglected is the influence of temperature, humidity and the vapor–liquid phase transition, which are important in the atmospheric clouds. These factors have a substantial impact on the polydispersity of small droplets (see, e.g., ). However, for monodisperse statistics considered here, they are likely to play a minor role (they will be more important when future works consider the full polydisperse problem).

One limitation of the theory stems from the assumption of St≪1 and its corollary that particle velocity statistics in this regime are St-independent , which limits the theory's applicability to real systems. This implies that MRV should be St-independent in this regime. Our DNS results (spanning 2 orders of magnitude in St) shown in Fig.  give some support to the latter. However, unlike the theoretical prediction for MRV of case St=0.054 (Fig. ), we have found that the prediction for St=0.11 is discernibly below the DNS result (figure in the Supplement). This could be due to the finite St effect not captured by the theory or could be for other reasons (details in the Supplement). Hence, a finite St extension of the theory is desirable to improve its applicability to real systems.