Also at ]Ministry of Education Key Laboratory of Tropical Atmosphere-Ocean System, Zhuhai, China

Sharp depletion of radial distribution function of particles due to collision and coagulation inside turbulent flow: a systematic study.

Xiaohui Meng Ewe-Wei Saw [email protected] [ School of Atmospheric Sciences and Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, Sun Yat-Sen University, Zhuhai, China

Abstract

The clustering (preferential concentration) and collision of particles in turbulent flow is a significant process in nature and technical applications. We perform direct numerical simulation (DNS) to study the clustering of small, heavy, monodisperse particles subject to collision-coagulation in turbulent flow (i.e., colliding particles always coagulate (coalesce) into large ones). We find that collision-coagulation causes the radial distribution function (RDF) of the particles to decrease strongly at particle separation distances $r$ close to the particle diameter $d$ . However, we observe that the RDF do not decrease indefinitely but approach a finite value in the limit of $r\to d$ . We study how the characteristics of this “depletion zone” relate to the particle Stokes number (St), particle diameter, and the Reynolds number of the turbulent flow. A collision-induced modulation factor $\gamma_{c}$ is defined to represent the degree of RDF depletion due to collisions-coagulation. In the region where $\gamma_{c}(r)$ is a quasi-power-law, we find that the corresponding power-law exponent $\tilde{c}_{1}$ only depends weakly on $St$ . We also find that the overall trend of $\tilde{c}_{1}$ with respect to $St$ is similar to that of the classical power-law exponent $c_{1}$ appearing in the RDF of non-colliding particles, i.e., the exponent increase at small $St$ , peak around $St\approx 0.7$ , and decrease thereafter. The same qualitative trend is also observed for the limiting values of $\gamma_{c}$ at $r\to d$ . A complementary investigation on the Stokes number trend of the full RDF in the depletion zone is conducted. The slope of RDF appear constant for $St\ll 1$ but is changing when $St$ is getting large. The position where the RDF starts to decrease is found to be $St$ -dependent. We found that he depletion zone is insensitive to the flow Reynolds number and $\gamma_{c}$ of different $Re_{\lambda}$ overlap. With changing particle diameter $d$ , the reduction of RDF occurs at scales that shift accordingly and always starts at around $2.4d-3d$ . We also find that the shape of $\gamma_{c}(r)$ is independent of changes in $d$ .

I Introduction

The fluctuation of particle concentration has a profound effect on inter-particle collision. This effect has a pivotal role in both natural sciences and industrial engineering. For example, the collision-coagulation process for small droplets (particles) determines their spatial and temporal size distribution. These microscale properties have significant influences on the formation of precipitation [1]. Small-scale turbulence in clouds makes an important contribution to the collision and coagulation of droplets [2] [1]. Another example is the formation of planetesimal. The collision of dust in protoplanetary disks sets the stage for planet formation. Research shows that the turbulent motion will concentrate dust in the dissipation scale, increasing the collision rate of dust particles [3] [4] [5].

The preferential concentration of inertia particle has become a prevalent research topic since the end of the 20th century. Squires and Eaton [6] found that the inertial particles concentrate preferentially in regions of low vorticity and high strain rate. The degree of particle clustering can be characterized via the radial distribution function (RDF), which is defined as the ratio of the probability of finding a particle pair at a distance $r$ normalized by the probability of the same event for random particle distribution. Reade & Collins [7] found that the clustering of small particles occurs on a scale that is much smaller than the Kolmogorov length scale and the RDF of particles follows a negative power law with the inter-particle separation distance $r$ . Chun et al. [8] developed a drift-diffusion theory to predict the RDF in turbulent flows for particles in the limit of small particle Stokes number. Their results indicate that RDF of particle is proportional to $c_{0}(r/\eta)^{c_{1}}$ , where $\eta$ is the Kolmogorov length scale. They also find that the exponent $c_{1}$ is proportional to $St^{2}$ . The Stokes number (St) is an important measurement for particle inertia, and is defined as the ratio of particle relaxation time $\tau_{p}$ and the Kolmogorov time scale $\tau_{\eta}$ . The dissipation-scale clustering of particles becomes stronger as the Stokes number increase and the RDF reaches a peak near the Stokes number of order unit [9]. This relationship between the RDF and the Stokes number is corroborated by both numerical and experimental studies [10] [11].

The preferential concentration of particle is expected to enhance particle collision but the calculation of collision kernel is still an open question. In the work of Sundaram and Collins [12], the RDF was first introduced into the formula of collision kernel [13]: $K=4\pi d^{2}g(d)\langle W(d)\rangle$ , where $d$ is particle diameter, $g(d)$ is the value of RDF at $r=d$ and $\langle W(d)\rangle$ is the mean radial relative velocity of particle at $r=d$ . The works on the RDF of particles mentioned thus far had used the ghost particle assumption and ignored the effect of collision and coagulation among particles. Saw and Meng [14] found that the RDF will drop profoundly at $r$ close to $d$ in the presence of collision-coagulation. This finding is interesting because it highlights the importance of accounting for actual particle collision and it also questions the formula of collision kernel mentioned above.

In this paper, we use direct numerical simulation (DNS) to study the RDF of the inertial, colliding, particles. DNS, which solves the Navier-Stokes equation fully resolving the spatial and temporal scales of the problem without using any turbulence modeling, is an efficient numerical tool to study the particle-laden turbulent flow. The RDF of particle considering the effect of the collision-coagulation is investigated. The influence of the particle and turbulent parameters on the decreasing of RDF is also studied in this paper. The paper is organized as follows: section 2 provides a summary of simulation methods and the relevant turbulent and particle parameters. The statistical results and discussion are in section 3 and section 4. Finally, the results of the influence of the turbulent and particle parameters on the RDF are summarized in section 5.

II Simulation method

We performed direct numerical simulation (DNS) of the particle-laden turbulent flow. The incompressible Navier-Stokes equations are shown below.

\frac{\partial\vec{u}}{\partial t}+\vec{u}\cdot\nabla\vec{u}=-\frac{1}{\rho}\nabla p+\nu\nabla^{2}\vec{u}+\vec{f}(\vec{x},t)

(1)

\nabla\cdot\vec{u}=0

(2)

The N-S equations are numerically solved on $N^{3}$ grids cube using a pseudo-spectral method on the periodic domain, the length of which is $2\pi$ . The turbulent velocity $\vec{u}$ is transformed from physical space to wavenumber space. The aliasing error rising from the convection part of N-S equation is removed by the 2/3-method [15]. $\vec{f}(\vec{x},t)$ in the N-S equation is an external forcing conducted at low-wavenumber in order to maintain statistically stationary [16]. In order to study the influence of the (Taylor scaled) Reynolds number on RDF, simulations with different $Re_{\lambda}$ are conducted: $Re_{\lambda}=84$ , $124$ and $189$ , the detailed turbulent parameters are shown in Table 1. For different Reynolds number, the grid size is $N=256^{3}$ (for $Re_{\lambda}=84$ and $124$ ) and $512^{3}$ (for $Re_{\lambda}=189$ ). The small scale resolution may be characterized by $k_{max}\eta=1.59$ , 1.21 and 1.38 respectively, where $k_{max}=N\sqrt{2}/3$ is the maximum resolved wavenumber magnitude. The 2-order Runge-Kutta method is used to conduct time advancement in N-S equation. The Courant number $C=0.0248$ , $0.0401$ and $0.0865$ .

Table 1: The DNS parameters and time-averaged statistics.

N

is the simulation grid size,

\nu

is the kinematic viscosity of turbulence,

\epsilon

is the dissipation rate of turbulent flow,

u^{\prime}

is the root-mean-square velocity of turbulent flow,

\lambda

is the Taylor length scale,

\eta

and

\tau_{\eta}

are the Kolmogorov length and time scale,

L

and

T_{L}

are the integral length and time scale,

Re_{\lambda}

is the Taylor scaled Reynolds number.

	$N$	$\nu$	$\epsilon$	$u^{\prime}$	$\lambda$	$\eta$	$\tau_{\eta}$	$L$	$T_{L}$	$Re_{\lambda}$
flow 1	256	0.001	0.0326	0.3519	0.2386	0.0132	0.1750	0.5073	1.4416	84
flow 2	256	0.001	0.1013	0.5684	0.2187	0.0100	0.0993	0.6151	1.0822	124
flow 2	512	0.001	0.9472	1.226	0.1544	0.0057	0.0325	0.7398	0.6034	189

The particles we take are small (the diameter of particle $d$ is smaller than the Kolmogorov length scale $\eta$ ) and heavy (the particle’s density is larger than the flow’s). The gravitational effect and inter-particle hydrodynamic interactions are not included in our DNS model because only basic questions are discussed in this paper. Under these circumstances, the particles are only subjected to viscous Stokes drag force, and the motion equation of particles is shown below [17]:

\frac{d\vec{v}}{dt}=\frac{\vec{u}-\vec{v}}{\tau_{p}}

(3)

In which, $\vec{v}$ is the particle velocity, $\vec{u}(\vec{x},t)$ is the fluid velocity at particle position. $\tau_{p}$ is the particle inertia response time, defined $\tau_{p}=\frac{1}{18}\frac{\rho_{p}}{\rho}\frac{d^{2}}{\nu}$ where $\rho_{p}$ and $\rho$ are density of particles and flow respectively, $d$ is the particle diameter and $\nu$ is dynamic viscosity of turbulent flow. The linear interpolation method is used for interpolating flow’s velocity in particle position and the 2-order Runge-Kutta method with “exponential integrators” is used for time advancement [18].

Spherical and mono-dispersed particles are randomly introduced in the simulation. Particles collide when their volumes overlap and a new particle is formed conserving volume and momentum. New particles are injected continuously and randomly in the system so that the particle system is in a steady state after a transient period. In order to study the influence of particle parameters on RDF, particles with different Stokes numbers from $0.01$ to $2.0$ are introduced in each simulation. The size of particle is another important parameter related to particle collision. Three different sizes of particles are introduced in each simulation: $d=\frac{1}{3}d_{*}$ , $d=d_{*}$ and $d=3d_{*}$ , $d$ is the diameter of particle and $d_{*}=9.4868\times 10^{-4}$ . The details of simulations are listed in Table 2. The statistics are calculated for mono-dispersed particles.

Table 2: Characteristics of the runs discussed here.

Re_{\lambda}

is the Taylor microscale Reynolds number of the flow.

d

is the particle diameter,

d_{*}=9.4868\times 10^{4}

St

is the particle Stokes number.

Run	$Re_{\lambda}$	$d$	$St$
1	124	$d_{*}$	0.01
2	124	$d_{*}$	0.05
3	124	$d_{*}$	0.1
4	124	$d_{*}$	0.2
5	124	$d_{*}$	0.5
6	124	$d_{*}$	0.7
7	124	$d_{*}$	1.0
8	124	$d_{*}$	2.0
9	124	$\frac{1}{3}d_{*}$	0.1
10	124	$3d_{*}$	0.1
11	84	$d_{*}$	0.1
12	189	$d_{*}$	0.1

III Results and Discussion

III.1 Stokes number dependence

The statistics from Run 1 to Run 8 in Table 2 are used to study the influence of Stokes number on RDF. The RDFs for particles with different Stokes numbers are shown in Figure 1. What is striking in this figure is the strong decrease of RDF when the particle separation distance $r$ is close to the particle diameter $d$ . Figure 1 also show that there is an increase in the slope and the magnitude of RDF at the scales $r/\eta\sim 1-10$ when $St$ is increased from 0.01 to 0.7. We note that the slope of the RDF for the $St=1.0$ case is smaller than that of the $St=0.7$ case, even though its values are everywhere larger. Beyond $St=1.0$ , the slope and magnitude are both decreasing from $St=1.0$ to $St=2.0$ . At larger scales ( $r/\eta\sim 10$ ), the RDF curves flatten and converge to 1. To elucidate the trend of RDF when the separation distance $r$ is close to the particle diameter $d$ , the RDFs are plotted as the function of $r-d$ in Figure 2. The most interesting aspect of this graph is that the relationship between RDF and $r-d$ exhibits a quasi-power-law trend in the range $4\times 10^{-5}\lesssim r-d\lesssim 3\times 10^{-4}$ . As $r$ continues to decrease towards the particle diameter, the slope of RDF gradually becomes smaller and approaches zero, i.e., the RDF approaches a plateau.

Refer to caption — Figure 1: The RDFs versus $r/\eta$ for particles with different Stokes number. The diameter of particle is $d=9.49\times 10^{-4}$ and the Taylor scaled Reynolds number is $Re_{\lambda}=124$ . The RDF drops significantly when $r$ is close to $d$ .

Figure 3 illustrates more closely the trend of RDF as $r$ decrease towards $d$ , using the cases of $St=0.1$ , $St=0.2$ , and $St=0.5$ as examples. This shows more clearly that after $r-d$ is less than $4\times 10^{-6}$ , RDF no longer decreases but levels off to a fixed value. The limiting value of the RDF at particle contact ( $r=d$ ) is proportional to the particle collision rate [12], thus a finite collision rate in our simulations implies that the value of RDF should be equal to a fixed value but not an infinitesimal value. The results shown above agree with this analysis.

We shall call the region where the RDF decreases, as seen in Fig. 3, the depletion zone. Subsequent discussions will mainly focus on this region.

III.2 The Collisional Modulation Factor

To further characterize this “depletion zone” of RDF due to particle collisions, following the work of (Saw and Meng, 2022) [14], we assume that the RDF could be factorized such that $g(r)=\gamma_{c}(r)g_{n}(r)$ , where $\gamma_{c}(r)$ represent the effect of collision-coagulation in the form of a “modulation factor”, while $g_{n}(r)$ is the RDF for non-colliding (ghost) particles under the same physical environment. It is well known that, for mono-dispersed particle, $g_{n}(r)$ is a power-law of $r$ [19][8]. However, as shown in Figure 4, when plotted against $r-d$ , $g_{n}$ level off to plateau as $r$ decrease towards $d$ as a result of finiteness of $g_{n}(d)$ . The collision induced modulation factor $\gamma_{c}$ which, by definition, equals $g(r)$ compensated (divided) by $g_{n}(r)$ , is calculated as such in each case and shown in Figure 5. As expected, at large $r$ , $\gamma_{c}$ universally converge to unity, signifying that collisional effect are only appreciable at $r\sim d$ . In the $r\sim d$ regime on the other hand, we see that as $St$ increases from a minute value (i.e., $0.01$ ), $\gamma_{c}$ gradually decreases, with smaller $r$ affected more strongly. The rate of this decrease (with respect to $St$ ) is at first very weak, consistent with the hypothesis in [14] that $\gamma_{c}$ is independent of $St$ in the limit of small $St$ . The observed rate of decrease becomes pronounced when $St$ increases from $0.1$ to $0.5$ . Beyond $St=0.5$ , $\gamma_{c}$ seem almost stagnant again until it reverse the trend and start to increases significantly when $St$ is greater than $1.0$ . This implies that when the Stokes number of particles is much smaller than 1.0, the influence of collision on the RDF is insensitive to the Stokes number. As the Stokes number increases, the influence of collision grows and peaks at $St=1.0$ , which is a trend similar to the one of the power ( $c_{1}$ ) of inertial clustering [19]. In order to see the trend of $\gamma_{c}$ at small $r$ more clearly, using the case of $St=0.1$ , 0.2, and 0.5 as examples, we plot in inset of Figure 5, $\gamma_{c}$ versus the gap distance $r-d$ . Again, we see that $\gamma_{c}$ follows a semi-power-law for $r-d$ in the range of $1-6\times 10^{-4}$ . At smaller $r-d$ , the curves flatten and level off to a finite value.

For the sake of comparison, let us recall from earlier works [7][8][19] that $g_{n}$ is a negative power law of $r$ in the regime of $r/\eta\lesssim 20$ , i.e., $g_{n}(r)=c_{0}(r/\eta)^{-c_{1}}$ , where $c_{1}$ scales as $St^{2}$ for $St\ll 1$ . We now attempt to derive an analogous relationship between $\gamma_{c}$ and the Stokes number. From Figure 5, $\gamma_{c}$ seems to follow a quasi-power-law for gap distances $(r-d)$ in the range $10^{-4}\lesssim r-d\lesssim 7\times 10^{-3}$ . We assume that in this region $\gamma_{c}=\tilde{{c}}_{0}(r-d)^{\tilde{c}_{1}}$ . The relationship between $\tilde{c}_{1}$ and $St$ is shown in Figure 6. For $St\ll 1$ , considering the level of statistical uncertainty, there is a tentatively trend of increasing $\tilde{c}_{1}$ with Stokes number ( $\tilde{c}_{1}$ follows a quasi-linear relationship with $St$ for $St\ll 1$ in linear axes). This positive trend is clearer at larger $St$ . $\tilde{c}_{1}$ reach a peak value at around $St=0.7$ and thereafter decreases slowly at larger $St$ .

For comparison, we also show $c_{1}(St)$ and $\gamma_{c}(r\to d)$ in Figure 6, where $\gamma_{c}(r\to d)$ is the limiting value of $\gamma_{c}$ at particle contact ( $r=d$ ). The latter is of interest since it is closely related to the collision rate (in practice, we use the value of $\gamma_{c}$ at $r-d\sim 2\times 10^{-6}$ as this limit). Comparing these three plots, we see that the general trend of both $\tilde{c}_{1}$ and $-\gamma_{c}(r\to d)$ as a function of Stokes number is similar to that of $c_{1}$ .

The above results indicate that when $St\ll 1.0$ , the drop of RDF caused by particle collision-coagulation is roughly independent of the Stokes number. While for large $St$ ( $>0.2$ ), the RDF in depletion zone is related to it. Furthermore, the relationship between the decrease of RDF and $St$ is similar to that between the preferential concentration of particle and $St$ . This implies that the Stokes number dependence of the RDF could not be complete decoupled from $\gamma_{c}$ and that this issue merit further investigations.

{comment}

III.2.1 Comparative study of the full RDF

In view of the significant Stokes number dependence of $\gamma_{c}$ , we also conducted a complementary investigation on the Stokes number trend of the full RDFs (i.e., $g(r)$ ) as a comparative study. The RDFs for various Stokes numbers are translated in vertically to overlap with the $g(r)$ for $St=0.05$ at $r-d\sim 2\times 10^{-4}$ to compare their shape. In order to show the influence on $St$ more clearly, the RDFs for $St=0.05$ and 0.7 are shown in Figure 10 and the RDFs for $St\ll 1$ are shown in the inset. It can be seen that the slope of RDF is almost constant for $St\ll 1$ but is changing when considering larger $St$ . Furthermore, the position of the peak of the RDF has a significant $St$ -dependence. This shows that both $\gamma_{c}(r)$ and $g(r)$ has nontrivial Stokes number dependence which should be further investigated in future works.

III.3 Reynolds number dependence

Statistics of Run 3, Run 11, and Run 12, listed in Table 2, are used to investigate the influence of the Taylor-scaled Reynolds number $Re_{\lambda}$ on RDF. The RDFs for different $Re_{\lambda}$ are shown in Figure 11. The Stokes number is 0.1 and the parameters for three simulations are shown in Table 1. We see that in the range of small $r$ , the RDFs of all cases overlap but for larger $r$ , the RDFs are separated. However, if $r$ is normalized by the Kolmogorov length scale $\eta$ , as shown in the inset of Figure 11, the RDFs now overlap at large $r$ . These results suggest that in the range of $Re_{\lambda}=84-189$ , the statistics of the depletion zone related to particle collision is not influenced by $Re_{\lambda}$ , while the power regime related to the classical inertial clustering depends on $Re_{\lambda}$ but only through the change in Kolmogorov length-scale $\eta$ .

The modulation factor $\gamma_{c}$ for these cases are shown in Figure 12. We find that they overlap within statistical uncertainty. This again suggests that the main characteristics of the depletion zone is insensitive to flow Reynolds number from $Re_{\lambda}=84$ to 189.

According to the Kolmogorov 1941 hypothesis [20], when the Reynolds number is large enough, the statistics of the small-scale of turbulent flow will not be influenced by of the large-scale. Collision occurs on a scale of particle diameter, which is here much smaller than the Kolmogorov length scale, therefore, the decrease of RDF is insensitive to $Re_{\lambda}$ . This view is consistent with the above findings.

III.4 Particle diameter dependence

The RDFs for particles with different diameters are shown in Figure 13, the statistics used are from Run 3, Run 9, and Run 10. What is striking in Figure 13 is that the position where RDF starts decreasing is consistent with particle diameter.

The modulation factor $\gamma_{c}$ for particles with different diameter as a function of the rescaled gap distance $(r-d)/d$ is shown in Figure 14. In this case, the modulation factors $\gamma_{c}$ are coincident.

The collision process of particles is strongly related to the particle size. Since we only consider the RDFs of mono-dispersed particles, we expected that the position where depletion zone begins is close to the particle diameter. The results shown here are in line with our expectations.

{comment}

IV Discussion

When the collision and coagulation occur, the RDF of particle drops significant at the separation $r$ is close to the particle diameter $d$ . The RDF is calculated by the probability of finding another particle at a separation distance $r$ away from the primary particle normalized by the same probability for randomly particle distribution. Because of the collision and coagulation, this probability decreases, which results in the decrease of the RDF. After few eddy turnover time, the preferential concentration and the collision of particle reach a dynamic equilibrium. The RDF will not decrease infinitely with the decreasing of $r$ . This is consistent with our results in figure x and y. The decreasing part of the RDF, which will be mainly discussed here, is named the depletion zone, in the range of $2\times 10^{-6}\lesssim r-d\lesssim 1\times 10^{-4}$ .

For particles with different Stokes number, the RDFs have the same shape in the depletion zone. And when particles in three different flows with three turbulent Reynolds number, the depletion zone of RDF will not be influenced by the Reynolds number of turbulent flow. According to the Kolmogorov’s hypothesis [20], when the Reynolds number is large enough, the statistics of the small-scale motions will not be influenced by of the large-scale. If the Stokes number of particle is small ( $St<O(10)$ ), the clustering of particles is mainly affected by the flow in small scale. The former research [9] [collins2004reynolds] also shows that the RDF versus $r/\eta$ is going to be saturation with increasing Reynolds number. Since the particles diameters in this paper are much smaller than the Kolmogorov length scale $\eta$ , the scale where particles collide is also much smaller than $\eta$ , and the Reynolds number will not affect the decline of RDF. When the diameters of the particles are different, the position of the RDF depletion zone is consistent with the particle diameter. From the above results, we can say that the depletion zone of RDF is self-similar.

In this article, we propose the modulation factor $\gamma_{c}$ in the RDF: $g(r)=\gamma_{c}(r)g_{n}(r)$ represents the impact of particle collision on the RDF. Unexpectedly, $\gamma_{c}$ is sensitive to the Stokes number, and the relationship with the Stokes number is opposite to the relationship with $g_{n}$ , indicating that the higher the degree of particle preferential concentration, the effect of collision on RDF bigger. In the different range of $r-d$ , the functional relationship between $\gamma_{c}$ and the Stokes number is also complicated. Furthermore, $\gamma_{c}$ is not influenced by the Reynolds number of flows, and its depletion zone is also consistent with the particle diameter. In the study of this paper, the function and significance of the modulation factor remains to be discovered, and whether it can help to study more factors that affect particle collision remains to be studied in the future.

V Conclusion

This paper studies the change of radial distribution function (RDF) of particles subjected to the collision and coagulation (coalescence) interaction. We investigate the relationship between the RDF and particle Stokes number, particle diameter, and Reynolds number. We find that the RDF diminishes significantly at small particle separation distances $r$ . When viewed as a function of $r-d$ , we found evidence that the RDF do not decrease infinitely, but levels off to a fixed value in the limit of $r\to d$ . To study the relationship between the degree of reduction of the RDF and the particle and turbulent parameters, we separate the RDF into two multiplicative parts i.e., $g=\gamma_{c}\,g_{n}$ , where $g_{n}$ is the RDF for non-colliding particles under equivalent environment and $\gamma_{c}$ is a collisional modulation factor that reflects the effect of particle collision on particle preferential concentration. We see that $g_{n}(r-d)$ levels off to a plateau as the argument $r-d$ approaches zero. On the other hand, the collisional factor $\gamma_{c}$ universally converges to unity at large $r$ and levels off to a fixed value at $r\approx d$ . We find that $\gamma_{c}$ is dependent on the Stokes number. Specifically, assuming a power-law model for $\gamma_{c}$ in the region $0.1d\lesssim r-d\lesssim 0.7d$ (i.e., $\gamma_{c}=\tilde{c}_{0}(r-d)^{\tilde{c}_{1}}$ ), we find that in the small Stokes number limit, the value of $\tilde{c}_{1}$ is only very weakly dependent on $St$ , while the overall Stokes number trend of $\tilde{c}_{1}$ is qualitatively similar to the power law exponent $c_{1}$ in the RDF of non-colliding (ghost) particles (i.e., $g_{n}(r))$ . The magnitude of $\gamma_{c}$ at the limit $r\to d$ varies with the Stokes number following a trend similar to that of $\tilde{c}_{1}(St)$ .

The preceding findings motivate a comparative investigation into the $St$ trend of the full RDF, which has the result that the slope of the RDF $g(r)$ in the depletion zone is the same for $St\ll 1$ but it is different for large Stokes number (i.e., $St\geq 0.05$ ). Besides this, the location of the peak of RDF is found to be significantly $St$ -dependent. These findings imply that the Stokes number dependence of the RDF could not be completely decoupled from $\gamma_{c}$ (nor from $g(r)$ ) except in the regime of $St\ll 1$ where $St$ dependence is weak.

We find that the shape of the RDF in the depletion zone ( $r\sim d$ ) do not change with the variation of the flow Reynolds number within the range studied i.e., $Re_{\lambda}=84-189$ and the collisional modulation factor $\gamma_{c}$ from different $Re_{\lambda}$ overlap.

On the effect of particle diameter $d$ , we find that larger $d$ leads to the falling edge in $\gamma_{c}$ occurring at larger values of $r-d$ (and $r$ ) such that the results coincide when $\gamma_{c}$ is plotted against $(r-d)/d$ .

Acknowledgements.

This work was supported by the National Natural Science Foundation of China (grant 11872382) and by the Thousand Young Talents Program of China.

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