4.1 Mean velocity and particle distribution

In figure 3(a,b), we display the mean streamwise fluid velocity profile for the five cases under investigation. We also include the velocity profile of an unladen turbulent channel flow at the same bulk Reynolds number, $Re_{b}=5600$ , indicated with the dashed line. We recall that all the simulations are performed by enforcing a constant mass flux. The fluid statistics are obtained by averaging in the streamwise and spanwise directions over the points outside the particles and by temporal averaging. Although the mean velocity profile for the particle-laden turbulent flows exhibit a strong modification with respect to the case of turbulent unladen flows, the difference between the particulate cases is not significant. In figure 3(a) we note that the mean velocity profile of the particulate flow is less blunt and more similar to a laminar flow with higher velocity toward the channel centre. This behaviour is slightly accentuated in suspensions with higher concentration of large particles. This suggests that the turbulent activity is reduced in the presence of the solid phase, the more so for larger particles. In figure 3(b) we present the mean velocity profile of particulate and unladen cases scaled in inner units, i.e. $U_{f}^{+}=U_{f}/u_{\unicode[STIX]{x1D70F}}$ and $y^{+}=yu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ with $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$ the friction velocity and $\unicode[STIX]{x1D70F}_{w}$ the wall shear stress. The inner-scaled velocity profiles of the particulate cases are lower than those of the unladen flow indicating an increase in the wall shear stress in the presence of particles. The log profile in the classical unladen turbulent flow reads $U^{+}=(1/\unicode[STIX]{x1D705})\text{ln}\,y^{+}+B$ where $\unicode[STIX]{x1D705}$ is the von Kármán constant and $B$ is an additive coefficient (Pope Reference Pope2000). Here we obtain, similar to Picano et al. (Reference Picano, Breugem and Brandt2015), that the slope of the log region increases while the additive constant reduces significantly, which results in an overall drag enhancement. No significant differences emerge among the different cases. We therefore conclude that the mean flow velocity of the turbulent channel flow is definitely controlled by the volume fraction of the solid phase, while the bi-dispersity of the suspensions plays a minor role.

Figure 3(c) shows the mean velocity of the dispersed solid phase. Particle statistics are obtained by spatial averaging over the points inside the particles and then by temporal averaging. The data in the figure reveal, overall, a similar behaviour for the different particle-laden flows. Comparing with the data in figure 3(a) one finds that slip between the fluid and particle velocities is evident close to the wall whereas this is almost zero in the core of the channel. The slip velocity is thus driving the dynamics near the wall, as further discussed below. As shown in the inset of figure 3(d), the particle slip velocity at the wall increases monotonically when increasing the ratio between the volume fraction of large-to-small particles. This is attributed to the fact that larger particles forming the near-wall layer reach larger distances from the wall where they are exposed to higher flow velocity (see Picano et al. Reference Picano, Breugem and Brandt2015; Costa et al. Reference Costa, Picano, Brandt and Breugem2016).

We employ a phase indicator function to calculate the local volume fraction profile across the channel. The indicator function assumes for each computational cell in the domain values of $\unicode[STIX]{x1D6F9}=1$ , $\unicode[STIX]{x1D6F9}=0$ or $0<\unicode[STIX]{x1D6F9}<1$ if the cell is located inside the particle, in the fluid phase or is cut by the interface. Taking average of the phase indicator function in the streamwise and spanwise (homogenous) directions followed by a time averaging, we obtain the particle concentration profile across the channel. We normalise the profile such that its mean value is equal to the total particle concentration. Although the particle centre cannot get closer than a particle radius to the wall, we still have non-zero values of the indicator function close to the wall, and thus non-zero local volume fraction. The wall-normal profile of local mean solid volume fraction $\unicode[STIX]{x1D719}(y)$ is displayed in figure 4(a) for the five suspensions considered here. Note that the horizontal axis is shown in logarithmic scale to ease the comparison in the near-wall region. For all cases the local concentration is characterised by a homogenous particle distribution in the bulk of the flow due to the action of turbulent mixing. A different behaviour is observed in the near-wall region where an inhomogeneous distribution occurs. In particular, we find a local maximum of $\unicode[STIX]{x1D719}(y)$ close to the wall, lower than the value of the bulk concentration, followed by a local minimum. In mono-disperse suspensions, the location of the maximum occurs at a wall distance slightly larger than a particle radius, while the minimum of $\unicode[STIX]{x1D719}$ occurs around one particle diameter away from the wall. This behaviour is attributed to the formation of a near-wall layer induced by the planar wall symmetry and the excluded finite volume of the rigid particles, as noted in Picano et al. (Reference Picano, Breugem and Brandt2015). Bi-disperse suspensions show a progressive modification from the case of mono-disperse suspensions of small particles to that of large particles; in particular the location of the first maximum of $\unicode[STIX]{x1D719}(y)$ indicates that smaller particles show a higher accommodation at the wall. The bi-dispersity however tends to smear out the near-wall volume fraction profile.

Figure 4.

Profile of local volume fraction for the five simulated suspensions: (a) total volume fraction, (b) volume fraction of small and large particles separately, (c) volume fraction of small and large particles normalised by their bulk value.

In figure 4(b) we show, separately, the concentration profile of small and large particles for all the cases. We note that the smaller particles tend always to form a layer, while layering is absent for the larger particles when these are in a bi-disperse suspensions and for lower values of their volume fraction. To better understand the effect of the bi-dispersity on the particle layering we show in figure 4(c) the local volume fraction of small and large particles normalised by their bulk values. It is clear from this figure that bi-dispersity tends to reduce or even destroy the wall layer of the larger particles while promoting the layering of the smaller particles near the wall. Hence for bi-disperse cases the near-wall dynamics tends to be controlled by the smaller particles that concentrate more at the wall. As we will discuss in analogy with the work by Costa et al. (Reference Costa, Picano, Brandt and Breugem2016), the near-wall layering dynamics plays a crucial role in determining the overall suspension drag.

Segregation of small and large particles in different spatial locations is a well-known phenomenon and has been already explored in the seminal experimental study of Bagnold (Reference Bagnold1954). Performing an experiment on the slope of falling sand with mixed sizes, it is observed that the smaller grains tend to migrate towards regions of highest shear while the larger particles accumulate in regions of lower shear. This has been attributed to the proportionality of the disperse phase pressure to the square of the grain diameter. Larger particles will be pushed away from regions of high shear rate by the stronger dispersive pressure (see Schlick et al. Reference Schlick, Isner, Freireich, Fan, Umbanhowar, Ottino and Lueptow2016, for recent theoretical analysis on this aspect). Even though a turbulent flow tends to redistribute the particles and to weaken this effect, we still observe segregation of small/large particles at the wall/centre region of the channel, an observation consistent with the findings by Bagnold. To further support this, we discuss the concentration profile of binary mixture suspensions in laminar flows, $Re_{b}=1000$ , in appendix appendix A.

Here, we also examine the particle number density across the channel as an alternative measure of the particle distribution. In this case, we consider only particle centres and compute the spatial and time average of their wall-normal positions. In figure 5(a), we show the profile of the mean particle number density, $\bar{n}$ , of small and large particles for the five turbulent cases studied. This is computed by counting the mean number of particle centres at each wall-normal station per unit volume. For each case, the peak of the mean number density occurs at a distance to the wall equal to the particle radius. Small particles are predominant close to the wall in cases 25–75 and 50–50. This is not true for the case 75–25 where the number of small and large particles close to the wall is of the same order. This result is in line with that of the local volume fraction reported in the previous figure. In panel (b) we normalise the data of panel (a) by the corresponding bulk value, $\bar{n}/n_{0}$ . Similarly the high traffic of small particles close to the wall is evident.

Figure 5.

(a) Profiles of the mean particle number density, $\bar{n}$ , (b) normalised mean particle number density, $\bar{n}/n_{0}$ , across the channel at $Re_{b}=5600$ .

4.2 Overall drag

The mean velocity profile and the overall drag of a turbulent channel flow are directly connected to the inhomogeneous multi-scale dynamics of the turbulent flow. In wall turbulence, it is common to define a bulk Reynolds number, $Re_{b}$ , representing a dimensionless form of the flow rate and the friction Reynolds number, $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}$ , as a dimensionless measure of the overall drag. Based on well-established empirical correlations, (e.g. Pope Reference Pope2000), it is possible to relate the two numbers for the case of the unladen turbulent plane channel flow,

When a turbulent flow with suspended particles is considered, the relation (4.1) needs to be modified to consider the rheological properties of the suspension. Laminar suspensions are characterised by a monotonic increase of their effective viscosity as a function of the volume fraction of the dispersed phase. The effective viscosity of the suspension in laminar condition has been first predicted in the seminal work by Einstein (Reference Einstein1906) for the dilute regime in the form of a linear correction, $\unicode[STIX]{x1D708}_{e}=\unicode[STIX]{x1D708}(1+2.5\unicode[STIX]{x1D6F7})$ . Later on Batchelor & Green (Reference Batchelor and Green1972) considered pair interactions and derived a second-order correction in the volume fraction valid for semi-dilute suspensions. In the dense regime, however, only empirical fits are available, among others the Eilers fit,

where $\unicode[STIX]{x1D6F7}_{max}\approx 0.6$ is the maximum packing fraction of particles (Stickel & Powell Reference Stickel and Powell2005). Note that Mwasame, Wagner & Beris (Reference Mwasame, Wagner and Beris2016) have recently introduced a weighting function $\unicode[STIX]{x1D6FD}$ , describing the effects of size ratio and volume fraction ratio on the effective viscosity of binary suspensions. Interestingly, this model shows that for concentrations $\unicode[STIX]{x1D6F7}<0.3$ , the effect of particle bi-dispersity on the effective viscosity of the suspension is negligible, i.e.  $\unicode[STIX]{x1D6FD}\approx 1$ . Thus, in our study at $\unicode[STIX]{x1D6F7}=0.2$ , we safely estimate the effective viscosity using (4.2). When inertia dominates the dynamics, deviations from the empirical fits valid in viscous regimes have been reported by several authors, e.g. inertial shear thickening (e.g. Kulkarni & Morris Reference Kulkarni and Morris2008; Yeo & Maxey Reference Yeo and Maxey2011; Picano et al. Reference Picano, Breugem, Mitra and Brandt2013).

The situation becomes even more complicated in turbulent flows laden with particles larger than the smallest hydrodynamic scales. As discussed in Picano et al. (Reference Picano, Breugem and Brandt2015), Prosperetti (Reference Prosperetti2015), the turbulent friction Reynolds number cannot be predicted by only taking into account the effective viscosity of the suspension,

where $Re_{\unicode[STIX]{x1D70F}}^{e,exp}$ is the expected suspension friction Reynolds number which accounts for the suspension dynamics only by considering the suspension effective viscosity. Applying the formula above, the friction Reynolds number pertaining the cases considered here would be $Re^{e,exp}=101.7$ , obtained using $\unicode[STIX]{x1D708}_{e}\simeq 1.9\unicode[STIX]{x1D708}$ . This should be compared with the values of the effective friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{e}=u_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}_{e}$ extracted from direct numerical simulation (DNS) of the present bi-disperse cases and reported in figure 6 where all values are higher than what is predicted from (4.3). In addition, the flow shows a dependence on the particle size which is not accounted for by (4.3). As shown here and in the previous works mentioned above, the presence of finite-sized spherical particles increases the overall drag more than what is predicted considering only the effective viscosity. An interesting exception is provided by the case of oblate disc-like particles, see Ardekani et al. (Reference Ardekani, Costa, Breugem, Picano and Brandt2016).

To explain these observations, a new theoretical framework has been recently proposed by Costa et al. (Reference Costa, Picano, Brandt and Breugem2016) to extend the law of the wall for the turbulent channel flow of finite-sized mono-disperse rigid spheres. The underlying idea is that the formation of a particle layer at the wall creates an additional source of drag. In the formula proposed by these authors, the overall stress is assumed to depend on the Reynolds number, the volume fraction and the particle size that together determine the wall layer thickness (for the details of the derivations we refer the readers to Costa et al. Reference Costa, Picano, Brandt and Breugem2016). The formula reads

where the particle layer thickness $\unicode[STIX]{x1D6FF}_{pw}=C(\unicode[STIX]{x1D6F7}/\unicode[STIX]{x1D6F7}_{max})^{(1/3)}D_{p}$ and $C\approx 1.5$ . This has been found to provide a good fit of data from simulations of particle-laden turbulent channel flow for a reasonable range of volume fractions and Reynolds numbers.

Figure 6.

Effective friction Reynolds number pertaining the different cases $Re_{\unicode[STIX]{x1D70F}}^{e}$ considered here and the predictions by the model of Costa et al. (Reference Costa, Picano, Brandt and Breugem2016).

To extend this formula to bi-disperse suspensions we need to provide an ‘effective’ particle size in order to characterise the wall layer. Examining the local volume fraction profiles in figure 4, we note that the distance between the local maximum and the wall is proportional to the particle diameter for mono-disperse cases. In our simulations the constant of proportionality is obtained equal to $1.37$ . We therefore assume the effective particle size, $D_{p}^{e}$ , of the binary mixtures as the product of the location of the local maximum in the concentration profile with the constant of proportionality, $1.37$ . In table 2 we report the values of the effective particle diameter pertaining each simulation normalised with half the channel height.

Table 2.

Effective particle diameter based on the location of local maximum in the profile of the local concentration.

Using this definition, we note that $D_{p}^{e}$ almost coincides with the smallest particle diameter for the cases 0–100, 25–75 and 50–50; it then increases and reaches the largest particle diameter for the 100–0 case, as expected. In figure 6, we report the effective friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}^{S}$ estimated from (4.4) with $C\simeq 1.33$ . We note that the predicted values are in very good agreement with the DNS data. This analysis also explains the trend for the variation of the overall drag observed in bi-disperse suspensions when increasing the percentage of large particles.

4.3 Turbulent statistics

Next we examine the statistics of the turbulent flow. Figure 7 shows the root mean square (r.m.s.) of the fluid velocity fluctuations for the five particulate cases in inner units, together with the corresponding statistics for the unladen turbulent channel flow at the same bulk Reynolds number, $Re_{b}=5600$ . Comparing the statistics of the particulate and the unladen flow we observe that the turbulence activity reduces in all the particulate flows, with lowered peaks. The mechanisms responsible for this have been discussed in detail in the previous studies by Lashgari et al. (Reference Lashgari, Picano, Breugem and Brandt2014) and Picano et al. (Reference Picano, Breugem and Brandt2015) for mono-disperse suspensions: this reduction is connected to the increasing importance of particle stresses with respect to Reynolds stresses in the momentum transfer. The profiles of the streamwise velocity fluctuations ${u^{\prime }}^{+}$ , displayed in figure 7(a), reveal that the wall-normal location of the peak is shifted toward the channel centreline for the particulate flows. This can be explained by the formation of the particle layers close to the wall that hinder the production of perturbation kinetic energy from the mean flow. The maximum streamwise velocity fluctuation is reduced for all cases and appear to be slightly higher for bi-disperse suspensions with larger percentages of small particles. The opposite behaviour is observed for the cross-stream velocity fluctuations, figure 7(b,c) where the peaks move toward the wall with respect to the unladen flow. For these components, we note the opposite trend as that of the streamwise fluctuations when varying the relative amount of large-to-small particles. We infer that large particles arriving to or departing from the wall layer induce a higher level of fluctuations in the cross-stream directions. Finally, we note that very close to the wall higher fluctuations are present in the particle-laden cases, which is attributed to the relatively large particle slip velocity and the squeezing motion of the fluid between the wall and the particles.

Figure 7.

Profiles of the intensity of the fluid velocity fluctuation components in (a) streamwise, (b) wall-normal and (c) spanwise directions scaled in inner units. The data pertain to the different cases under investigation as indicated in the legend.

The r.m.s. velocity fluctuations of the particle phase are shown in figure 8 in inner units in order to have a direct comparison with those pertaining the fluid phase. First, we observe that the fluctuations do not vanish at the wall, unlike those of the fluid phase with the only exception of the wall-normal component. In the region close to the wall where the first layer of particles is formed the streamwise and cross-stream fluctuations of the case with all large/small particles are the largest/smallest, see figure 8(a–c). This is attributed to the higher/lower slip velocity of the large/small particles in the near-wall region. The opposite behaviour is observed in the core of the channel where small particles are subjected to more agitation due to the turbulent activity. Away from the near-wall region, the level of fluctuations of the particle phase is generally lower than that of the fluid phase.

Figure 8.

Profiles of the intensity of the particle velocity fluctuation components in (a) streamwise, (b) wall-normal and (c) spanwise directions scaled in inner units. The data pertain to the different cases under investigation as indicated in the legend.

Figure 9.

(a) Particle mean square displacement $\langle \unicode[STIX]{x0394}z_{p}^{2}\rangle$ for the different cases under investigation. (b) Dispersion coefficient, $D_{zz}$ , in the spanwise direction versus the relative particle volume fraction.

4.4 Particle dynamics

In this last part we focus on the particle dispersion dynamics by measuring the particle lateral displacement, which results from particle–particle and particle–fluid interactions. In this work we examine only the particle mean square displacement in the spanwise direction to avoid inhomogeneous and mean flow effects that determine the motion in the wall-normal and streamwise directions. We analyse separately the behaviour of small and large particles in the binary suspensions. The mean square displacement in the spanwise direction is defined by $\langle \unicode[STIX]{x0394}z_{p}^{2}\rangle (\unicode[STIX]{x0394}t)=\langle [\boldsymbol{z}_{p}(t+\unicode[STIX]{x0394}t)-\boldsymbol{z}_{p}(t)]^{2}\rangle _{p,t}$ where $z_{p}$ is the vector containing the spanwise position of the particle centres and $\unicode[STIX]{x0394}t$ is the time interval. The ensemble average, $\langle \rangle _{p,t}$ is taken over all the particles and times after a fully developed flow is established. For more details about particle dispersion and diffusion we refer the readers to the works by Da Cunha & Hinch (Reference Da Cunha and Hinch1996) and Sierou & Brady (Reference Sierou and Brady2004).

The particle mean square displacement is shown in figure 9(a) versus time. Note that the mean square displacement, $\langle \unicode[STIX]{x0394}z_{p}^{2}\rangle$ , is normalised by $(2h)^{2}$ whereas the time is expressed in units of $t_{ref}=d_{l}/U_{b}$ . For all the cases, as expected, the particle mean square displacement varies initially quadratically in time, $\langle \unicode[STIX]{x0394}z_{p}^{2}(\unicode[STIX]{x0394}t)\rangle \propto \unicode[STIX]{x0394}t^{2}$ , indicating a high correlation in the particle trajectories at small intervals. Later on, the classical diffusive behaviour takes over: the particle trajectories de-correlate and the mean square displacement varies linearly with time. In the inset of the figure we report the regime behaviour, the linear diffusive part, for the different particle-laden flows.

To better highlight the differences we provide in figure 9(b), the value of the dispersion coefficient, $D_{zz}$ , obtained by fitting the diffusive, long-time regime to a straight line. As appreciated from the figure, a non-monotonic behaviour is observed. Mono-disperse suspensions of either small or large particles show a similar value of the diffusion coefficient. In bi-disperse suspensions, interestingly, the diffusion coefficient of large and small particles is always close, with a slightly higher value for the largest particles. We explain this non-monotonic behaviour by the following arguments. Adding a small amount of bigger particles in a mono-disperse suspension of small particles, as in the 25–75 case, we expect that the disturbances induced by the large particles enhance the particle dispersion. Conversely, adding a small amount of smaller particles to a mono-disperse suspension of large particles, as in the 75–25 case, will reduce the overall dispersion because of the reduced mobility induced by the smaller particles that are positioned among the larger ones. The peculiar picture of the diffusion coefficient for bi-disperse suspensions can be explained considering that these opposite trends need to match when the concentration of the two particles are similar. Finally, we note that the non-monotonic behaviour of the dispersion coefficient is observed for the particle size distribution studied here and may need further investigations for different size distributions.

One interesting aspect of dense bi-disperse suspensions is the probability of collision, which, in turns, depends on the particle-pair relative distribution and the first-order velocity structure functions. For binary mixtures, we report the particle-pair statistics by considering pairs consisting of (i) only two small particles, (ii) only two large particles and (iii) one small and one large particle. The first step is to examine the radial distribution function, $g(r)$ , as a function of the distance between the centres of the particle pairs, $r$ . The radial distribution function is a measure of the non-uniformity of the particle distribution and is obtained by counting the average number of particle pairs whose centres are at distance $r$ with respect to the value of a corresponding random distribution (see for more details Reade & Collins Reference Reade and Collins2000; Gualtieri et al. Reference Gualtieri, Picano, Sardina and Casciola2012). In mathematical form, $g(r)=1/An_{0}dN_{r}/dr$ , where $A=4\unicode[STIX]{x03C0}r^{2}$ is the area of the shell around the reference particle, $N_{r}$ is the number of the particle pairs within a sphere of radius $r$ and $n_{0}=N_{p}(N_{p}-1)/2V$ the density of the particle pairs in the volume $V$ with $N_{p}$ the total number of particles.

Figure 10.

(a) Radial distribution function (RDF) $g(r)$ and (b) RDF at pair contact for the five cases under consideration.

The relative position between the particles in the flow determines part of the collision dynamics, but the full scenario becomes clear when considering also the particle-pair relative velocity in the normal direction. The normal relative velocity between the particle pair as a function of the distance $r$ reads, $\text{d}v_{n}(r)=(\boldsymbol{u}_{i}-\boldsymbol{u}_{j})\boldsymbol{\cdot }\boldsymbol{r}_{i}-\boldsymbol{r}_{j}/|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|$ , where $\boldsymbol{u}_{i}$ and $\boldsymbol{r}_{i}$ are the velocity and position vector of particle $i$ . The multiplication of the ensemble average of the negative part of the normal relative velocity, $\text{d}v_{n}^{-}(r)=\text{d}v_{n}(r)|_{{<}0}$ , and the radial distribution function gives the collision kernel: $\unicode[STIX]{x1D705}(r)=\langle \text{d}v_{n}^{-}(r)\rangle \cdot g(r)$ . Note that the collision kernel is properly defined at the pair contact; however here we use the extended formula as a function of $r$ to understand how the approaching dynamics governs the collisions.

We display the radial distribution function (RDF) pertaining the different bi-disperse cases under investigation in figure 10(a). In this work, the particle-pair statistics are reported only for the region in the middle of the channel, in the range $0.25<y/2h<0.75$ , in order to exclude wall effects. We thus compare the particle dynamics in the homogenous region of the channel. Note that here the average density of particle pairs in this region is used in the calculation of $n_{0}$ . All the profiles are characterised by strong segregation at $r\simeq 2a$ where $a$ is defined by the distance between the particle pairs at contact in each case; i.e. all the profiles start at $r=2a$ corresponding to the contact between the pairs. As $r$ increases $g(r)$ decreases and tends to unity when $r\gg 2a$ denoting a random distribution. As it can be seen in figure 10(b) where the RDF is shown at $r=2a$ , increasing the percentage of large particles generally increases the segregation at contact for all possible pairs. We note that for the mono-disperse cases, the RDF at contact is higher for larger particles. For bi-disperse suspensions, the pairs formed by large–large and small–small particles show higher segregation than mixed pairs. Hence particles tend to cluster more with those of the same size, probably because their dynamics is more similar.

Figure 11.

(a) Collision kernel at contact, (b) particle relative normal velocity and (c) extended collision kernel of the different particle pairs as a function of the distance between the particle surfaces.

The collision kernels for particles at contact are depicted in figure 11(a). The trend is not the same as for the particle-pair segregation, indicating the importance of the relative velocity for a collision to occur. In particular, we note that the normalised probability of collisions is the highest for pairs of large particles, followed by the probability of collisions between particles of different sizes and finally lowest for pairs of only small particles. Since the collision kernel is determined by the RDF times the average approaching velocity at contact, the observed differences are due to the pair dynamics. In figure 11(b) we therefore show the absolute value of the negative normal relative velocity as a function of the distance between particles. As expected, the absolute relative velocities increases monotonically with $r$ , see also Lashgari et al. (Reference Lashgari, Picano, Breugem and Brandt2016); however the rate is different for different particle sizes. Large particles exhibit the highest normal relative velocities at all separation distances while the opposite is true for small particles. This result is not surprising since large particles have more inertia and are expected to collide even when starting at larger separations, and the dynamics at larger separation is dominated by the most energetic large-scale turbulent fluctuations. As mentioned above, the product of the relative approaching velocity and the RDF determines the collision kernel, whose scale-by-scale behaviour is shown in figure 11(c). This quantity reduces decreasing inter-particle distances until it reaches the value at contact shown in the panel (a). Finally we note that the negative (approaching) relative velocity of pair particles at $r\approx 2a$ is small but not negligible. The relative velocities are reduced by the strong lubrication force acting when the distance between the two particles is small. On the other hand, the probability of finding pair particles at contact is high as shown in the plot of the radial distribution function. Thus, on average, a third particle approaching a pair feels the variation in the particle dynamics at $r=4a$ . This explains the change in the slope of the relative velocity and collision kernel around $r=4a$ .