3.1.1 Unladen air flow

As a baseline, we first present the wall-normal profiles for the unladen fluid velocity. These are plotted in inner units in figure 3, showing the expected logarithmic behaviour above $y^{+}\sim 30$ . This is used to determine the friction velocity using a Clauser chart method (Clauser Reference Clauser1956; Wei, Schmidt & McMurtry Reference Wei, Schmidt and McMurtry2005). The profiles for the mean velocity, streamwise and wall-normal r.m.s. fluctuation and Reynolds shear stress are plotted in figure 4 in outer units, i.e. normalized by the channel half-height and the centreline velocity. The agreement with DNS of spanwise-periodic channels at comparable Reynolds numbers (e.g. Moser, Kim & Mansour Reference Moser, Kim and Mansour1999) suggests that the flow in the central part of the channel is not significantly impacted by the finite width of the cross-section.

Figure 4. Wall-normal profiles for the unladen flow at both considered Reynolds numbers, normalized in outer units: (a) mean velocity; (b) streamwise r.m.s. velocity fluctuation; (c) wall-normal r.m.s. velocity fluctuation; (d) Reynolds shear stress.

3.1.2 Particle concentration

Figure 5 displays the mean profiles of normalized particle concentration $C/C_{0}$ for the four particle–laden cases. Here and in the following wall-normal profile plots, data points in the profiles are plotted at the $y/h$ location at the centre of the respective wall-normal bin, each bin having a width of 0.25 mm (about 9 pixels). Data points measured for the HiVF cases along wall-parallel planes are also shown and found to agree closely with the wall-normal imaging results. Both LoVF and HiVF cases display a peak of concentration in the near-wall region, confirming that turbophoresis is active in the present regime. However, in the more dilute cases the peak is mild, and away from the wall the concentration gradually increases towards another local maximum at the centreline. On the other hand, the higher-loading cases display a much stronger peak of concentration near the wall, and the profile is essentially flat in the core region. The peaks appear to be at a finite standoff distance from the wall, which is however hard to quantify precisely.

As mentioned in the Introduction, previous measurements of near-wall segregation in similar regimes are lacking, and a comparison with past numerical simulations is in order. We refer to point-particle DNS studies, which are free from issues associated with turbulence modelling. Among those, several one-way-coupled simulations yielded near-wall concentration peaks two or more orders of magnitude above the channel mean, and mostly contained within the viscous sublayer (Marchioli & Soldati Reference Marchioli and Soldati2002; Marchioli et al. Reference Marchioli, Soldati, Kuerten, Arcen, Taniere, Goldensoph, Squires, Cargnelutti and Portela2008; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a ; Bernardini Reference Bernardini2014). Those results, while insightful, are influenced by the fact that point particles can amass to arbitrary densities. In two-way-coupled simulations, the momentum back-reaction from the particles reduced the near-wall segregation, as did the interparticle collisions; see Li et al. (Reference Li, McLaughlin, Kontomaris and Portela2001), Vreman (Reference Vreman2007) and Nasr et al. (Reference Nasr, Ahmadi and McLaughlin2009). Those authors did show concentrations reaching a minimum adjacent to the near-wall peak and increasing up to a centreline maximum, similarly to our LoVF profiles. However, they found that such reduction of the near-wall peak (and the simultaneous appearance of a centreline maximum) occurred for increasing mass loading, in contrast with the present results. On the other hand, the concentration profiles we observe for LoVF are consistent with the argument of Young & Leeming (Reference Young and Leeming1997) that turbophoresis is driven by the gradient of fluid $V_{rms}$ (figure 4 c). The concentration is maximized in correspondence to the concentration minimum, decays steeply towards the wall and more mildly towards the centreline, following approximately the same trend as the wall-normal gradient of $V_{rms}$ . We remark that the centreline concentration maximum was observed in several, but not all, one-way-coupled simulations, and was found to depend on the flow orientation: for example, Nilsen et al. (Reference Nilsen, Andersson and Zhao2013) found it for downward and no-gravity flow, but not for upward flow.

Figure 5. Mean particle concentration profiles normalized by the global concentration for (a) LoSt cases and (b) HiSt cases. Here and in the following plots, WP indicates data points from wall-parallel measurements, with the horizontal error bars indicating uncertainty on laser sheet position.

3.1.3 Particle velocity

Figure 6 displays mean velocity profiles compared to the unladen air velocity. The data are presented both in outer (figure 6 a,b) and inner (figure 6 c,d) units, normalizing by the unladen velocity scales. In the viscous and buffer layer, the particles travel faster than the unladen air. This is a consequence of fast-moving particles retaining part of their momentum when transported towards the wall by turbulent fluctuations, without being constrained by the no-slip boundary condition. Such behaviour was already highlighted by Kulick et al. (Reference Kulick, Fessler and Eaton1994) and in several later experimental and numerical studies (e.g. Taniere et al. Reference Taniere, Oesterle and Monnier1997; Rouson & Eaton Reference Rouson and Eaton2001; Vreman Reference Vreman2007; Li et al. Reference Li, Wang, Liu, Chen and Zheng2012). Righetti & Romano (Reference Righetti and Romano2004) explicitly commented on an effective slip boundary condition for the particle field. Further away from the wall, in the LoVF cases the particles travel at approximately the same speed as the unladen air, while in the HiVF cases they lag in the logarithmic and buffer layers, recovering to the unladen air velocity in the channel core. A decrease of mean velocity with increasing mass loading was also reported by Kulick et al. (Reference Kulick, Fessler and Eaton1994), although for higher $St^{+}$ . In that case the lag was visible up to the centreline, but this was likely due to the wall roughness (Benson et al. Reference Benson, Tanaka and Eaton2005). In general, we observe less flat velocity profiles than in previous experiments; see, for example, Kulick et al. (Reference Kulick, Fessler and Eaton1994), Paris (Reference Paris2001), Caraman et al. (Reference Caraman, Borée and Simonin2003) and Benson et al. (Reference Benson, Tanaka and Eaton2005). Vreman (Reference Vreman2015) argued that those were again influenced by some wall roughness that enhanced the wall-normal particle velocity fluctuations and in turn flattened the mean velocity profiles.

Figure 6. Profiles of mean streamwise particle velocity for (a,c) LoSt and (b,d) HiSt cases, normalized in outer units (a,b) and inner units (c,d). Unladen fluid profiles are plotted for comparison.

Figure 7 displays profiles of mean wall-normal particle velocity. In a fully developed state, this should be identically zero. This is the case (within error bounds) for the unladen fluid, while the particles do show some residual drift towards the wall. This is likely caused by turbophoresis; the effect is exemplified in the LoVF case where the peak at $y/h\sim 0.2$ approximately corresponds to the maximum of unladen fluid $V_{rms}$ and to the minimum of particle concentration, consistent with the theory of Young & Leeming (Reference Young and Leeming1997) (see also Capecelatro, Desjardins & Fox Reference Capecelatro, Desjardins and Fox2016). Numerical simulations at similar $St^{+}$ indicated that the turbophoretic drift continues to modify the particle field during $\mathit{O}(10^{4})$ viscous time scales, which over the considered range of $Re_{\unicode[STIX]{x1D70F}}$ corresponds to $\mathit{O}(10^{3})$ channel heights (Marchioli et al. Reference Marchioli, Soldati, Kuerten, Arcen, Taniere, Goldensoph, Squires, Cargnelutti and Portela2008; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a ; Bernardini Reference Bernardini2014). While these estimates are influenced by the one-way coupled nature of the modelling, they clearly indicate that the particle field requires a much greater development length than the sole hydrodynamics. However, we also remark that the observed wall-normal mean velocities are about 1 % of the streamwise velocity, and smaller than the r.m.s. fluctuation in the same direction (reported below). Moreover, all statistics show no visible trend over the imaging windows (about $1.7h$ and $3.7h$ in the streamwise direction for the wall-normal and wall-parallel measurements, respectively). Thus, also considering that the particles are expected to have reached terminal velocity much before entering the imaging section, the influence of the partial streamwise development is unlikely to qualitatively impact the reported trends.

Figure 7. Profiles of mean wall-normal particle velocity for (a) LoSt and (b) HiSt cases. Unladen fluid profiles are plotted for comparison.

Figure 8. Profiles of r.m.s. streamwise particle velocity for (a) LoSt and (b) HiSt cases. Unladen fluid profiles are plotted for comparison.

Figure 9. Profiles of r.m.s. wall-normal particle velocity for (a) LoSt and (b) HiSt cases. Unladen fluid profiles are plotted for comparison.

The r.m.s. streamwise fluctuations of the particle velocity are plotted in figure 8. The LoVF cases display profiles similar to those of the unladen flow in the channel core, and significantly more intense fluctuations (up to 20 % higher than the fluid) in the near-wall region, with little differences between LoSt and HiSt. Previous studies have found particles with velocity fluctuations stronger than the carrier fluid in several configurations, including particle–laden jets and homogeneous turbulence (e.g. Hardalupas, Taylor & Whitelaw Reference Hardalupas, Taylor and Whitelaw1989; Petersen et al. Reference Petersen, Baker and Coletti2019). Specifically in channel flows, the observed trend agrees with one-way coupled simulations as reported by Marchioli et al. (Reference Marchioli, Soldati, Kuerten, Arcen, Taniere, Goldensoph, Squires, Cargnelutti and Portela2008) and Nasr et al. (Reference Nasr, Ahmadi and McLaughlin2009) for lower but still turbophoretic Stokes numbers. Following Taniere et al. (Reference Taniere, Oesterle and Monnier1997), the increase in r.m.s. velocities may be interpreted as a consequence of the spread in momentum of particles with different history: the ones arriving to a near-wall interrogation window from more distant locations retain some of their relatively high speed due to inertia; while those coming from a rebound on the wall have lost some of their kinetic energy in the collision. Remarkably, the HiVF cases show a significant increase of particle velocity fluctuations even at $y/h\sim 0.4$ , which is even more dramatic for HiSt. The near-wall peak is about the same as in LoVF, but the cross-section-average r.m.s. fluctuation is substantially augmented for the higher loading. This is in contrast with past two-way and four-way coupled point-particle DNS: considering mass loadings higher than but comparable to the current HiVF cases, Li et al. (Reference Li, McLaughlin, Kontomaris and Portela2001), Vreman (Reference Vreman2007) and Nasr et al. (Reference Nasr, Ahmadi and McLaughlin2009) found a decrease in streamwise r.m.s. fluctuations (although the trend with increasing $\unicode[STIX]{x1D719}_{v}$ reported by Vreman was locally not monotonic at high loadings).

Figure 9 shows profiles of the wall-normal r.m.s. fluctuations of the particle velocities. For LoVF, the particle $V_{rms}$ is lower than the unladen fluid $V_{rms}$ in the channel core, but it remains fairly flat across the channel and largely exceeds the unladen fluid levels for $y/h<0.1$ . The effect of $St^{+}$ in the considered range is minor. Moreover, while the fluctuation level decreases approaching the wall, it does not appear to vanish. This is in contrast with one-way coupled simulations where the particle $V_{rms}$ is consistently lower than the fluid $V_{rms}$ (thus vanishing at the wall; Marchioli et al. Reference Marchioli, Soldati, Kuerten, Arcen, Taniere, Goldensoph, Squires, Cargnelutti and Portela2008), but is consistent with previous experiments with particles of similar $St^{+}$ (Li et al. Reference Li, Wang, Liu, Chen and Zheng2012). The approximately even redistribution of the lateral kinetic energy across the channel cross-section may partly be due to particle inertia, and partly to collisions with the wall (interparticle collision being relatively unlikely at the lower volume fraction). For HiVF, $V_{rms}$ increases more significantly approaching the wall, and the tendency is stronger for HiSt. This behaviour is similar as for $U_{rms}$ , and indicates again that the particle–fluid dynamics has been altered: by the modification of the underlying turbulent flow and/or by the increase in particle–particle/wall–particle collisions.

The wall-normal velocity is tightly related to the particle flux towards the wall, which eventually may lead to particle deposition. The flux can be expressed as the rate of particles per unit area crossing a control plane, $J=(\text{d}N/\text{d}t)/A_{s}$ , where $N$ is the number of particles on either side of the plane, $t$ indicates time and $A_{s}$ is the surface area of the control plane. Normalizing by the global concentration $C_{0}$ yields a characteristic velocity $k=J/C_{0}$ ; taking the control plane at the wall gives the commonly used deposition velocity $k_{d}$ , which in turn can be made non-dimensional with a velocity scale usually taken as the friction velocity, $k_{d}^{+}=k_{d}/u_{\unicode[STIX]{x1D70F}}$ (Liu & Agarwal Reference Liu and Agarwal1974; Young & Leeming Reference Young and Leeming1997; Bernardini Reference Bernardini2014). Here the spanwise direction is assumed homogeneous and the above definitions are adapted to the two-dimensional measurements: the concentration is areal rather than volumic, and wall-parallel lines act as the control planes. Figure 10 shows the non-dimensional characteristic velocity $k^{+}$ as a function of wall distance for the LoSt case (which shows similar trends to the HiSt case). We plot separately the fluxes towards and away from the wall. Because there is no net particle deposition, at the wall both fluxes are in balance (within experimental scatter). Wall collision cannot be directly detected with the present set-up, but the absolute value of $k^{+}$ at the measurement location closest to the wall (i.e. control plane at $y^{+}\sim 4$ for the LoSt case) is taken as a proxy of $k_{d}^{+}$ . For the LoVF case, one retrieves $k_{d}^{+}=\mathit{O}(0.1)$ , in agreement with previous observations (see, for example, the collection of data in Young & Leeming (Reference Young and Leeming1997)). On the other hand, the HiVF case shows a sharp increase of flux in the inner layer and a much higher deposition velocity $k_{d}^{+}=\mathit{O}(1)$ . This is consistent with the high $V_{rms}$ levels reported above, and indicates that the change in particle transport properties at high loading greatly impacts wall collision and (for a non-reflective wall) deposition.

Figure 10. Profiles of characteristic flux velocity $k^{+}$ based on fluxes towards and away from the wall, for (a) LoSt-LoVF and (b) LoSt-HiVF cases.

In figure 11 we present profiles of the cross-correlation between the particle streamwise and wall-normal fluctuations, referred to as particle Reynolds shear stress, along with the unladen fluid counterpart. For LoVF, these are found to follow the trend of the unladen fluid in the channel core up to about $y/h=0.2$ , but visibly exceed those values in the near-wall region. A similar behaviour was reported in the vertical pipe flow of Caraman et al. (Reference Caraman, Borée and Simonin2003), whereas in horizontal flow studies such as Li et al. (Reference Li, Wang, Liu, Chen and Zheng2012) particle Reynolds stresses were above/below the fluid levels in the core/near-wall region. These discrepancies stress once more the consequential differences between configurations, in particular as pertains to the gravity force direction. The HiVF cases show again an earlier departure from the unladen fluid statistics and a more dramatic increase in correlation magnitude.

Figure 11. Profiles of particle Reynolds shear stress for (a) LoSt and (b) HiSt cases. Unladen fluid profiles are plotted for comparison.

To explore this dynamic further, we perform a quadrant analysis in the ( $u,v$ ) plane. Following classic notation utilized in wall turbulence studies, we label events belonging to the four quadrants as Q1 ( $u>,v>0$ ), Q2 ( $u<0,v>0$ ), Q3 ( $u<0,v<0$ ) and Q4 ( $u>,v<0$ ). We report on the LoSt cases, which behave similarly to the HiSt cases. Figure 12(a) shows, for reference, the contributions to the Reynolds stresses for the unladen fluid velocity at the same Reynolds number. This highlights the predominance of the Q2 and Q4 events which contribute to positive turbulence production, with Q4 prevailing over Q2 for $y/h<0.06$ (or $y^{+}<15$ ), and vice versa further from the wall (Kim, Moin & Moser Reference Kim, Moin and Moser1987). The particles (figure 12 b,c) follow a similar trend, but with noteworthy differences. The prevalence of Q4 events in the near-wall region is much more pronounced, which is consistent with sweeps being crucial in the process of trapping the particles near the wall (Marchioli & Soldati Reference Marchioli and Soldati2002). This is in stark contrast with the result of Li et al. (Reference Li, Wang, Liu, Chen and Zheng2012): they found overwhelmingly higher probability of Q2 events near the floor of their horizontal channel, where gravity caused much more frequent wall rebounds. For HiVF, both Q2 and Q4 contributions are similarly enhanced, but the cross-over point is farther from the wall compared to LoVF: the region where particles are swept towards the wall is wider, which corresponds to a more intense turbophoretic drift, a stronger near-wall peak of concentration and a depletion of the centreline peak (see figure 5). For this case also Q3 is remarkably large near the wall, probably a consequence of particles colliding with each other and with the wall (Righetti & Romano Reference Righetti and Romano2004).

Figure 12. Contribution of each quadrant of the ( $u,v$ ) plane to the Reynolds shear stresses for (a) unladen fluid, and inertial particles for (b) the LoSt-LoVF case and (c) the LoSt-HiVF case.

For completeness, we present in figure 13 the r.m.s. of the particle spanwise velocity fluctuations from the wall-parallel measurements. At $y/h=0.2$ and 1, the values are consistent with the fluid $W_{rms}$ in the DNS of Moser et al. (Reference Moser, Kim and Mansour1999) for $Re_{\unicode[STIX]{x1D70F}}=180$ and 395 at the same wall-normal locations, which are expected to be close to the unladen fluid values in the present case. The sharp increase at $y/h=0.11$ for the LoSt-HiVF case indicates again an augmented fluctuation of the particle velocity near the wall with higher loading, at odds with previous two-way coupled simulations (Li et al. Reference Li, McLaughlin, Kontomaris and Portela2001; Nasr et al. Reference Nasr, Ahmadi and McLaughlin2009).

Figure 13. Profile of r.m.s. spanwise particle velocity for LoSt-HiVF and HiSt-HiVF cases, obtained from the wall-parallel measurements expanded in § 3.2.

We conclude this section by considering the skewness of the streamwise velocity fluctuations for the inertial particles, in comparison with the unladen fluid. Figure 14 shows data for the LoSt cases (HiSt cases displaying the same trend). To improve convergence, the particle profiles are binned in four regions, each displaying fairly homogeneous behaviour and roughly corresponding to the inner layer ( $y^{+}<10$ ), buffer layer ( $10<y^{+}<30$ ), log layer ( $30<y^{+}<100$ ) and outer layer ( $100<y^{+}<235$ ). The unladen fluid streamwise fluctuations have positive skewness in the inner layer and part of the buffer layer, and negative elsewhere as expected (e.g. Kim et al. Reference Kim, Moin and Moser1987). On the other hand, the inertial particles show positive skewness across the channel height, irrespective of volume fraction. Considering the flow is in the direction of gravity, this may be due to a tendency of the particles in the channel core to favour the downward side of turbulent eddies, as is known to happen in homogeneous turbulence (Wang & Maxey Reference Wang and Maxey1993). We will return to this point in § 3.2.2.

Figure 14. Profile of the skewness of the particle streamwise velocity fluctuations in the LoSt-LoVF and LoSt-HiVF cases. Unladen fluid profile is plotted for comparison. The vertical dashed lines demarcate the regions over which the particle–laden data are averaged.

3.2.1 Two-point statistics

We use RDFs to describe the scale-by-scale concentration in the area surrounding a generic particle, compared to a uniform distribution (Sundaram & Collins Reference Sundaram and Collins1997). For two-dimensional distributions such as those obtained by planar imaging, this can be written as (e.g. Wood, Hwang & Eaton Reference Wood, Hwang and Eaton2005)

(3.1) $$\begin{eqnarray}g(r)=\frac{\langle N_{r}/A_{r}\rangle }{N/A},\end{eqnarray}$$

where $N_{r}$ represents the number of particles within an annulus of radius $r$ and area $A_{r}$ , and $N$ is the total number of particles within the planar domain of area $A$ . In the presence of clustering, the RDF is expected to increase above unity for decreasing $r$ , and the range over which it remains significantly greater than unity approximately indicates the length scale over which clustering occurs. We compute RDFs by binning particle pairs in equally spaced annuli of radial width 0.5 mm ( $0.03h$ ). An edge-correction strategy is needed for particles near the image boundaries. Omitting annuli that cross the image boundary limits the maximum separation to the radius of the domain-inscribed circle, reducing the number of usable particle pairs with increasing separations and thus affecting the large-scale characterization. We instead mirror the particle field across the image boundaries, so that the same number of annuli is used for each particle location. The maximum separation then equals the full image size, introducing only small biases near the boundaries (Salazar et al. Reference Salazar, De Jong, Cao, Woodward, Meng and Collins2008; de Jong et al. Reference de Jong, Salazar, Woodward, Collins and Meng2010; Petersen et al. Reference Petersen, Baker and Coletti2019). To avoid projection biases at separations below the illuminated volume thickness (Holtzer & Collins Reference Holtzer and Collins2002), we only present $g(r)$ for $r>1.1~\text{mm}$ .

This ‘global’ (i.e. omnidirectional) definition of RDF does not discriminate between different directions of the separation $r$ . We also calculate ‘directional’ RDFs, in which the separations are oriented either streamwise or spanwise. This allows us to characterize the streamwise and spanwise extent of the highly concentrated particle structures. Additionally, we calculate the angular distribution function (ADF; see Gualtieri, Picano & Casciola Reference Gualtieri, Picano and Casciola2009; Nicolai, Jacob & Piva Reference Nicolai, Jacob and Piva2013) which is obtained by binning the planar domain in polar coordinates ( $r,\unicode[STIX]{x1D703}$ ):

(3.2) $$\begin{eqnarray}g(r,\unicode[STIX]{x1D703})=\left.\left\langle \frac{N_{r,\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703})}{A_{r,\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703})}\right\rangle \right/\left(\frac{N}{A}\right).\end{eqnarray}$$

Here $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ correspond to spanwise and streamwise directions, respectively. We use equally spaced annuli of radial width $0.03h$ and divide each of them in 24 azimuthal sectors of area $A$ (in which we count $N$ particles). Streamwise and spanwise homogeneities are leveraged to limit the analysis to one quarter of the $(r,\unicode[STIX]{x1D703})$ circle.

We first consider the wall-parallel plane at the centreplane. Figure 15 shows RDFs (global and directional) for LoSt and HiSt cases. The global RDFs indicate that clustering extends over similar length scales for both cases, but it is significantly more pronounced for LoSt. This is not unexpected since the latter is closer to the condition $St_{\unicode[STIX]{x1D702}}\sim 1$ , which was shown to produce more intense clustering in homogeneous turbulence (Wang & Maxey Reference Wang and Maxey1993; Wood et al. Reference Wood, Hwang and Eaton2005) and at the centreplane of channel flows (Fessler et al. Reference Fessler, Kulick and Eaton1994). The directional RDFs also indicate that the clusters are more elongated in the streamwise than in the spanwise direction. The spanwise RDF remains somewhat above unity throughout the field, indicating that some structure in the particle distribution persists over large scales in that direction. This general picture is confirmed by the ADFs in figure 16, which also show how the particle field becomes more quickly decorrelated for separations in direction $\unicode[STIX]{x1D703}\sim 45^{\circ }$ .

Figure 15. Global and directional RDFs along the centreplane for (a) LoSt-HiVF and (b) HiSt-HiVF. The vertical dashed line indicates the laser sheet thickness.

Figure 16. Angular distribution functions along the centreplane for (a) LoSt-HiVF and (b) HiSt-HiVF.

Figure 17. Two-point correlation of streamwise velocity fluctuations with separations in streamwise and spanwise directions along the centreplane, for (a) LoSt-HiVF and (b) HiSt-HiVF. The vertical dashed line indicates the laser sheet thickness.

We next consider the two-point Eulerian velocity correlations, which provide information on the level of spatial coherence of the particle motion. We follow Fevrier, Simonin & Squires (Reference Fevrier, Simonin and Squires2005), who in turn borrowed the formalism proposed by Sundaram & Collins (Reference Sundaram and Collins1999), and write the general expression for the correlation between the streamwise velocity fluctuations of particles $m$ and $n$ , normalized by their velocity variance:

(3.3) $$\begin{eqnarray}R_{uu}(r)=\frac{\langle u^{(m)}u^{(n)}\mid \boldsymbol{x}=\boldsymbol{x}_{\boldsymbol{p}}^{(\boldsymbol{m})};\boldsymbol{x}+\boldsymbol{r}=\boldsymbol{x}_{\boldsymbol{p}}^{(\boldsymbol{n})}\rangle }{\langle u^{2}\rangle }.\end{eqnarray}$$

Here $\boldsymbol{x}$ is the location within the measurement plane, $\boldsymbol{x}_{\boldsymbol{p}}^{(\boldsymbol{i})}$ is the position of the generic $\boldsymbol{i}$ th particle, $\boldsymbol{r}$ is the separation vector connecting the particle pair ( $m,n$ ) and angle brackets represent ensemble-averaging over all particle pairs. Boldface denotes vectorial quantities. The calculation is implemented with the same processing routine used for the RDFs and ADFs (which contains the information on the particle pair mutual positions). Again, we calculate both ‘directional’ correlations, in which the separation vector is either streamwise or spanwise, and polar correlations, which span the ( $r,\unicode[STIX]{x1D703}$ ) space.

Figure 17 displays the directional velocity correlations evaluated at the centreplane. For both LoSt and HiSt cases, the normalized values do not approach unity for vanishingly small separations. (This is also confirmed by data points for separations smaller than the laser sheet thickness, not shown because inherently less accurate.) This indicates that a significant portion of the particle velocity is not spatially correlated. This is in line with the mesoscopic Eulerian formalism introduced by Fevrier et al. (Reference Fevrier, Simonin and Squires2005), according to which inertial particle motion consists of two components: a contribution from the underlying turbulent velocity field, spatially correlated; and a quasi-Brownian velocity distribution, random and as such spatially uncorrelated. The latter is rooted in the particle inertia, in particular the memory of interactions with distant eddies. This results in different velocities of arbitrarily close particles, possibly enhancing collision rates, and is consistent with the concepts of caustics and sling effect (Wilkinson & Mehlig Reference Wilkinson and Mehlig2005; Bewley, Saw & Bodenschatz Reference Bewley, Saw and Bodenschatz2013; Reeks Reference Reeks2014). The gap between unity and $R_{uu}$ for vanishing separations is a measure of the fraction of random uncorrelated motion (Fevrier et al. Reference Fevrier, Simonin and Squires2005; Vance, Squires & Simonin Reference Vance, Squires and Simonin2006). This framework has been employed in numerous theoretical and numerical studies to analyse and model different particle–laden flows, from turbulent channels (Vance et al. Reference Vance, Squires and Simonin2006) to homogeneous turbulence (Meneguz & Reeks Reference Meneguz and Reeks2011) and planar jets (Masi et al. Reference Masi, Simonin, Riber, Sierra and Gicquel2014). However, experimental observations of Eulerian particle velocity correlations have been rarely reported, Khalitov & Longmire (Reference Khalitov and Longmire2003) and Sahu et al. (Reference Sahu, Hardalupas and Taylor2014) being notable exceptions. Figure 17 indicates that HiSt particles display a larger uncorrelated component of the motion than LoSt, consistent with the mesoscopic Eulerian formalism.

From figure 17 one also observes that more inertial particles show a slower decay of velocity correlation with increasing separation, according to the picture of high- $St$ particles responding to larger turbulent scales. Moreover, the streamwise fluctuations are significantly more correlated in streamwise than spanwise directions. This is consistent with the idea that the correlated particle motion is dictated by the turbulent flow. Indeed, if one defines integral scales of the fluctuating particle velocity based on the separation at which the correlation drops by 50 %, the transverse scale appears to be roughly half the longitudinal one, similar to the expected behaviour of the underlying turbulence. The polar diagrams of $R_{uu}$ in figure 18 confirm this picture, and further suggest that the particle motion is organized in large streamwise-elongated structures, whose half-width is about $0.5h$ . This is consistent with the spatial particle distributions as deduced from the RDFs.

Figure 18. Polar map of streamwise velocity two-point correlation along the centreplane, for (a) LoSt-HiVF and (b) HiSt-HiVF.

Figure 19. (a) Global and directional RDFs and (b) ADF for the LoSt-HiVF case along the near-wall plane.

Figure 20. (a) Two-point correlation of streamwise velocity fluctuations with separations in the near-wall plane. (b) Spatial velocity correlation map for streamwise velocity fluctuations of inertial particles in the near-wall plane.

After considering two-point statistics at the centreplane, we move our attention to the near-wall plane. We focus on the LoSt case, for which more significant preferential concentration is expected. Figure 19 displays global and directional RDFs and ADFs, which indicate how the particles are arranged in elongated streaks, multiple channel heights in length. Indeed, due to the highly anisotropic spatial distribution of the particles in this region, the global RDFs provide limited insight compared to the directional representations. The amplitude of the peak is significantly smaller than at the centreplane, indicating generally weaker clustering. This is consistent with the fact that particles have much larger response times than the near-wall turbulent scales. As mentioned in the Introduction, several authors used point-particle simulations to investigate the near-wall structure of the particle distributions in regimes for which turbophoresis and preferential concentration are intense (McLaughlin Reference McLaughlin1989; Zhang & Ahmadi Reference Zhang and Ahmadi2000; Rouson & Eaton Reference Rouson and Eaton2001; Marchioli & Soldati Reference Marchioli and Soldati2002; Soldati & Marchioli Reference Soldati and Marchioli2009; Sardina et al. Reference Sardina, Schlatter, Brandt, Picano and Casciola2012a ; Bernardini Reference Bernardini2014). They found thin streaks separated by $\mathit{O}(100)$ wall units, which roughly correspond to fluid-phase low-speed streaks in wall-bounded flows (Robinson Reference Robinson1991), and are even longer than the fluid streaks. Experimental observations of particle streaks have been sporadic, and mostly limited to snapshot realizations (Kaftori et al. Reference Kaftori, Hetsroni and Banerjee1995a ; Niño & Garcia Reference Niño and Garcia1996). The present measurements provide quantitative information on such structures: the spanwise RDF shows a minimum at separations of ${\sim}0.3h$ or 70 wall units (which can be interpreted as a measure of the streak width) and recovery to a local maximum at ${\sim}0.75h$ or 175 wall units (a measure of the streak spacing). These values are somewhat larger than those found in numerical studies at similar regimes. Moreover, the RDF amplitude we observe at small separations is much smaller than in computations, as will be confirmed by instantaneous particle distributions shown later. Besides the above-mentioned limitations of the point-particle modelling approach, the differences can be partly attributed to the location and thickness of the observation region. Most numerical studies report on streaks along thin slices within the viscous sublayer, which are challenging to isolate in laboratory experiments. Here the imaged particles are contained in a slab of thickness ${\sim}1.1~\text{mm}$ centred at $y/h\sim 0.11$ , thus in the approximate range $y^{+}=17{-}34$ . The projection through this thickness may significantly influence the apparent concentration in a region of large wall-normal gradients. Future quantitative comparisons with simulations should take into account such finite thickness of the illumination volume.

When compared to the centreplane, the two-point velocity correlations at $y/h=0.11$ show an even stronger uncorrelated component of the motion as shown in figure 20. The uncorrelated velocity component near the wall is expected to increase with increasing particle inertia (Fevrier et al. Reference Fevrier, Simonin and Squires2005), and indeed in the near-wall region the particle response time is much larger than the local time scale of the turbulence (i.e. $St^{+}\gg 1$ ). Besides inertia, interparticle and wall–particle collisions may also contribute to the random particle motion (Vance et al. Reference Vance, Squires and Simonin2006). In the HiVF regime considered, both near-wall concentration and deposition velocity are relatively high (§ 3.1), thus collisions may play a significant role in the observed partitioning between correlated and uncorrelated velocity. The negative lobe of velocity correlation along the spanwise direction indicates that the particles are arranged in a streaky fashion, alternating positive and negative streamwise velocity fluctuations. The longitudinal extent of those features cannot be precisely assessed from the present measurements, but the long tail of the correlation function in the streamwise direction suggests they can extend beyond the field of view. The trends in figure 20 are quantitatively similar to the RDFs and ADF in figure 19, implying that the fluctuations of particle velocity and concentration are simultaneous. We will elaborate on this point in the next subsection.

3.2.2 Domain tessellation

In order to further investigate the instantaneous distribution of the inertial particle positions and velocities, we apply domain tessellation methods along the wall-parallel planes. These have been widely used to study clustering of inertial particles in turbulence (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2012). These approaches should be considered complementary to RDFs, since the latter are strictly two-point quantities, while tessellations are sensitive to the multi-particle arrangement. The simplest method is perhaps box-counting, which consists of dividing the domain into boxes of equal size, counting the particles in each box, and comparing the probability distribution function (p.d.f.) of the number of particles per box against the Poisson distribution expected for randomly distributed particles. This technique provides a simple scalar measure of the amount of clustering and has been fruitfully exploited in experimental studies (Fessler et al. Reference Fessler, Kulick and Eaton1994; Aliseda et al. Reference Aliseda, Cartellier, Hainaux and Lasheras2002). In recent years, the Voronoi tessellation method (Monchaux, Bourgoin & Cartellier Reference Monchaux, Bourgoin and Cartellier2010) has gained broader favour: the domain (in our case the two-dimensional image) is divided into cells associated with individual particles, each cell containing the set of points closer to that particle than to any other. The inverse of the area of each cell equals the local instantaneous concentration, $C=1/A_{cell}$ . The method has been used in several experimental and numerical studies of wall-bounded particle–laden flows (Garcia-Villalba et al. Reference Garcia-Villalba, Kidanemariam and Uhlmann2012; Nilsen et al. Reference Nilsen, Andersson and Zhao2013; Nicolai et al. Reference Nicolai, Jacob and Piva2013; Rabencov et al. Reference Rabencov, Arca and van Hout2014). Compared to the box-counting method, it has the advantage of not requiring an extrinsic/arbitrary length scale (the box size).

Here we adopt the Voronoi tessellation to investigate the particle distribution along the centreplane. Figure 21(a) shows a sample instantaneous realization for the LoSt-HiVF case, with Voronoi cells drawn around each particle. In figure 21(c) the p.d.f. of the cell areas (normalized by its ensemble average) is plotted. As typical of inertial particles clustered by turbulence, the distribution is much wider compared to a random Poisson process, which is well approximated by a $\unicode[STIX]{x1D6E4}$ distribution (Ferenc & Néda Reference Ferenc and Néda2007). The p.d.f. of the Voronoi cells is found to closely follow a log-normal distribution (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010; Petersen et al. Reference Petersen, Baker and Coletti2019), which allows us to characterize the curve by its standard deviation $\unicode[STIX]{x1D70E}_{A}$ . The latter is a metric of the amount of clustering: LoSt and HiSt cases are found to have $\unicode[STIX]{x1D70E}_{A}/\langle A_{cell}\rangle =0.78$ and 0.70, respectively, confirming that the former has stronger tendency to produce clusters.

Figure 21. (a) Voronoi tessellation diagram in a sample centreplane realization, with (b) highlighted clusters (coherent clusters in cyan). (c) Probability distribution function of the Voronoi cell areas $A_{cell}$ along the centreplane (red circles), compared with a $\unicode[STIX]{x1D6E4}$ distribution (blue dashed line); the vertical dashed line indicates the threshold $A_{cell}^{\ast }$ .

The topology and behaviour of clusters of highly concentrated particles are relevant to the interphase coupling, especially in view of significant collective back-reaction of the dispersed phase on the carrier fluid. We therefore analyse individual clusters (coloured in figure 21 b), defined as connected groups of particles whose Voronoi cell areas are smaller than a threshold value $A_{cell}^{\ast }$ (figure 21 c): the latter is taken as the value below which the probability of finding small cell areas is higher than for randomly distributed particles (Monchaux et al. Reference Monchaux, Bourgoin and Cartellier2010). To avoid spurious edge effects, we apply the additional constraint that the area of the cells neighbouring a cluster also be smaller than $A_{cell}^{\ast }$ (Zamansky et al. Reference Zamansky, Coletti, Massot and Mani2016). The sum of the areas of all cells belonging to each cluster is taken as its ‘cluster area’, $A_{C}$ .

Figure 22. The p.d.f.s of the cluster areas for the LoSt-HiVF and HiSt-HiVF cases in the channel centreplane, normalized by (a) the Kolmogorov scale corresponding to the flow velocity and (b) the square of the channel half-height, $h^{2}$ .

Figure 22 shows the probability distribution of cluster areas $P(A_{C})$ along the centreplane for LoSt and HiSt, the p.d.f. in figure 22(a) normalized by the corresponding Kolmogorov length scale and in figure 22(b) normalized by the square of the channel half-height, $h^{2}$ . Consistent with previous experimental studies, typical sizes are $\mathit{O}(10\unicode[STIX]{x1D702})$ (Aliseda et al. Reference Aliseda, Cartellier, Hainaux and Lasheras2002), although such estimates may be affected by the number of particles in the system (Petersen et al. Reference Petersen, Baker and Coletti2019). HiSt particles tend to cluster over larger sets, consistent with their limitation of only responding to the larger scales of turbulence. Above a certain size the probability distributions approach a power-law decay, which is a feature of fractality or geometric self-similarity, indicating a fractal-like formation process due to turbulence (Baker et al. Reference Baker, Frankel, Mani and Coletti2017). This was clearer in the homogeneous turbulence studies of Sumbekova et al. (Reference Sumbekova, Cartellier, Aliseda and Bourgoin2017) and Petersen et al. (Reference Petersen, Baker and Coletti2019), probably due to a combination of limited number of particles in the field of view and limited dynamic spatial range at the present Reynolds numbers. Figure 22(b) shows that normalizing by the channel half-height produces a remarkable collapse of the LoSt and HiSt cases. This may suggest that, while particle clustering at the present $St$ is influenced by small-scale turbulence, the energetic scales of fluid motion also play a major role (as recently argued, for example, by Petersen et al. (Reference Petersen, Baker and Coletti2019)). Moreover, this may indicate that the cluster size is significantly influenced by the channel geometry. Indeed, an a priori estimate of the controlling effect of channel walls on clustering is often deduced from the ratio of the channel height to a characteristic cluster size, $L_{c}=\unicode[STIX]{x1D70F}_{p}^{2}g$ (Capecelatro et al. Reference Capecelatro, Pepiot and Desjardins2014). Here $L_{c}=2.8~\text{mm}$ and $2h/L_{c}=10.6$ , and the value of $L_{c}$ is close to the peak of p.d.f. ( $A_{c}$ ). We remark, however, that the definition of $L_{c}$ is usually adopted in much denser regimes than the present one, being independent of the air-flow characteristics. Further studies with different particle properties and turbulence conditions should discriminate whether $L_{c}$ is an appropriate scale for highly dilute systems.

Following Baker et al. (Reference Baker, Frankel, Mani and Coletti2017), we define ‘coherent clusters’ as those objects large enough to display a scale-invariant topology, i.e. in the range of $P(A_{C})$ that approximates the power-law decay. Smaller objects are considered as randomly occurring groups of particles, not necessarily brought together by the underlying turbulent flow. We conventionally set the cutoff at the respective maxima of $P(A_{C})$ for both LoSt and HiSt, noting that the choice of twice larger cutoffs does not qualitatively change the observed trends. Besides the physical interpretation discussed in Baker et al. (Reference Baker, Frankel, Mani and Coletti2017), this step allows us to discard clusters formed by only a few particles (too small for a meaningful topological description).

Figure 23. The p.d.f.s of (a) the SVD-based aspect ratio and (b) the angle between the primary axis and the vertical for the LoSt-HiVF and HiSt-HiVF cases.

We use the singular value decomposition (SVD) method introduced by Baker et al. (Reference Baker, Frankel, Mani and Coletti2017) to probe the shape and spatial orientation of the coherent clusters. The SVD provides the principal axes and corresponding singular values for a particle set: the primary axis lies along the direction of greatest particle spread from the cluster centroid, the secondary axis being orthogonal to it. The corresponding singular values $s_{1}$ and $s_{2}$ measure the spread along the respective axes. In figure 23(a) the p.d.f. of the aspect ratio $s_{2}/s_{1}$ is plotted for LoSt (differences with HiSt are marginal). The limit values 0 and 1 correspond to particles arranged in a straight line and in a perfect circle, respectively. The distribution is quantitatively similar to that reported by Petersen et al. (Reference Petersen, Baker and Coletti2019) for clusters settling in homogeneous turbulence. The peak ratio between 0.4 and 0.55 reflects a tendency to form somewhat elongated objects. Figure 23(b) illustrates the probability distribution of the primary axis orientation, measured by the angle of the latter with the vertical ( $\unicode[STIX]{x1D703}_{g}$ ). The peak at $\unicode[STIX]{x1D703}_{g}=0$ indicates a tendency of the (coherent) clusters to be aligned with gravity (and thus the direction of motion). This behaviour was also reported in homogeneous turbulence studies (Baker et al. Reference Baker, Frankel, Mani and Coletti2017; Petersen et al. Reference Petersen, Baker and Coletti2019) and is consistent with the directional RDFs and ADFs presented above.

Figure 24. The p.d.f.s of normalized in-cluster (red squares) and global (black circles) concentrations for (a) LoSt-HiVF and (b) HiSt-HiVF.

The local particle concentration within clusters can be significantly higher compared to the global value $C_{0}$ (Baker et al. Reference Baker, Frankel, Mani and Coletti2017). Figure 24 shows p.d.f.s of $C/C_{0}$ associated with the particles within coherent clusters, compared to the unconditional distribution: the peaks in the p.d.f. for the clustered particles are about four times higher for HiSt and almost five times higher for LoSt. Besides reaffirming that the latter case displays more intense clustering, these plots indicate how the in-cluster concentration can be substantial, such that two-way-coupling (and possibly four-way-coupling) effects may be at play.

Figure 25. (a) The same instantaneous realization as in figure 21, with the Voronoi cell colour-coded by the local particle streamwise velocity. (b) Joint p.d.f. of streamwise velocity and concentration at the centreplane for the LoSt-HiVF case. The dashed lines indicate the velocity and concentration values averaged along the centreplane. The black contour line indicates the p.d.f. level at 0.005.

Figure 26. The p.d.f.s of normalized in-cluster (red squares) and global (black circles) velocities for (a) LoSt-HiVF and (b) HiSt-HiVF in the channel centreplane.

The travelling velocity of the clustered particles is also important to the transport process. Figure 25(a) depicts the same instantaneous realization as in figure 21, with the Voronoi cells now colour-coded by the streamwise velocity of the respective particles. The more concentrated regions appear associated with higher velocities, as confirmed by the joint p.d.f. for LoSt-HiVF (figure 25 b): the local particle concentration and streamwise velocity are positively correlated. For a more quantitative account, shown in figure 26 are the p.d.f.s of the streamwise velocity for particles belonging to coherent clusters along the centreplane, as well as for all particles in the field of view. The clustered particles travel downward measurably faster than the generic particles. The explanations may be two-fold: on the one hand, particles may be favouring the downwash side of turbulent eddies, according to the picture of preferential sweeping originally proposed by Maxey (Reference Maxey1987) and later demonstrated by the simulations of Wang & Maxey (Reference Wang and Maxey1993) and recently by the experiments of Petersen et al. (Reference Petersen, Baker and Coletti2019); on the other hand, the highly concentrated clusters may be exerting a collective drag force on the fluid, in turn enhancing their vertical velocity as shown in the numerical simulations of Bosse, Kleiser & Meiburg (Reference Bosse, Kleiser and Meiburg2006) and Frankel et al. (Reference Frankel, Pouransari, Coletti and Mani2016).

In principle, the Voronoi method may also be used to identify the highly concentrated structures near the wall. However, figure 27(a) shows how the p.d.f. of the Voronoi cell areas measured along the near-wall plane is in fact narrower than for randomly distributed particles. This may be an artifact due to the relatively high concentration near the wall: particles very close to each other might be identified as one, reducing the probability of detecting small cell areas. Alternatively, the actual topology of the particle field, expected to be organized in streaks, could result in a ‘crystallized’ pattern with a relatively regular arrangement of cells (and thus a narrow p.d.f. of their area). Either way, the Voronoi tessellation method (in its standard form) does not appear as a suitable tool to study clustering in the present near-wall particle fields. We therefore resort to the box-counting approach: we tessellate the domain with square boxes of size 60 wall units or ${\sim}0.18h$ , and in figure 27(b) we plot the p.d.f. of the concentration in each box $C_{box}$ , comparing it with the Poisson distribution expected for randomly located particles. The relatively broad distribution indicates that the particles are indeed clustered over scales of the order of the box size. The choice of the latter is informed by the width of the particle streaks as estimated from the RDF analysis in the previous subsection, although it is verified that varying it by a factor of two yields similar conclusions. The box index $BI=(\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70E}_{Poisson})/\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70E}$ indicate mean and standard deviation of the distribution, respectively, and $\unicode[STIX]{x1D70E}_{Poisson}$ is the standard deviation of the Poisson distribution) is a comparative measure of clustering. Near the wall we find $BI=0.05$ , while $BI=0.2$ at the centreplane. The latter (consistent with the results of Fessler et al. (Reference Fessler, Kulick and Eaton1994)) confirms the indication from the RDF analysis that clustering in the channel core is significantly more intense.

Figure 27. (a) The p.d.f. of the Voronoi cell areas $A_{cell}$ along the near-wall plane (red circles), compared with a $\unicode[STIX]{x1D6E4}$ distribution (blue dashed line). (b) The p.d.f. of particle concentration in square boxes of size $0.18h$ used to tessellate the near-wall plane (red circles) compared to a Poisson distribution (blue dashed line).

Figure 28(a) shows an instantaneous realization of the near-wall particle field, colour-coded with the mean particle velocity in each square box of the tessellation. The visual impression of elongated slow-velocity streaks is in line with the RDF results. In figure 28(b) we plot a joint p.d.f. of local particle concentration and streamwise velocity, based on values averaged in each box. The apparent negative correlation contrasts with the centreplane trend in figure 25(b), indicating a tendency of the highly concentrated particles to travel slower than the average. Combined with the RDFs, ADFs and two-point correlations reported above, this confirms the picture of particles accumulating in slow-moving streaks.

Figure 28. (a) Sample realization at the near-wall plane, tessellated by $0.18h\times 0.18h$ boxes and colour-coded by the mean local streamwise particle velocity within the boxes. (b) Joint p.d.f. of streamwise velocity and box-based concentration at the near-wall plane for the LoSt-HiVF case. The dashed lines indicate the velocity and concentration averaged along the near-wall plane. The black contour line indicates the p.d.f. level at 0.005.