4.1.1 Theoretical framework for particle relative velocities

As in Part 1, we define the relative position and velocity as $\boldsymbol{r}^{p}(t)$ and $\boldsymbol{w}^{p}(t)$ , respectively, and ${\bf\Delta}\boldsymbol{u}(\boldsymbol{r}^{p}(t),t)$ as the difference in fluid velocities at the particle locations. By subtracting (2.3) for two particles at a given instant in time, we obtain the following equation for the relative position and velocity of the two particles

Notice that the gravity vector precisely cancels out of this equation, making it appear as though gravity does not influence the relative motion of the particle pair; however, this is not true, as gravity has an implicit effect on the statistics of ${\bf\Delta}\boldsymbol{u}(\boldsymbol{r}^{p}(t),t)$ , and this substantially modifies the relative position and velocity statistics of the particles (as will be shown below).

Following the nomenclature in Part 1, we define the second-order particle relative velocity structure function tensor as

where $\langle \cdot \rangle _{\boldsymbol{r}}$ denotes an ensemble average conditioned on $\boldsymbol{r}^{p}(t)=\boldsymbol{r}$ . We now use the formal solution of (4.1) to construct the exact expression for $\boldsymbol{S}_{2}^{p}(\boldsymbol{r})$ . This will allow us to explain the expected trends in the structure functions with inertia and gravity. The exact expression for $\boldsymbol{S}_{2}^{p}(\boldsymbol{r})$ is given by (e.g. see Pan & Padoan Reference Pan and Padoan2010)

where $\hat{Z}$ denotes a variable $Z$ normalized by Kolmogorov scales.

We first consider (4.3) in the absence of gravity. From (4.3), we observe that particle relative velocities are affected by the fluid velocity difference ${\bf\Delta}\boldsymbol{u}(\boldsymbol{r}^{p}(t),t)$ along their trajectories. For small values of $St$ , the high damping of the exponential term in (4.3) causes contributions to the integral from the particle history to be negligible, and hence the particle relative velocities in the equation become equivalent to the instantaneous fluid velocity difference at the current particle position, reflecting the fact that preferential sampling is the dominant mechanism affecting relative particle motion. As shown in Part 1, preferential sampling causes inertial particles to experience larger fluid velocity differences than those of fluid particles in directions parallel to the particle separation vector and smaller fluid velocity differences in directions perpendicular to the particle separation vector. As $St$ increases, in the absence of gravity, the exponential damping diminishes and the influence of the particle history becomes increasingly important. At higher $St$ , particles retain a memory of their interactions with the turbulence, and it is the fluid velocities at larger separations along their path histories that dominate the contribution to the particle velocity dynamics, leading to particle relative velocities that are larger than the local fluid velocity difference. This phenomenon has been referred to as ‘caustics’ (e.g. see Wilkinson & Mehlig Reference Wilkinson and Mehlig2005; Wilkinson et al. Reference Wilkinson, Mehlig and Bezuglyy2006). Refer to Bragg & Collins (Reference Bragg and Collins2014b ) for a more complete theoretical description of this phenomenon, and to Part 1 for supporting DNS evidence.

We now introduce gravity and discuss its effect on the relative velocities. Recall in § 3.1 that gravity generally reduces the degree of preferential sampling. With gravity, we therefore expect the relative velocities of small- $St$ particles to be affected less by preferential sampling, and therefore to have dynamics closer to that of the fluid particles. At higher values of $St$ , gravity alters the inertial particle relative velocities through its influence on the path-history mechanism. We observed in § 3.1 that gravity reduces the correlation time scales of the flow along particle trajectories. This implies that gravity causes the fluid velocity differences in (4.3) to become decorrelated more rapidly, and thus it reduces the correlation radius over which the relative velocities are influenced by their path-history interactions with the fluid. We therefore expect the particle relative velocities to be reduced by gravity. A similar explanation is given in Fouxon et al. (Reference Fouxon, Park, Harduf and Lee2015), where the authors also argue that path-history effects are reduced in the presence of gravity.

In addition to the magnitude changes caused by gravity, its directional nature causes the symmetry of the particle field to become anisotropic. We define the mean inward, parallel and perpendicular relative velocity statistics as

where $w_{\Vert }$ and $w_{\bot }$ are the longitudinal (i.e. parallel to the particle separation vector) and transverse (i.e. perpendicular to the particle separation vector) components of the relative velocity, respectively, and $p(w_{\Vert }\mid \boldsymbol{r})$ and $p(w_{\bot }\mid \boldsymbol{r})$ are their respective probability density functions conditioned on the separation vector $\boldsymbol{r}$ . In the absence of gravity, $S_{2\Vert }^{p}$ , $S_{2\bot }^{p}$ , and $S_{-\Vert }^{p}$ are functions of the separation distance $r\equiv |\boldsymbol{r}|$ only. With gravity, these statistics become functions of the angle between the separation vector $\boldsymbol{r}$ and the gravity vector $\boldsymbol{\mathfrak{g}}$ , here denoted as ${\it\theta}$ . A convenient way to express this dependence is to expand these functions in a spherical harmonic series

where $C_{2\ell }^{0}(r)$ , $D_{2\ell }^{0}(r)$ and $E_{2\ell }^{0}(r)$ , are the spherical harmonic coefficients, and $Y_{2\ell }^{0}$ are the spherical harmonic functions. In discussing the results below, we will quantify the degree of anisotropy introduced by gravity by analysing the relative magnitudes of the spherical harmonic coefficients. Similar expansions will be used to describe the anisotropic radial distribution function that results from gravity.

4.1.2 Relative velocity results

We now analyse the results from the DNS using the decompositions described in § 4.1.1. To study how gravity affects the magnitude of the relative velocity, we first consider the zeroth-order coefficients, $C_{0}^{0}(r)\equiv S_{-\Vert }^{p}(r)$ , $D_{0}^{0}(r)\equiv S_{2\Vert }^{p}(r)$ and $E_{0}^{0}(r)\equiv S_{2\bot }^{p}(r)$ , where we have taken $Y_{0}^{0}=1$ . In effect, these coefficients represent the spherical average over the anisotropic particle field.

Figure 10 shows the probability density function (PDF) for the parallel component of the relative velocity for $0\leqslant r/{\it\eta}\leqslant 2$ , at different Stokes numbers and a fixed Reynolds number of 398. The panel on the left-hand side shows the case of no gravity and the panel on the right-hand side the case of $Fr=0.052$ . It is immediately apparent that gravity has a profound effect on the PDFs, causing a dramatic reduction in the tails of the distribution, particularly for particles with larger Stokes numbers. This is consistent with the explanation given in § 4.1.1, which argued that gravity suppresses path-history effects, thereby reducing the relative velocities.

Figure 10.

PDFs of the radial relative velocities without gravity (a) and with gravity ( $Fr=0.052$ ) (b) at $R_{{\it\lambda}}=398$ and $0\leqslant r/{\it\eta}\leqslant 2$ . The solid line denotes data for fluid ( $St=0$ ) particles.

We next consider the relative velocity variances as a function of the separation distance. Figure 11 shows the spherically averaged quantities $S_{2\Vert }^{p}(r)$ and $S_{2\bot }^{p}(r)$ . For $St\geqslant 1$ (corresponding to $Sv\geqslant 19.3$ ), we observe that gravity strongly decreases the relative velocities, by orders of magnitude in some cases. The explanation is that gravity reduces the effect of the path-history interactions, causing a reduction in the relative velocities (cf. § 4.1.1).

Figure 11.

The variance of the particle relative velocities parallel to the separation vector (a) and perpendicular to the separation vector (b) for $R_{{\it\lambda}}=398$ , plotted as a function of $r/{\it\eta}$ for different $St$ . The thin solid lines indicate data with gravity ( $Fr=0.052$ ), the dashed lines indicate data without gravity, and the thick dotted line indicates fluid ( $St=0$ ) particles. The Stokes numbers are indicated by the line labels, and the fluid velocity scalings are indicated by thick solid lines. The main plots show $St=0$ and $St\geqslant 1$ , while the insets show $St=0$ and $St=0.3$ .

Data at $St=0.3$ are shown in the insets in figure 11. For $St=0.3$ and very small separations ( $r/{\it\eta}\lesssim 1$ ), the relative velocities show evidence of path-history effects (see also Part 1), causing the relative velocities to exceed those of $St=0$ particles in both the parallel and perpendicular directions. However, at larger separations ( $1\lesssim r/{\it\eta}\lesssim 10$ ), the perpendicular relative velocities become less than those of $St=0$ particles, as expected, as a result of preferential sampling. We also see that gravity causes the relative velocities to approach those of the underlying fluid, since it decreases preferential sampling effects. These trends are consistent with the discussion in § 4.1.1. (Note that since particles with $St<0.3$ have only weak gravitational forces ( $Sv<5.79$ ), the effect of gravity on the relative velocities is less apparent and data from these particle classes are therefore not shown.)

The decreased influence of path-history interactions is also apparent in the scaling of the relative velocity variances in figure 11. In the dissipation range, the fluid relative velocity variances are proportional to $r^{2}$ . In the absence of gravity, the relative velocities for $St\geqslant 1$ do not exhibit $r^{2}$ -scaling over any portion of the dissipation range, as was noted in Part 1. However, with gravity, we see a clearer $r^{2}$ -scaling extending much deeper into the dissipation range for $1\lesssim r/{\it\eta}\lesssim 10$ . Compared to the case without gravity, one has to go to smaller separations to observe deviations from $r^{2}$ scaling, as gravity reduces the path-history effect responsible for the formation of caustics.

Of particular interest to the collision kernel is the mean inward velocity (cf. § 4.3). In general, the mean inward velocity is more difficult to analyse theoretically than the relative velocity variance, but qualitatively we expect both statistics to follow the same trends. In figure 12, we show $S_{-\Vert }^{p}$ at $R_{{\it\lambda}}=398$ , both with and without gravity. We see that the mean inward velocities, like the variances, decrease with the addition of gravity at large $St$ due to the reduced influence of the path-history mechanism and at small $St$ due to the reduced influence of the preferential sampling mechanism. Interestingly, with gravity, the mean inward relative velocities (in contrast to the relative velocity variances) follow the local fluid scaling almost perfectly throughout the entire dissipation range. This result was also noted in Bec et al. (Reference Bec, Homann and Ray2014). One plausible explanation is that since the mean inward velocity is a lower-order statistic than the velocity variance, it is less affected by path-history interactions and is more affected by the local fluid turbulence, causing the scaling exponents to increase (see Part 1).

Figure 12.

The mean inward particle relative velocity for $R_{{\it\lambda}}=398$ , plotted as a function of $r/{\it\eta}$ for different $St$ . The thin solid lines indicate data with gravity ( $Fr=0.052$ ), the dashed lines indicate data without gravity and the thick dotted line indicates fluid ( $St=0$ ) particles. The Stokes numbers are indicated by the line labels and the fluid velocity scalings are shown by thick solid lines. The main plot shows $St=0$ and $St\geqslant 1$ , while the inset shows $St=0$ and $St=0.3$ .

Figure 13.

Filled contours of $S_{-\Vert }^{p}(r/{\it\eta})/u_{{\it\eta}}$ (a) and $S_{2\Vert }^{p}(r/{\it\eta})/u_{{\it\eta}}^{2}$ (b) evaluated at $r/{\it\eta}=0.25$ and $R_{{\it\lambda}}=227$ for different values of $St$ and $Sv$ . The contours are logarithmically scaled, and the colour bar labels indicate the exponents of the decade. The diagonal lines denote three different values of $Fr$ , corresponding to conditions representative of stratiform clouds ( $Fr=0.01$ ), cumulus clouds ( $Fr=0.05$ ) and cumulonimbus clouds ( $Fr=0.3$ ).

Figure 14.

(a) $S_{-\Vert }^{p}/u_{{\it\eta}}$ , (b) $S_{2\Vert }^{p}/u_{{\it\eta}}^{2}$ and (c) $S_{2\bot }^{p}/u_{{\it\eta}}^{2}$ , plotted as a function of $R_{{\it\lambda}}$ for various $St$ . The relative velocities are shown at $r/{\it\eta}=0.25$ and the quantities at a given Reynolds number are divided by their corresponding value at $R_{{\it\lambda}}=90$ . Shaded contours denote 95 % confidence intervals. Note that (a) has a much smaller ordinate scale than (b) and (c).

To further verify our arguments in § 4.1.1, it is helpful to decouple the effects of gravity and inertia by varying each independently. We do so in figure 13, where we show $S_{-\Vert }^{p}$ and $S_{2\Vert }^{p}$ for various $St$ and $Sv$ . While the results at high $Sv$ are likely artificially affected by the periodic boundary conditions (see appendix A), we can still use these data to discuss the qualitative trends in the relative velocities at different values of $St$ and $Sv$ . We see that $S_{-\Vert }^{p}$ and $S_{2\Vert }^{p}$ have similar qualitative trends. For $Sv\lesssim 10$ , both quantities increase with increasing $St$ , either due to preferential sampling effects (at low $St$ ) or path-history effects (at high $St$ ). The relative velocities also decrease with increasing $Sv$ , since gravity causes both effects to be less significant, as discussed in § 4.1.1. Finally, we observe that the relative velocities are the smallest for the largest values of $Sv$ and $St\sim 1$ . At smaller $St$ , preferential concentration effects are more significant, while at larger $St$ , path-history effects are more significant. Both effects lead to an increase in the relative velocities. An implication of these results for clouds is that for a given value of $St$ , droplets in stratiform clouds ( $Fr=0.01$ ) will generally have smaller relative velocities than droplets in cumulonimbus clouds ( $Fr=0.3$ ).

We have thus far examined and explained relative velocity statistics for fixed Reynolds numbers. We now consider how these statistics are affected by changes in $R_{{\it\lambda}}$ . In figure 14, we show $S_{-\Vert }^{p}/u_{{\it\eta}}$ , $S_{2\Vert }^{p}/u_{{\it\eta}}^{2}$ and $S_{2\bot }^{p}/u_{{\it\eta}}^{2}$ . In Part 1, we noted that $S_{2\Vert }^{p}/u_{{\it\eta}}^{2}$ increases weakly with increasing $R_{{\it\lambda}}$ in the absence of gravity for $0.3\lesssim St\lesssim 1$ , and attributed this trend to the increased scale separation with increasing $R_{{\it\lambda}}$ , which causes a few inertial particles to be affected by their memory of increasingly energetic turbulence. (We also suggested that increased intermittency effects at higher Reynolds numbers could contribute to the observed trends.) While we expected the increased scale separation of the turbulence to also increase the relative velocities for higher- $St$ particles, we instead found that the relative velocities decreased with increasing $R_{{\it\lambda}}$ for $1\lesssim St\lesssim 3$ . We argued that this trend was caused by a corresponding decrease in the rotation time scales $T_{\unicode[STIX]{x1D619}\unicode[STIX]{x1D619}}^{p}/{\it\tau}_{{\it\eta}}$ , which in turn reduced the influence of path-history effects and decreased the relative velocities. However, with gravity, we find that $T_{\unicode[STIX]{x1D619}\unicode[STIX]{x1D619}}^{p}/{\it\tau}_{{\it\eta}}$ is generally independent of $R_{{\it\lambda}}$ (see § 3.1), and thus we expect that this allows the increased scale separation of the turbulence to cause $S_{2\Vert }^{p}/u_{{\it\eta}}^{2}$ to uniformly increase with increasing $R_{{\it\lambda}}$ for $St\gtrsim 0.3$ . Our results in figure 14(b) confirm this expectation. We also see that $S_{-\Vert }^{p}/u_{{\it\eta}}$ is generally unaffected by changes in $R_{{\it\lambda}}$ (figure 14 a), since it is less influenced by the relatively infrequent occurrences of the effects discussed above. We are also interested in the effect of the Reynolds number on the perpendicular relative velocities, since these statistics will also affect the degree of particle clustering (see § 4.2). From figure 14(c), we see that the trends with $R_{{\it\lambda}}$ in the perpendicular direction are identical to those in the parallel direction.

Figure 15.

$S_{2\bot }^{p}/S_{2\Vert }^{p}$ , evaluated at $r/{\it\eta}=0.25$ , and plotted as a function of $St$ for different values of $R_{{\it\lambda}}$ . Open symbols denote the case with gravity ( $Fr=0.052$ ) and the filled symbols denote the case without gravity.

In Part 1, we found that without gravity, the parallel and perpendicular relative velocities became equivalent at small separations for $St\gtrsim 0.3$ . The physical explanation is that the path-history contribution to their relative velocities decreases the coherence of the pair motion, and in the ballistic limit where the pairs move independently of each other, the parallel and perpendicular components are equal. However, with gravity, path-history effects are weaker, and thus the parallel and perpendicular velocities may not be the same in this regime.

We compare these two quantities in figure 15 by plotting $S_{2\bot }^{p}/S_{2\Vert }^{p}$ at $r/{\it\eta}=0.25$ . We see that both with and without gravity, the ratio $S_{2\bot }^{p}/S_{2\Vert }^{p}$ approaches 2 at low Stokes numbers, the value for fluid particles (e.g. see Pope Reference Pope2000). For $St\gtrsim 0.3$ without gravity, $S_{2\bot }^{p}$ and $S_{2\Vert }^{p}$ are equivalent, as expected. At high values of $St$ with gravity, however, the parallel and perpendicular components are not equivalent, since path-history effects are weakened by gravity. As $R_{{\it\lambda}}$ increases, the particle relative velocities are affected by increasingly energetic turbulence along their path histories. As a result, the relative velocities are larger and the particles move more ballistically, causing the ratio $S_{2\bot }^{p}/S_{2\Vert }^{p}$ to decrease with increasing $R_{{\it\lambda}}$ .

We now examine the effects of Reynolds number on the scaling of the relative velocities. As in Part 1, we compute the scaling exponents of the mean inward relative velocity ( ${\it\zeta}_{\Vert }^{-}$ ) and the relative velocity variance ( ${\it\zeta}_{\Vert }^{2}$ ) by performing linear least-squares power-law fits over separations $0.75\leqslant r/{\it\eta}\leqslant 2.75$ . While the results in figure 11 suggest that the power-law exponent may vary considerably over this range, we are forced to use this relatively large range to gather sufficient statistics at the smaller separations. Thus, while these scaling exponents allow us to assess the trends with $R_{{\it\lambda}}$ , they only provide a qualitative understanding, as the scaling varies considerably throughout the dissipation range and this variation is not completely captured by this analysis.

The scaling exponents are shown in figure 16. The trends at a given value of $R_{{\it\lambda}}$ are in agreement with our observations from figures 11 and 12, which are explained above. We also see that with gravity, the scaling exponents tend to decrease with increasing $R_{{\it\lambda}}$ , since the particles are more influenced by their memory of increasingly energetic turbulence and therefore tend to move ballistically, as noted above.

Figure 16.

The scaling exponents of the mean inward relative velocities (a) and relative velocity variances (b) plotted as a function of $St$ for different values of $R_{{\it\lambda}}$ . The open symbols denote the case with gravity ( $Fr=0.052$ ) and the filled symbols denote the case without gravity. The scaling exponents are computed using linear least-squares regression over the range $0.75\leqslant r/{\it\eta}\leqslant 2.75$ .

Angular distributions of $S_{-\Vert }^{p}(\boldsymbol{r})/S_{-\Vert }^{p}(r)$ are presented in figure 17 for a range of Stokes numbers, $R_{{\it\lambda}}=398$ , and $r<{\it\eta}$ . The anisotropy of the distribution is reflected in the colour variation over the surface of the sphere. We see immediately that the asymmetry in the relative velocities follows opposite trends at small and large $St$ . That is, at small $St$ , the relative velocities are largest for particles separated in the vertical direction, whereas for $St\gtrsim 1$ , the opposite is true. The explanation is as follows. At small separations and $St\lesssim 0.3$ , particle pairs that are separated vertically will have parallel relative velocities that are proportional to $\unicode[STIX]{x1D61A}_{33}(\boldsymbol{x}^{p}(t),t)$ , the longitudinal velocity gradient in the vertical direction as sampled by inertial particles (see § 3.1). Particle pairs that are separated along the horizontal direction, however, will have parallel relative velocities that are proportional to $\unicode[STIX]{x1D61A}_{11}(\boldsymbol{x}^{p}(t),t)$ , the longitudinal velocity gradient in the horizontal direction. In § 3.1, we observed that particles tend to preferentially sample flow where the vertical velocity gradients are larger and the horizontal velocity gradients are smaller. As a result, the relative velocities are expected to be larger for particles that are separated in the vertical direction and smaller for particles that are separated in the horizontal directions. At large $St$ , however, the relative velocities are smallest for particles that are separated vertically. The physical explanation is that the correlation time scales of the fluid velocity gradients along this direction are the smallest (see § 3.1) and thus the particles have a weaker response to vertical fluid velocity fluctuations and correspondingly lower particle relative velocities along this direction.

Figure 17.

The directionally dependent mean inward relative velocity $S_{-\Vert }^{p}(\boldsymbol{r})$ , normalized by the spherically averaged mean inward relative velocity $S_{-\Vert }^{p}(r)$ , shown on a unit sphere for $R_{{\it\lambda}}=398$ and $r<{\it\eta}$ with gravity ( $Fr=0.052$ ). The different columns correspond to different values of $St$ . (a) Shows the projection where gravity is directed into the page and (b) shows the projection where gravity is directed downward.

Figure 18.

The second (a) and fourth (b) spherical harmonic coefficients of $S_{-\Vert }^{p}(\boldsymbol{r})$ , normalized by the zeroth spherical harmonic coefficient, plotted as a function of $r/{\it\eta}$ for different $St$ and $R_{{\it\lambda}}$ with gravity ( $Fr=0.052$ ). The different values of $St$ considered ( $0.3$ , $0.5$ , $1$ , $3$ ) are shown in black, red, blue and green, respectively, and the Stokes numbers are indicated by the line labels.

Next, we consider the dependence of the anisotropy on the separation distance by plotting the spherical harmonic coefficients of order 2 and 4 as a function of $r$ in figure 18. (Coefficients above order 4 are too small to measure accurately and thus are not shown.) We see that the anisotropy in the relative velocities generally decreases with increasing separation. The physical explanation is that with increasing separation distance, the particle motions are increasingly dependent on larger eddies, and the relative velocities induced by these isotropic eddies are increasingly energetic relative to the anisotropic velocities induced by gravity.

For small values of $St$ , $C_{2}^{0}$ and $C_{4}^{0}$ are positive, indicating that the particle relative velocities are strongest for particles that are separated vertically. The opposite is true at large $St$ . These observations are in agreement with the trends shown in figure 17. We also see that $C_{2}^{0}$ and $C_{4}^{0}$ tend to become more isotropic at high $St$ and less isotropic at low $St$ as $R_{{\it\lambda}}$ increases. While the physical explanation for the trend at low $St$ is unclear, at high $St$ , we expect that this increase in isotropy is linked to the increase in the relative velocities with increasing $R_{{\it\lambda}}$ . That is, as the overall relative velocities increase, the anisotropic velocities induced by gravity will be comparatively weaker, and thus the particle relative velocities will be more isotropic.

4.2.1 Theoretical framework for particle clustering

We use the angular distribution function $g(\boldsymbol{r})$ (hereafter ‘ADF,’ see Gualtieri, Picano & Casciola Reference Gualtieri, Picano and Casciola2009) to quantify the degree and orientation of particle clustering. We define $g(\boldsymbol{r})$ as

In this equation, $N(r,{\it\theta},{\it\phi})$ denotes the number of particle pairs in a truncated spherical cone with nominal radius $r$ , polar angle ${\it\theta}$ and azimuthal angle ${\it\phi}$ . The volume of the truncated spherical cone $V(r,{\it\theta},{\it\phi})$ is given by

where ${\rm\Delta}r$ is the radial width, ${\rm\Delta}{\it\theta}$ is the extent of the polar angle and ${\rm\Delta}{\it\phi}$ is the extent of the azimuthal angle.

As discussed in § 4.1.1, the equation governing the relative motion of like particles, (4.1) is the same with and without gravity. Hence the theory of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009), developed without consideration of gravitational settling, is not explicitly changed by the presence of gravity; instead gravity modifies the equation through implicit changes to its coefficients. From Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) (see also Bragg & Collins Reference Bragg and Collins2014a ), the equation describing $g(\boldsymbol{r})$ at steady state is

where ${\bf\lambda}$ is the dispersion tensor describing the influence of the fluid velocity difference field on the dispersion of the particles (see Bragg & Collins Reference Bragg and Collins2014a ).

Since $\boldsymbol{S}_{2}^{p}$ is anisotropic with gravity (cf. § 4.1.2), so too is the ADF $g(\boldsymbol{r})$ , as is evident in (4.12). We similarly adopt an expansion of the ADF in terms of spherical harmonic functions

where $\mathscr{C}_{2\ell }^{0}$ are the spherical harmonic coefficients for the ADF. We only attempt to model the zeroth-order term, which we denote as

where $Y_{0}^{0}\equiv 1$ . Note that $g(r)$ is just the spherical average of the ADF $g(\boldsymbol{r})$ , or the equivalent of the radial distribution function (RDF) defined in Part 1. While this approach captures the leading-order contribution of gravity to the ADF, it does not predict the gravity-induced anisotropy. (Note that a similar approach was adopted by Alipchenkov & Beketov (Reference Alipchenkov and Beketov2013) to model particle clustering in a homogeneous turbulent shear flow.) We will assess the accuracy of this approach in § 4.2.2.

We therefore consider the isotropic form of (4.12), which is given as

where $\hat{Z}$ denotes a variable $Z$ normalized by Kolmogorov scales. ${\it\lambda}_{\Vert }$ denotes the projection of the dispersion tensor ${\bf\lambda}$ along a direction parallel to the particle separation vector. We note that the first term on the right-hand side is associated with outward diffusion (which acts to reduce the RDF), while the second is associated with an inward drift (which acts to increase the RDF). The corresponding diffusion and drift coefficients are denoted as $\mathfrak{D}$ and $\mathfrak{d}$ , respectively (see Bragg & Collins Reference Bragg and Collins2014a ).

As explained in Bragg & Collins (Reference Bragg and Collins2014a ), the overall degree of clustering is determined by the relative strength of the drift and diffusion coefficients, $\mathfrak{d}/\mathfrak{D}$ . Increases (decreases) in this ratio are associated with increased (decreased) clustering. To understand how gravity affects the clustering, we therefore need to understand its effect upon the ratio $\mathfrak{d}/\mathfrak{D}$ .

Bragg & Collins (Reference Bragg and Collins2014a ) demonstrated that in the limit $St\ll 1$ , $\mathfrak{D}$ and $\mathfrak{d}$ simplify to

and

respectively, where $B_{nl}$ is the non-local coefficient defined in Chun et al. (Reference Chun, Koch, Rani, Ahluwalia and Collins2005) that is independent of the particle Stokes number. As is evident in (4.16), the diffusion term is independent of the Stokes number – hence $\mathfrak{D}$ is unaffected by gravity in this limit. In contrast, $\mathfrak{d}$ is proportional to both $St$ and $\langle \hat{\unicode[STIX]{x1D61A}}^{2}\rangle ^{p}-\langle \hat{\unicode[STIX]{x1D619}}^{2}\rangle ^{p}$ . Gravity reduces the correlation between the particles and the strain and rotation fields, and thereby reduces $\mathfrak{d}$ . The net effect of gravity in this limit is therefore to reduce the ratio $\mathfrak{d}/\mathfrak{D}$ and the degree of clustering. We also note that at low $St$ , the trends in the strain and rotation rates with $R_{{\it\lambda}}$ are very weak. Consistent with Part 1, these trends imply the clustering to be nearly independent of $R_{{\it\lambda}}$ in the limit $St\ll 1$ , with or without gravity.

For inertial particles with intermediate Stokes numbers in the range $0.2\lesssim St\lesssim 0.7$ (note that this range is approximate and is likely sensitive to changes in Reynolds number, as mentioned in Part 1), we no longer have simple relationships for the drift and diffusion terms. However, we can still make qualitative predictions for the trends in the clustering based on our physical understanding and the explanations in Bragg & Collins (Reference Bragg and Collins2014a ). Over this range of $St$ , Bragg & Collins (Reference Bragg and Collins2014a ) argued that both preferential sampling effects and path-history effects act to increase clustering, with the path-history effects becoming increasingly dominant as $St$ increases. Since gravity decreases both preferential sampling and path-history effects, we therefore expect it to reduce clustering over the approximate range of $0.2\lesssim St\lesssim 0.7$ .

We next use (4.15) to understand clustering at larger values of $St$ . In Part 1, we argued that without gravity, ${\hat{S}}_{2\Vert }^{p}\approx {\hat{S}}_{2\bot }^{p}$ for $St\gtrsim 0.3$ , and we were thus able to neglect the term $2\hat{r}^{-1}({\hat{S}}_{2\Vert }^{p}-{\hat{S}}_{2\bot }^{p})$ in (4.15) at high $St$ . However, with gravity, the parallel and perpendicular relative velocity variances are generally not equal (as discussed in § 4.1.2), and hence this term must be retained. We are still able to neglect $\hat{{\it\lambda}}_{\Vert }$ at small separations and high $St$ , since this term is inversely proportional to $St$ and decreases as the time scales of the fluid velocity seen by the particles decrease (see Bragg & Collins Reference Bragg and Collins2014a ).

The simplified form of (4.15) at high $St$ is therefore

Taking the ratio between the drift and diffusion coefficients gives us

where ${\it\zeta}_{\Vert }^{2}$ is the scaling exponent of the parallel relative velocity variance (see § 4.1.2).

From § 4.1.2, we see that with gravity, ${\it\zeta}_{\Vert }^{2}$ increases very strongly, while $2{\hat{S}}_{2\bot }^{p}/{\hat{S}}_{2\Vert }^{p}$ increases weakly. We expect that ${\it\zeta}_{\Vert }^{2}-2{\hat{S}}_{2\bot }^{p}/{\hat{S}}_{2\Vert }^{p}$ will therefore increase with gravity, leading to an increase in the clustering at high $St$ . We attribute these trends to the fact that gravity reduces the influence of path-history effects on particle relative motion in the dissipation range at high $St$ . Path-history effects primarily act to diminish clustering for $St\gtrsim 1$ (see Bragg & Collins Reference Bragg and Collins2014a ), and thus gravity reduces their influence and therefore acts to increase $g(\hat{r})$ . We emphasize that this does not necessarily imply that gravity increases the inward drift and decreases the outward diffusion. In fact, our data indicate that gravity decreases both $\mathfrak{d}$ and $\mathfrak{D}$ . The reduction in $\mathfrak{D}$ with gravity, however, is stronger than the corresponding reduction in $\mathfrak{d}$ , causing the ratio $\mathfrak{d}/\mathfrak{D}$ , and therefore the clustering strength, to increase. We also see from § 4.1.2 that increasing $R_{{\it\lambda}}$ , with gravity, generally leads to a decrease in both ${\it\zeta}_{\Vert }^{2}$ and ${\hat{S}}_{2\bot }^{p}/{\hat{S}}_{2\Vert }^{p}$ , due to the increased influence of path-history effects on particle relative motion at higher Reynolds numbers. While both quantities seem to be decreased by about the same amount with increasing $R_{{\it\lambda}}$ , the latter quantity has a greater effect on the clustering since it is multiplied by a factor of two in (4.19). This explains why $\mathfrak{d}/\mathfrak{D}$ decreases as $R_{{\it\lambda}}$ increases, leading to a decrease in the clustering.

In summary, gravity reduces clustering at low and intermediate values of $St$ , but increases clustering at high values of $St$ . Gravity reduces both preferential sampling and path-history effects. Preferential sampling effects increase clustering at low values of $St$ , and thus gravity reduces clustering by weakening the preferential sampling drift mechanism. At intermediate values of $St$ , both preferential sampling and path-history effects act to increase clustering, and thus gravity, by weakening both effects, reduces clustering. At higher values of $St$ , path-history effects act to diminish clustering, and gravity increases clustering by decreasing path-history effects.

This explanation is consistent with arguments put forth by Franklin et al. (Reference Franklin, Vaillancourt and Yau2007), Ayala et al. (Reference Ayala, Rosa, Wang and Grabowski2008), Onishi et al. (Reference Onishi, Takahashi and Komori2009), Woittiez et al. (Reference Woittiez, Jonker and Portela2009), Rosa et al. (Reference Rosa, Parishani, Ayala, Grabowski and Wang2013) for low-Stokes-number particles. However, at higher Stokes numbers, several of these authors argued that gravity facilitates interactions with large-scale turbulent eddies, which in turn leads to increased particle clustering (Franklin et al. Reference Franklin, Vaillancourt and Yau2007; Woittiez et al. Reference Woittiez, Jonker and Portela2009; Rosa et al. Reference Rosa, Parishani, Ayala, Grabowski and Wang2013), an hypothesis stated without supporting data, theory or a physical argument. In contrast, we argue that gravity reduces the temporal correlation radius over which the particles are affected by their path-history interactions with the turbulence, and therefore causes the particle relative motion in the dissipation range to be less affected by their interaction with larger-scale turbulence. Our hypothesis is supported by our DNS results for the relative velocities (§ 4.1.2) and also by the extension of the theoretical model of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) shown above. We will show in § 4.2.2 that this model is able to reproduce the DNS clustering results very well.

Another recent paper by Park & Lee (Reference Park and Lee2014) suggests the increase in particle clustering with gravity at high Stokes numbers is linked to the skewness of the vertical velocity gradients of the underlying fluid. This proposed explanation, however, seems to be flawed, and we suggest that their finding reflects a certain correlation in the system, but not a causal relation. This conclusion is supported by the recent work of Gustavsson et al. (Reference Gustavsson, Vajedi and Mehlig2014), who observed qualitatively similarly strong clustering of high-Stokes-number particles in the presence of gravity in their random Gaussian flow field, whose fluid velocity, by definition, has no skewness.

Figure 19.

Instantaneous particle locations in $1000{\it\eta}\times 1000{\it\eta}\times 10{\it\eta}$ slices of the domain for different values of $St$ (0.3 (a,d), 1 (b,e), 3 (c,f)) and $R_{{\it\lambda}}=398$ . Panels (a–c) are for the case without gravity, and (d–f) are for the case with gravity ( $Fr=0.052$ ), with the gravitational vector pointing downward.

4.2.2 Particle clustering results

We begin by first examining instantaneous snapshots of particle positions in the simulations in figure 19. From these visualizations, it is evident that gravity alters both the degree and orientation of the clusters and generally causes them to be aligned with the gravity vector. Recall that the clustering observed at $St\gtrsim 1$ is not related to the Maxey centrifuge mechanism (Maxey Reference Maxey1987), but rather is due to the history effect (Bragg & Collins Reference Bragg and Collins2014a ; Bragg, Ireland & Collins Reference Bragg, Ireland and Collins2015a ,Reference Bragg, Ireland and Collins b ). This explains why higher-Stokes-number particles show signs of clustering even though they nearly uniformly sample the strain and rotation fields (cf. figure 2).

To quantify the clustering, we first consider the spherically averaged RDFs $g(r)$ to analyse the overall effect of gravity on the degree of clustering. Figure 20 shows plots of the RDFs both with gravity ( $Fr=0.052$ ) and without gravity. The results show gravity causes the RDFs to decrease at low $St$ and to increase at high $St$ , in agreement with our arguments in § 4.2.1 and earlier findings (Franklin et al. Reference Franklin, Vaillancourt and Yau2007; Ayala et al. Reference Ayala, Rosa, Wang and Grabowski2008; Onishi et al. Reference Onishi, Takahashi and Komori2009; Woittiez et al. Reference Woittiez, Jonker and Portela2009; Rosa et al. Reference Rosa, Parishani, Ayala, Grabowski and Wang2013; Bec et al. Reference Bec, Homann and Ray2014; Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014). For sufficiently large separations, the RDFs with and without gravity approach each other.

Figure 20.

DNS (symbols) and theoretical (‘ZT’, lines) data for the RDFs $g$ plotted as a function of $r/{\it\eta}$ at $R_{{\it\lambda}}=398$ both with gravity (open symbols and solid lines) and without gravity (filled symbols and dashed lines). (a) Shows RDFs for $St=0.3$ , and (b) for $St=3$ . The theoretical predictions are calculated from the equations in Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) with the non-local correction from Bragg & Collins (Reference Bragg and Collins2014a ), using our DNS data to specify $T_{\unicode[STIX]{x1D61A}\unicode[STIX]{x1D61A}}^{p}$ , $S_{2\Vert }^{p}$ and $S_{2\bot }^{p}$ .

To test the theory in § 4.2.1, we compute $g(r)$ from (4.15) using the relative velocity statistics $S_{2\Vert }^{p}$ and $S_{2\bot }^{p}$ obtained from DNS. In addition, we use the directionally averaged strain time scales $T_{\unicode[STIX]{x1D61A}\unicode[STIX]{x1D61A}}^{p}$ from the DNS (see § 3.1) and the non-local closure proposed in Bragg & Collins (Reference Bragg and Collins2014a ) to compute the dispersion tensor. This allows us to test the formulation of (4.15) and the theoretical arguments in § 4.2.1. Figure 20 shows that the theory captures the quantitative results in the DNS well, indicating that (4.15) is an accurate model even for an anisotropic particle phase, and thus verifies the physical explanations presented in § 4.2.1.

Next, we consider the dependence of the RDFs on $R_{{\it\lambda}}$ in figure 21. As was the case without gravity (see Part 1), the RDFs are largely independent of $R_{{\it\lambda}}$ for $St\lesssim 1$ with gravity. This is in agreement with our arguments in § 4.2.1 and implies that small- $St$ clustering is a small-scale phenomenon (both with and without gravity) that is generally unaffected by the intermittency of the turbulence. For $St>1$ the RDFs generally increase with increasing $R_{{\it\lambda}}$ , since the ratio between the drift and diffusion increases, as discussed in § 4.2.1. The RDFs seem to approach a constant value, however, suggesting that any Reynolds-number dependencies may eventually saturate in the limit $R_{{\it\lambda}}\rightarrow \infty$ .

Figure 21.

The normalized RDFs at $r/{\it\eta}=0.25$ with gravity ( $Fr=0.052$ ), plotted as a function of $R_{{\it\lambda}}$ . The RDFs at a given Reynolds number are normalized by their corresponding value at $R_{{\it\lambda}}=90$ to highlight the Reynolds-number dependence. Shaded contours denote 95 % confidence intervals for the RDFs.

We can also quantify the degree of small-scale clustering by performing a power-law fit of the RDFs (Reade & Collins Reference Reade and Collins2000) at small separations,

as discussed in Part 1. The power-law fits are performed over the range $0.75\leqslant r/{\it\eta}\leqslant 2.75$ , and the calculated values of $c_{0}$ and $c_{1}$ are plotted in figure 22. We observe that both the power-law coefficient $c_{0}$ and the exponent $c_{1}$ decrease when gravity is introduced for $St\lesssim 1.5$ and increase when gravity is introduced for $St\gtrsim 1.5$ , consistent with our explanations in § 4.2.1. The theoretical model for $g(r/{\it\eta})$ from (4.15) (with the relative velocities and strain time scales specified from the DNS) is in good agreement with the DNS data, both with and without gravity. We also note that our DNS results for the exponent $c_{1}$ agree well with those of Rosa et al. (Reference Rosa, Parishani, Ayala, Grabowski and Wang2013) and Bec et al. (Reference Bec, Homann and Ray2014) (not shown).

Figure 22.

The (a) prefactor $c_{0}$ and the (b) exponent $c_{1}$ of the power-law fits of the RDFs using (4.20). Open symbols denote data with gravity ( $Fr=0.052$ ) and filled symbols denote data without gravity. The predictions from Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) for $R_{{\it\lambda}}=398$ (‘ZT’, where DNS data are used to specify the relative velocities and the strain time scales here) are shown with solid lines (gravity) and dashed lines (no gravity). The theoretical high- $St$ predictions from Fouxon et al. (Reference Fouxon, Park, Harduf and Lee2015) (‘FT’) for $R_{{\it\lambda}}=398$ are also included in (b).

Figure 23.

Filled contours of the RDF evaluated at $r/{\it\eta}=0.25$ (a) and $r/{\it\eta}=9.75$ (b) for different values of $St$ and $Sv$ , for $R_{{\it\lambda}}=227$ . The diagonal lines denote three different values of $Fr$ , corresponding to conditions representative of stratiform clouds ( $Fr=0.01$ ), cumulus clouds ( $Fr=0.05$ ) and cumulonimbus clouds ( $Fr=0.3$ ).

In Fouxon et al. (Reference Fouxon, Park, Harduf and Lee2015), the authors state that $c_{1}$ should be independent of $St$ for large $St$ and small $Fr$ . This is generally consistent with our DNS results for $St\gtrsim 1$ in figure 22(b). They also argue that $c_{1}$ can be determined entirely from the fluid field as

We include comparisons between our DNS results and the theoretical predictions of Fouxon et al. (Reference Fouxon, Park, Harduf and Lee2015) in figure 22(b). We see that (4.21) predicts $c_{1}$ to within about 50 % at high $St$ . The model from (4.15), however, gives much more accurate quantitative predictions, but does require the second-order structure functions and the Lagrangian time scales for the inertial particles from the DNS.

To further investigate the two-parameter space of particle inertia and gravity, we consider the RDFs at $R_{{\it\lambda}}=227$ for various $St$ and $Sv$ in figure 23. While the particle clustering behaviour here is generally complex and varies strongly with $St$ and $Sv$ , we are able to provide physical explanations for several of the observed trends. For $St\gg 1$ , the RDFs generally decrease with increasing $St$ , since the particles become unresponsive to almost all of the underlying turbulence, as expected. Also, for $St\lesssim 1$ , the RDFs tend to decrease with increasing $Sv$ , since the preferential sampling and path-history mechanisms both act to increase clustering here, and they are both reduced by gravity. We also see that for $1<St<10$ , the RDFs generally increase with increasing $Sv$ , as expected, since gravity increases the ratio between the drift and the diffusion, as discussed in § 4.2.1.

We now investigate the anisotropy of the particle field by introducing higher-order spherical harmonic functions,

in figure 24 for various values of $St$ with gravity ( $Fr=0.052$ ) at $R_{{\it\lambda}}=398$ .

Figure 24.

The ADF $g(\boldsymbol{r})$ (normalized by the RDF $g(r)$ ) shown on a unit sphere for $R_{{\it\lambda}}=398$ and separations $r<{\it\eta}$ with gravity ( $Fr=0.052$ ). The different columns correspond to different values of $St$ . (a) Shows the projection where gravity is directed into the page and (b) shows the projection where gravity is directed downward.

Consistent with the qualitative observations from figure 19, we see that particle clustering is strongest along the vertical direction, in agreement with earlier findings of Dejoan & Monchaux (Reference Dejoan and Monchaux2013), Bec et al. (Reference Bec, Homann and Ray2014) and Park & Lee (Reference Park and Lee2014). At low $St$ , the anisotropic clustering is caused by the fact that particles tend to preferentially sample downward-moving flow (see § 3.2). Bec et al. (Reference Bec, Homann and Ray2014) showed that this preferential sampling causes particles to form vertical clusters when $St$ is small.

When $St$ is large, the effects of preferential sampling vanish, and the anisotropy is related to the way gravity influences the path-history effect. Since the analysis presented in § 4.2.1 is restricted to spherically averaged clustering, the results cannot be used to predict the trends in anisotropy. However, we expect that the clustering anisotropy is linked to both the anisotropy in the relative velocities (see § 4.1.2) and the anisotropy in the strain and rotation time scales (see § 3.1). That is, directional variations in the relative velocities and the strain and rotation time scales will lead to directional variations in the ratio between drift and diffusion. This in turn will cause the particle clustering to be anisotropic. Indeed, the recent paper of Fouxon et al. (Reference Fouxon, Park, Harduf and Lee2015) argues that vertical clusters form at high $St$ with gravity because the velocities of particle pairs have different correlation times in the horizontal and vertical directions. We hope that this and similar theories can be further developed and tested by using the anisotropic strain and rotation time scale results presented in § 3.1.

We also note that the anisotropy increases with increasing $St$ . The physical explanation is that as $St$ (and thus $Sv$ ) increases, gravitational forces become more significant, causing the particle motion to be more anisotropic. However, we expect that at some sufficiently large values of $St$ and $Sv$ , the particles are unaffected by the fluid turbulence, and thus clustering (and the associated anisotropy) will vanish.

We explore the degree of anisotropy as a function of the separation distance $r$ and the Reynolds number by plotting the spherical harmonic coefficients $\mathscr{C}_{2}^{0}$ and $\mathscr{C}_{4}^{0}$ in figure 25 (coefficients above order four are too small to be statistically significant, and hence are not shown). In agreement with figure 19, we see that the anisotropy increases with increasing $St$ . We also observe that as $r/{\it\eta}$ increases, the anisotropy approaches zero, since both the clustering and the clustering anisotropy vanish in the limit $r/{\it\eta}\rightarrow \infty$ .

Figure 25.

The second (a) and fourth (b) spherical harmonic coefficients of the angular distribution function normalized by the zeroth spherical harmonic coefficient, plotted as a function of $r/{\it\eta}$ , for different $St$ and $R_{{\it\lambda}}$ with gravity ( $Fr=0.052$ ). The different values of $St$ considered ( $0.3$ , $0.5$ , $1$ , $3$ ) are shown in black, red, blue and green, respectively, and the Stokes numbers are indicated by the line labels.

The degree of anisotropy is also Reynolds-number dependent and decreases with increasing $R_{{\it\lambda}}$ for larger values of $St$ . Since the relative velocities of high- $St$ particles generally also tend to become more isotropic as $R_{{\it\lambda}}$ increases (see § 4.1.2), it is likely that the reduction in the anisotropy of the relative velocities will cause a similar reduction in the anisotropy of the clustering.