2.1 Three closure forms for diffusivity $\unicode[STIX]{x1D60B}_{UU}$

Since our primary interest is in the statistics of pair relative motion, we will consider a more tractable, lower-dimensional PDF $\unicode[STIX]{x1D6FA}(\boldsymbol{r},\boldsymbol{U})=\int \langle P\rangle (\boldsymbol{r},\boldsymbol{U},\boldsymbol{x},\boldsymbol{V};t)\,\text{d}\boldsymbol{V}$ , where the dependence on $\boldsymbol{x}$ was dropped due to homogeneity. In Rani et al. (Reference Rani, Dhariwal and Koch2014), we showed that (2.9) yields the following equation (in dimensional form) for $\unicode[STIX]{x1D6FA}(\boldsymbol{r},\boldsymbol{U})$ :

where $\unicode[STIX]{x1D70F}_{v}$ is the particle viscous relaxation time and $\boldsymbol{r}$ and $\boldsymbol{U}$ are the pair separation and relative velocity, respectively. The diffusivity $\unicode[STIX]{x1D60B}_{UU}$ is given by

where the integrand is the Eulerian two-time correlation of fluid relative velocities, with both pair separation $\boldsymbol{r}$ and centre-of-mass position $\boldsymbol{x}$ held fixed. This formulation of diffusivity is valid under the conditions $St_{r}\gg 1$ and $St_{I}\gg 1$ .

In this study, $\unicode[STIX]{x1D60B}_{UU}$ in (2.11) is resolved using three approaches, giving rise to three closure forms. In the first closure form (CF1), the two-time correlation in (2.11) is directly computed from DNS of forced isotropic turbulence with stationary particles, and integrated in time to yield $\unicode[STIX]{x1D60B}_{UU}$ . In the second closure form (CF2), the Eulerian two-time correlation of relative velocities is converted into two-point correlations of fluid velocities, allowing us to derive an expression in terms of an integral over the wavenumber. In the third closure form (CF3), $\boldsymbol{r}$ remains fixed during flow time scales, but $\boldsymbol{x}$ responds to integral-scale eddies, i.e.  $\boldsymbol{x}$ changes during integral time scales. Thus, CF3 attempts to improve upon and extend CF2’s validity to $St_{I}\sim 1$ . The effects of CF3, however, are anticipated to be seen only at the lower end of the Stokes number range considered. At higher Stokes numbers, CF2 and CF3 should show very similar behaviour.

One may regard CF1 as the most accurate among the three closure forms since a DNS-based evaluation of the Eulerian two-time correlation would entail no further approximations. But, CF2 and CF3, the latter in spite of its improvements, contain approximations made to obtain analytical expressions for diffusivity. The differences between CF1 and CF2/CF3, and their implications for pair statistics are extensively analysed in this study.

In the following discussion, we present the salient features of the derivation of CF2 and CF3 (details are in Rani et al. Reference Rani, Dhariwal and Koch2014). The computational details of the evaluation of $\unicode[STIX]{x1D60B}_{UU}$ for CF1 are discussed in § 3.1.

2.2 CF2 and CF3

In arriving at (2.11), Rani et al. (Reference Rani, Dhariwal and Koch2014) considered the limits $St_{r}\gg 1$ and $St_{I}\gg 1$ . In these Stokes number regimes, particles are nearly stationary so that the temporal change in $\unicode[STIX]{x0394}\boldsymbol{u}$ experienced by pairs is primarily due to the evolution of turbulent scales and not due to pair (relative) motion itself. Rani et al. (Reference Rani, Dhariwal and Koch2014) approximated the temporal evolution of $\unicode[STIX]{x0394}\boldsymbol{u}$ at two positions separated by $\boldsymbol{r}$ as arising due to the passive advection of eddies of size $r$ by larger, integral-scale eddies. Hence, one may replace the Eulerian two-time correlation in (2.11) by a correlation of relative velocities seen by two pairs with the same $\boldsymbol{r}$ , but with the centres of mass separated by $\boldsymbol{u}_{I}t$ , where $\boldsymbol{u}_{I}$ is the large-scale fluid velocity. This would give us closure form 2 (CF2) for $\unicode[STIX]{x1D60B}_{UU}$ :

Relaxing the $St_{I}\gg 1$ criterion to $St_{I}\sim 1$ enables us to account for the change in centre-of-mass position due to interactions with eddies of time scales ${\sim}\unicode[STIX]{x1D70F}_{v}$ , the centre-of-mass response time. This is done by replacing $\boldsymbol{u}_{I}$ with the relative velocity $\boldsymbol{W}$ between the large-scale eddies and the centre of mass, yielding closure form 3 (CF3):

In the discussion that follows, we will focus on CF3, since the final expressions for CF2 and CF3 differ only by a constant factor.

The CF3 diffusivity can then be expressed as an average over all values of $\boldsymbol{W}$ as:

where $\unicode[STIX]{x1D6F7}(\boldsymbol{W})$ is the PDF of $\boldsymbol{W}$ , and

Since $\boldsymbol{W}$ is principally influenced by the large-scale fluid eddies, its PDF can be considered as Gaussian (Batchelor Reference Batchelor1953):

where $W_{rms}$ is the r.m.s. fluctuating velocity of $\boldsymbol{W}$ . The two-point velocity correlations in (2.15) may be expressed in terms of the velocity spectrum tensor $\unicode[STIX]{x1D619}(\boldsymbol{k})$ , allowing us to further analytically resolve this equation.

For isotropic turbulence, we can write $\unicode[STIX]{x1D60B}_{UU}$ as:

For CF3, Rani et al. (Reference Rani, Dhariwal and Koch2014) derived the following forms for $\unicode[STIX]{x1D60B}_{UU,\bot }$ and $\unicode[STIX]{x1D60B}_{UU,||}$ :

where $E(\unicode[STIX]{x1D709})$ is the energy spectrum, $\unicode[STIX]{x1D709}$ is the wavenumber and the analytical expression for $W_{rms}$ is presented in § 2.3. The corresponding forms for CF2 can be obtained by simply replacing $W_{rms}$ with $u_{rms}$ in (2.18) and (2.19), where $u_{rms}$ is the fluid fluctuating r.m.s. velocity. It is anticipated that at high Stokes numbers, $W_{rms}\rightarrow u_{rms}$ so that CF2 and CF3 approach each other.

2.3 Expression for $W_{rms}$

$W_{rms}$ is the r.m.s. of the relative velocity between large-scale eddies and the centre-of-mass velocity. It is given by

where $V_{i}$ is the velocity of the pair centre of mass, $u_{i}$ is the fluid velocity with which eddies of size $r$ are advected past the pair by larger eddies and $\langle u_{i}^{2}\rangle =\langle u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\rangle$ ( $\langle V_{i}^{2}\rangle$ follows a similar notation).

The velocity $V_{i}$ is governed by

where $u_{cm,i}$ was defined in (2.3). Multiplying (2.21) with $V_{i}$ and ensemble averaging yields

where $\langle u_{cm,i}V_{i}\rangle$ is approximated with $\langle u_{i}V_{i}\rangle$ on the right-hand side of (2.22). This is reasonable since both $u_{i}-V_{i}$ and $u_{cm,i}-V_{i}$ are determined by eddies with sizes of the order of or smaller than $r$ , so that these two quantities are expected to have similar statistics.

At steady state, $\langle V_{i}u_{i}\rangle =\langle V_{i}^{2}\rangle$ . This means that from (2.21)

where $\langle u_{i}^{2}\rangle /3={u^{\prime }}^{2}$ , and similarly $\langle V_{i}^{2}\rangle /3={V^{\prime }}^{2}$ . To close $W_{rms}$ , one needs expressions for $u^{\prime }$ and the ratio $V^{\prime }/u^{\prime }$ . The latter was obtained from Jung, Yeo & Lee (Reference Jung, Yeo and Lee2008):

where $T_{L}$ is the fluid Lagrangian integral time scale, $T^{\prime }=T_{fp}-\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale and $T_{fp}$ is the Lagrangian integral time scale of fluid velocities seen by the particles. Jung et al. (Reference Jung, Yeo and Lee2008) provided an expression for $T_{fp}$ in terms of fluid Eulerian and Lagrangian integral time scales. Further, it can be inferred from their study that ${u^{\prime }}^{2}=u_{rms}^{2}$ for high $St$ particles, i.e. variance of fluid velocity seen by high-inertia particles is nearly equal to the variance of turbulent velocity fluctuations.

3.1 DNS

3.2 Langevin simulations

Using the CF1, CF2 and CF3 closures for $\unicode[STIX]{x1D60B}_{UU}$ discussed in § 2, three sets of Langevin stochastic simulations were performed to evolve pair separations $\boldsymbol{r}$ and relative velocities $\boldsymbol{U}$ . The governing equations for $\boldsymbol{r}$ and $\boldsymbol{U}$ are:

Here, $\unicode[STIX]{x1D652}$ represents a Wiener process, and the diffusion matrix $\unicode[STIX]{x1D63D}$ can be written in terms of $\unicode[STIX]{x1D60B}_{UU}$ as:

where $\unicode[STIX]{x1D63D}^{\text{T}}$ is the transpose of $\unicode[STIX]{x1D63D}$ , and $\unicode[STIX]{x1D63D}$ is computed from a Cholesky decomposition of $\unicode[STIX]{x1D60B}_{UU}(r)$ . Simulation of Langevin equations (3.7) and (3.8) is statistically equivalent to solving the Fokker–Plank equation (2.10) for the PDF $\unicode[STIX]{x1D6FA}(\boldsymbol{r},\boldsymbol{U})$ .

To be able to consistently compare pair statistics from the Langevin and DNS runs, it is important that the Langevin simulations use the same turbulence parameters as those in statistically stationary DNS. Hence, inputs to Langevin runs such as the Kolmogorov and integral length scales, dissipation rate, kinematic viscosity, $u_{rms}$ and $Re_{\unicode[STIX]{x1D706}}$ are all identical to those in table 1. In particular, one also has to ensure that the model energy spectrum used in CF2 and CF3 closely matches the DNS energy spectrum. This was achieved by suitably selecting the parametric inputs to the model spectrum provided in Pope (Reference Pope2000), as follows:

where $\unicode[STIX]{x1D6FD}=5.2$ and $p_{0}=2$ (Pope Reference Pope2000).

The parameters $c_{L}$ and $c_{\unicode[STIX]{x1D702}}$ are determined from the following constraints:

where $\unicode[STIX]{x1D716}$ is the dissipation rate, and the wavenumber limits $[1,\unicode[STIX]{x1D705}_{max}]$ are the same as in DNS. These wavenumber limits are also used in (2.18) and (2.19) for CF2 and CF3. The parameters $c_{L}$ and $c_{\unicode[STIX]{x1D702}}$ are numerically evaluated using the DNS values of $u_{rms}$ , $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D708}$ from table 1. The resulting values are shown in table 2. In figure 2, the model spectra calculated from (3.10)–(3.12) are compared with the DNS energy spectra for $Re_{\unicode[STIX]{x1D706}}=76$ and $Re_{\unicode[STIX]{x1D706}}=131$ . Good agreement is seen between the model and DNS spectra.

Figure 2.

Comparison of the DNS and model energy spectra at $Re_{\unicode[STIX]{x1D706}}=76$ and $Re_{\unicode[STIX]{x1D706}}=131$ . The model spectrum is used to compute the CF1 and CF2 diffusivities.

Table 2.

Parameters for the model energy spectrum. After determining $c_{L}$ and $c_{\unicode[STIX]{x1D702}}$ , the parameter $C$ was adjusted to match the DNS energy spectrum. Pope (Reference Pope2000) suggested $C=1.5$ .

The computational domain in the Langevin simulations is a sphere of diameter $8L$ , where $L$ is the integral length scale. This domain size is sufficiently large, since particle pairs become decorrelated at separations of $O(L)$ for all the Stokes numbers considered in this study. A specular reflective boundary condition was imposed at the outer boundary of the domain. This meant that a particle colliding with the outer boundary is reflected back into the domain, with its velocity component tangential to the boundary unaffected, and the velocity component normal to the boundary reversed.

For each $St_{\unicode[STIX]{x1D702}}$ , pair statistics are averaged for at least 1000 $\unicode[STIX]{x1D70F}_{v}$ . Such a large averaging time was necessary due to the strong dependence of the statistical errors on the sample size in Langevin simulations. Further, in order to increase the sample size at separations of the order of Kolmogorov length scale, a single pair is split into multiple, equally weighted fractional pairs whenever the separation of a pair goes below a certain value (Rani et al. Reference Rani, Dhariwal and Koch2014). When a parent pair is split, initially the fractional pairs have the same position and velocity vectors as the parent. Each of the fractional pairs is then evolved independently, except that it only makes a fractional contribution when computing the statistics. In our simulations, splitting is executed at three different radial locations, $r=2\unicode[STIX]{x1D702},5\unicode[STIX]{x1D702},10\unicode[STIX]{x1D702}$ . However, fractional pairs are not split again. We found that splitting a pair into 10 equally weighted fractional pairs gave us sufficient data at the smaller separations without excessively increasing the number of pairs to be tracked. Recombination of fractional pairs when their separations exceeded the specified radial distances was not undertaken.

4.1 Diffusivity tensor

We first compare the diffusivity tensors from the three closure forms considered in this study. In addition, the CF1 and CF2 diffusivities are analysed in greater detail for pair separations in the integral range, where one can derive analytical estimates for these. Subsequently, we compare CF1 with the diffusivity closure of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003, Reference Zaichik and Alipchenkov2007). As CF1 is computed through DNS, it is essentially exact for $St_{r}\gg 1$ , presenting us an opportunity to assess the Zaichik & Alipchenkov closure in this limit. Such an analysis of their diffusivity had not been undertaken previously.

Figure 3.

The transverse component, $\unicode[STIX]{x1D60B}_{UU,\bot }(r)$ , and the longitudinal component, $\unicode[STIX]{x1D60B}_{UU,||}(r)$ , of the diffusivity tensor for $St_{\unicode[STIX]{x1D702}}=10$ as a function of dimensionless pair separation $r/L$ . Diffusivity tensor components for CF1, CF2 and CF3 are shown. Upper black solid line denotes CF3 transverse component, and lower grey solid line denotes CF2 transverse component. (a) $Re_{\unicode[STIX]{x1D706}}=76$ , and (b) $Re_{\unicode[STIX]{x1D706}}=131$ . Transverse and longitudinal components of $\unicode[STIX]{x1D60B}_{UU}$ for CF1, CF2 and CF3 in the viscous range at $Re_{\unicode[STIX]{x1D706}}=76$ and 131 are shown in table 3.

Table 3.

Transverse and longitudinal components of $\unicode[STIX]{x1D60B}_{UU}$ for CF1, CF2 and CF3 in the viscous range at $Re_{\unicode[STIX]{x1D706}}=76$ and 131. The values shown are for $St_{\unicode[STIX]{x1D702}}=10$ . The following notation is used: $\unicode[STIX]{x1D60B}_{UU,||}^{[1,2,3],\star }=[\unicode[STIX]{x1D60B}_{UU,||}^{[1,2,3]}\times \unicode[STIX]{x1D70F}_{v}^{2}/(u_{rms}^{2}\times T_{eddy})]/(r/L)^{2}$ .

In figure 3, the longitudinal and transverse components of the diffusivity for CF1, CF2 and CF3 are plotted as a function of the dimensionless separation $r/L$ ( $L$ is the integral length scale). Shown in figure 3(a,b) are the diffusivity components at $Re_{\unicode[STIX]{x1D706}}=76$ and 131, respectively, for the $St_{\unicode[STIX]{x1D702}}=10$ pairs. It may be noted that $\unicode[STIX]{x1D60B}_{UU}\times \unicode[STIX]{x1D70F}_{v}^{2}$ is independent of the Stokes number for CF1 and CF2, but not for CF3 due to the presence of $W_{rms}$ on the right-hand side of (2.18) and (2.19). The diffusivities are shown for the lowest Stokes number $St_{\unicode[STIX]{x1D702}}=10$ , since the differences between CF2 and CF3 are more pronounced at low Stokes numbers. For Kolmogorov-scale separations, i.e. $r\sim \unicode[STIX]{x1D702}$ , $\unicode[STIX]{x1D60B}_{UU,\bot }(r)$ and $\unicode[STIX]{x1D60B}_{UU,||}(r)$ show an $r^{2}$ scaling for all three closures (numerical values of the scaling coefficients are shown in table 3). This scaling arises because for $r<\unicode[STIX]{x1D702}$ , the fluid velocity field may be regarded as locally linear, i.e.  $\unicode[STIX]{x0394}\boldsymbol{u}\approx \boldsymbol{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}$ , which in conjunction with (2.11) gives rise to the $r^{2}$ scaling. At both $Re_{\unicode[STIX]{x1D706}}=76$ and 131, it is seen that CF3 is in reasonable agreement with CF1 for inertial range separations. However, in the transition region between the inertial and integral ranges, as well as in the integral range, CF1 is higher than CF3. The diffusivity components of CF1 exceed those of CF2 at all separations. These trends are to be expected, especially at $St_{\unicode[STIX]{x1D702}}=10$ , since CF3 does a better job at lower Stokes numbers than does CF2.

For integral-scale separations, i.e.  $r\gtrsim L$ , one can perform a more detailed comparison of CF1 and CF2, since one can derive estimates for these closures in this region. It may recalled that in deriving the CF2 diffusivity expression, we assumed that the temporal change of the fluid relative velocities seen by a pair is primarily due to the passive advection of size $r$ eddies past the pair by large-scale eddies with velocity $u_{I}$ . This physical picture is valid only when $r/u_{I}\ll r/u_{r}=\unicode[STIX]{x1D70F}_{r}$ , i.e. when $r$ is small enough such that the time taken to advect the size $r$ eddies past the pair is smaller than their turnover time. Therefore, one expects CF2 to perform poorly at large separations.

For $r\gtrsim L$ , the particle pairs are effectively uncorrelated and behave like two independent particles. Consequently, the pair diffusivity is equal to twice the single particle diffusivity. Using this principle, we can write for CF1:

where $T_{E}$ is the Eulerian integral time scale.

In the case of CF2, noting that $\unicode[STIX]{x1D60B}_{UU,||}^{[2]}=\unicode[STIX]{x1D60B}_{UU,\bot }^{[2]}$ when $r\gtrsim L$ , we have

where we replaced $W_{rms}$ with $u_{rms}$ in (2.18) and (2.19), and the right-hand side of (4.2) is the limiting value of the right-hand side of (2.18) as $r\rightarrow \infty$ . We have also used the identity $\int _{0}^{\infty }(E(\unicode[STIX]{x1D709})/\unicode[STIX]{x1D709})\,\text{d}\unicode[STIX]{x1D709}=(u_{rms}^{2}/\unicode[STIX]{x03C0})L$ and $T_{eddy}=L/u_{rms}$ , where $L$ is the integral length scale. The ratio $T_{eddy}/T_{E}\sim 1.2{-}1.5$ so that from (4.1) and (4.4), the ratio $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[2]}$ is now estimated to be ${\sim}2.5{-}3.2$ for $r\gtrsim L$ .

In figure 3, for $r\gtrsim L$ , we see that $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[2]}\approx 2.05$ when $Re_{\unicode[STIX]{x1D706}}=76$ , and ${\approx}3.45$ when $Re_{\unicode[STIX]{x1D706}}=131$ , in accordance with the preceding scaling estimate. Further, $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[3]}\approx 1.5$ when $Re_{\unicode[STIX]{x1D706}}=76$ , and ${\approx}2.29$ when $Re_{\unicode[STIX]{x1D706}}=131$ . The lower values of $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[3]}$ are because CF3 exceeds CF2 for smaller $St_{\unicode[STIX]{x1D702}}$ . We will also see in § 4.3 that for $r\gtrsim L$ , the pair relative-velocity variance computed using CF1 is in good agreement with an analytical expression for the relative-velocity variance of uncorrelated pairs (Pan & Padoan Reference Pan and Padoan2013). However, both CF2 and CF3 underpredict this analytical variance by nearly the same factors as $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[2]}$ and $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[3]}$ for $r\gtrsim L$ in figure 3.

In figure 4, we compare only CF2 and CF3 for the highest Stokes number considered. Figures 4(a,b) show the diffusion tensor components for the $St_{\unicode[STIX]{x1D702}}=80$ particles at $Re_{\unicode[STIX]{x1D706}}=76$ and 131, respectively. At high Stokes numbers, $W_{rms}\approx u_{rms}$ , so that one expects CF2 and CF3 to have similar diffusivities, which is confirmed in figure 4.

Figure 4.

Transverse component, $\unicode[STIX]{x1D60B}_{UU,\bot }(r)$ , and longitudinal component, $\unicode[STIX]{x1D60B}_{UU,||}(r)$ , of the diffusion coefficient tensor for $St_{\unicode[STIX]{x1D702}}=80$ as a function of dimensionless pair separation $r/L$ at: (a) $Re_{\unicode[STIX]{x1D706}}=76$ and (b) $Re_{\unicode[STIX]{x1D706}}=131$ . CF2 and CF3 diffusivities are compared.

We now present a comparative analysis of CF1 and the Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003, Reference Zaichik and Alipchenkov2007) closures. As already mentioned, CF1 may be regarded as ‘exact’ for $St_{r}\gg 1$ . In this limit, the Zaichik & Alipchenkov diffusivity in $\boldsymbol{U}$ -space may be written as:

where $\unicode[STIX]{x1D61A}(\boldsymbol{r})$ is the Eulerian structure function tensor, and $T_{Lr}$ is the Lagrangian two-point time scale at separation $r$ . We can now calculate $\unicode[STIX]{x1D60B}_{UU}^{Zaichik}$ using the well-known scaling expressions of $\unicode[STIX]{x1D61A}(r)$ and $T_{Lr}$ for separations $r$ in the dissipative, inertial and integral ranges. The expressions for the longitudinal and transverse components of $\unicode[STIX]{x1D61A}(r)$ are (Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2003):

where $C_{1}=2.0$ .

The corresponding expressions for $T_{Lr}(r)$ are also provided in Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003). It is, however, relevant to elaborate on $T_{Lr}$ in the dissipative range. To determine $T_{Lr}$ in this range, Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003) approximated the fluid relative velocity as being linear in the separation vector: $\unicode[STIX]{x0394}\boldsymbol{u}\approx \boldsymbol{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}$ . Accordingly, the viscous $T_{Lr}$ would need to be found in terms of the correlation time scales, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ , of the strain-rate and rotation-rate tensors constituting the velocity gradient. Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) considered $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=A_{1}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale. Subsequently, Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2007) reconsidered this analysis with separate forms for $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ . We can now write the expressions for $T_{Lr}(r)$ as:

where $A_{1}=\sqrt{5}$ (Lundgren Reference Lundgren1981), $A_{\unicode[STIX]{x1D70E}}=2.3$ , $A_{\unicode[STIX]{x1D714}}=7.2$ , $A_{2}=1/\sqrt{6}$ and $T_{L}$ is the Lagrangian integral time scale. The relationship between the viscous range $T_{Lr,\bot }$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ in (4.10) was obtained from the Brunk, Koch & Lion (Reference Brunk, Koch and Lion1997) study of tracer pair diffusion and coagulation in isotropic random velocity fields.

Figure 5.

Comparison of CF1 of $\unicode[STIX]{x1D60B}_{UU}$ with the diffusivity of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003, Reference Zaichik and Alipchenkov2007) in the limit $St_{r}\gg 1$ . $\unicode[STIX]{x1D60B}_{UU,||}(r)$ and $\unicode[STIX]{x1D60B}_{UU,\bot }(r)$ are the longitudinal and transverse components of $\unicode[STIX]{x1D60B}_{UU}$ , respectively. ‘Zaichik’ refers to the diffusivity calculated from (4.13)–(4.15). Diffusion coefficient is plotted as a function of dimensionless pair separation $r/L$ at: (a) $Re_{\unicode[STIX]{x1D706}}=76$ , and (b) $Re_{\unicode[STIX]{x1D706}}=131$ . Also shown in panel (a) are the diffusivities obtained using the two scaling expressions for the viscous time scale: (4.6) and (4.9), and (4.6) and (4.10). In (b) we show diffusivities obtained from the scaling expressions in the inertial subrange: (4.7) and (4.11). For the viscous range, two forms of scaling expressions are plotted: one in which strain rate and rotation rate have identical time scales ( $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ ), and the second in which they are different ( $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}\neq \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ ). For the latter, these time scales are obtained from Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2007).

Conveniently, the scaling expressions given in Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) can be combined into unified forms for the entire range of turbulent scales. These are (Zaichik, Simonin & Alipchenkov Reference Zaichik, Simonin and Alipchenkov2006; Pan & Padoan Reference Pan and Padoan2013):

where $u_{\unicode[STIX]{x1D702}}$ is the Kolmogorov velocity scale, $C_{K}=2$ , $C_{Kn}\approx 2.5$ and $C_{T}=0.4$ .

We now compare CF1 with $\unicode[STIX]{x1D60B}_{UU}^{Zaichik}(r)$ in figure 5. The latter is computed from (4.5) in conjunction with (4.13)–(4.15). The agreement between CF1 and $\unicode[STIX]{x1D60B}_{UU}^{Zaichik}$ is good at $Re_{\unicode[STIX]{x1D706}}=76$ and reasonable at $Re_{\unicode[STIX]{x1D706}}=131$ . In the dissipative range, $\unicode[STIX]{x1D60B}_{UU,||}^{Zaichik}$ and $\unicode[STIX]{x1D60B}_{UU,\bot }^{Zaichik}$ are higher than their CF1 counterparts at $Re_{\unicode[STIX]{x1D706}}=131$ . It is to be noted that in the integral range $T_{L}\approx T_{E}/1.1$ is used (Pan & Padoan Reference Pan and Padoan2013), where $T_{E}$ is the Eulerian integral time scale evaluated from DNS using $T_{E}=u_{rms}^{2}\times \int _{0}^{\infty }\unicode[STIX]{x1D70C}(t)\,\text{d}t$ . Here $\unicode[STIX]{x1D70C}(t)$ is the Eulerian autocorrelation of fluid velocities. The unified expressions (4.13)–(4.15) result in a rather broad inertial region at $Re_{\unicode[STIX]{x1D706}}=76$ . Moreover, at $Re_{\unicode[STIX]{x1D706}}=131$ , it is surprising that the Zaichik transverse component falls below even the CF1 longitudinal component in the inertial region. These trends suggest that the combined expressions need improvement in the inertial range, as well as in the transition region between the inertial and integral ranges. To elaborate on this aspect, in figure 5(b), we also plot the scaling expressions $\unicode[STIX]{x1D61A}_{||}(r)\times T_{Lr}$ and $\unicode[STIX]{x1D61A}_{\bot }(r)\times T_{Lr}$ for inertial range $r$ , computed from (4.7) and (4.11). These are in good agreement with $\unicode[STIX]{x1D60B}_{UU,||}^{[1]}$ and $\unicode[STIX]{x1D60B}_{UU,\bot }^{[1]}$ in the inertial region, suggesting that the scaling laws for the inertial subrange are accurate, but the unified expressions fail to accurately capture the transition from the inertial- to integral-scale separations. The effects of the unified expressions on the RDF predictions of Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) are elaborated in § 4.2.

We also explored the differences between the viscous time scale of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003) (4.9), and of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2007) (4.10). The transverse and longitudinal diffusivities obtained from these two forms of the viscous time scale are plotted in figure 5(a). It can be seen that the longitudinal diffusivities from the two Zaichik & Alipchenkov studies are identical, and are in good agreement with $\unicode[STIX]{x1D60B}_{UU,||}^{[1]}(r)$ . The transverse diffusivity corresponding to (4.9) is also in good agreement with $\unicode[STIX]{x1D60B}_{UU,\bot }^{[1]}(r)$ , but that computed using (4.10) significantly overpredicts the current $\unicode[STIX]{x1D60B}_{UU,\bot }^{[1]}(r)$ . The transverse diffusivity based on a Lagrangian time scale (Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2007) exceeding that based on an Eulerian time scale (CF1) is evidently problematic. The viscous time scale $T_{Lr,\bot }=(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D70E}}/5+\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}/3)$ arises in the context of tracer pair diffusion in isotropic random velocity fields (Brunk et al. Reference Brunk, Koch and Lion1997). It is not clear if this time scale can be applied for isotropic turbulence. We believe that the rather simple scaling $T_{Lr}=A_{1}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ with $A_{1}=\sqrt{2}$ (Lundgren Reference Lundgren1981) is sufficiently accurate, obviating the need for the more complex form presented in Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2007) at least for Stokes numbers greater than one.

4.2 Radial distribution function

Figure 6.

Radial distribution function as a function of $St_{\unicode[STIX]{x1D702}}$ at specific separations: (a) $r/\unicode[STIX]{x1D702}=6$ , (b) $r/\unicode[STIX]{x1D702}=12$ , (c) $r/\unicode[STIX]{x1D702}=18$ and (d) $r/\unicode[STIX]{x1D702}=24$ . In each plot, squares and circles represent data from CF1 and current DNS at $Re_{\unicode[STIX]{x1D706}}=76$ ; triangles represent DNS data of Février, Simonin & Legendre (Reference Février, Simonin and Legendre2001) at $Re_{\unicode[STIX]{x1D706}}=69$ . Solid line represents data from Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) theory for $Re_{\unicode[STIX]{x1D706}}=69$ .

Figure 7.

Radial distribution function versus $St_{\unicode[STIX]{x1D702}}$ at separation $r/\unicode[STIX]{x1D702}=1$ . Squares and circles represent data from CF1 and current DNS at $Re_{\unicode[STIX]{x1D706}}=76$ . Curve 1 represents Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) theory for $Re_{\unicode[STIX]{x1D706}}=69$ ; and Curve 2 represents Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) theory for $Re_{\unicode[STIX]{x1D706}}=75$ .

The RDF is a well-established measure of particle clustering. In figure 6, the RDF is presented as a function of $St_{\unicode[STIX]{x1D702}}$ at four separations $r/\unicode[STIX]{x1D702}=6,12,18$ and 24. The results from the CF1-based Langevin simulations are compared with the data from the current DNS, the Février et al. (Reference Février, Simonin and Legendre2001) DNS, and also with the results from the Zaichik et al. (Reference Zaichik, Simonin and Alipchenkov2003) theory. The Février et al. (Reference Février, Simonin and Legendre2001) data were for $Re_{\unicode[STIX]{x1D706}}=69$ , while the current DNS data are for $Re_{\unicode[STIX]{x1D706}}=76$ . There is excellent agreement between the CF1 RDF and the two sets of DNS RDFs at all four separations, particularly for $St_{\unicode[STIX]{x1D702}}>10$ . The RDFs obtained from the Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003) theory are significantly higher than the DNS values at all separations.

In subsequent studies (Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2007, Reference Zaichik and Alipchenkov2009), they endeavoured to improve the theory, principally by dropping their earlier assumption that the Lagrangian correlation time scales of the strain-rate and rotation-rate tensors are equal. For $St_{\unicode[STIX]{x1D702}}<1$ , the power-law exponent $C_{3}$ of the RDF ( ${\sim}(\unicode[STIX]{x1D702}/r)^{C_{3}}$ for $r\ll \unicode[STIX]{x1D702}$ ) obtained using the modified theory showed good agreement with the RDF exponents computed using DNS and the theory of Chun et al. (Reference Chun, Koch, Rani, Ahluwalia and Collins2005). In figure 7, we compare the RDFs from CF1 and the current DNS with those from the Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) theory at separation $r/\unicode[STIX]{x1D702}=1$ . Also shown are the RDF values from Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003). We observe that the RDFs computed using their modified theory move closer to, but still overpredict, the current DNS data for $St_{\unicode[STIX]{x1D702}}\gtrsim 10$ .

We attribute this overprediction to two features of the Zaichik & Alipchenkov theory. First, we believe that the discrepancy may principally be due to the way in which the RDFs were computed in their study, i.e. through the solution of the transport equations for the moments of the PDF $\unicode[STIX]{x1D6FA}(\boldsymbol{r},\boldsymbol{U})$ (see (2.10)). This aspect is elaborated in the following discussion.

Considering the pair PDF $\unicode[STIX]{x1D6FA}(\boldsymbol{r},\boldsymbol{U})$ , the moments of interest are

The governing equation for the marginal PDF $\unicode[STIX]{x1D714}(\boldsymbol{r})$ is obtained by integrating (2.10) over the $\boldsymbol{U}$ space, and that for $\langle U_{i}\rangle$ is obtained by premultiplying (2.10) with $U_{i}/\unicode[STIX]{x1D714}(\boldsymbol{r})$ and then integrating over $\boldsymbol{U}$ . Similarly, one also obtains the transport equation for $\langle U_{i}U_{j}\rangle$ by taking the second relative-velocity moments of (2.10). It may be noted that $\unicode[STIX]{x1D714}(r)$ and the RDF $g(r)$ are related through $g(r)=\unicode[STIX]{x1D714}(r)/\unicode[STIX]{x1D714}(r\rightarrow \infty )$ .

The transport equation for $\unicode[STIX]{x1D714}(\boldsymbol{r})$ is given by (Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2003)

which only tells us that at steady state, in isotropic turbulence $\langle U_{i}\rangle =0$ . In fact, $\unicode[STIX]{x1D714}(\boldsymbol{r})$ has to be obtained from the $\langle U_{i}\rangle$ equation. However, when one attempts to solve the equations for the first or higher moments (of $U_{i}$ ), one encounters additional closure problems. For instance, the equation for $\langle U_{i}\rangle$ contains the unclosed moment $\langle U_{i}U_{j}\rangle$ (Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2003). To obtain $\langle U_{i}U_{j}\rangle$ , one writes the transport equation for $\langle U_{i}U_{j}\rangle$ , which in turn involves $\langle U_{i}U_{j}U_{k}\rangle$ , and so on. This leads to an infinite hierarchy of moments equations, which is typically broken by introducing further closure approximations. For example, in Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2003), a ‘quasi-Gaussian’ approximation (QGA) was introduced for $U_{i}$ , allowing them to approximate the fourth-order moments of relative velocities in terms of the second-order moments. Even for $St_{\unicode[STIX]{x1D702}}\gg 1$ particles, QGA may be problematic for separations $r<L$ , where the relative-velocity PDF is far from Gaussian and the second moments may be transported over distances much larger than $r$ . Further, as pointed out by Bragg & Collins (Reference Bragg and Collins2014b ), for larger Stokes numbers, the PDF of $\boldsymbol{U}$ in the dissipation range may be extremely intermittent. These closure approximations may be an important contributing factor to the errors in the RDFs of Zaichik & Alipchenkov. A more detailed analysis of the predictions of the Zaichik & Alipchenkov theory may be found in Bragg & Collins (Reference Bragg and Collins2014a ,Reference Bragg and Collins b ).

The second reason for the differences between the DNS and Zaichik RDFs in figure 6 may be the use of the unified expressions for the diffusivity in their theory. As shown in figure 5(b) and in the discussion following (4.5), the unified expressions used for $\unicode[STIX]{x1D61A}$ and $T_{Lr}$ need improvement in the inertial region, as well as the transition region between the inertial and integral ranges. The discrepancies in the diffusivity in these regions may have contributed to the overprediction of RDFs by their theory, since the high $St_{\unicode[STIX]{x1D702}}$ particles preferentially respond to these scales.

Figure 8.

RDFs from Langevin simulations (CF1, CF2 and CF3) and from DNS as a function of dimensionless pair separation $r/\unicode[STIX]{x1D702}$ for the indicated values of $St_{\unicode[STIX]{x1D702}}=10$ , 80. (a,b)  $Re_{\unicode[STIX]{x1D706}}=76$ , and (c,d) $Re_{\unicode[STIX]{x1D706}}=131$ .

Figure 8(a,b) compare the RDFs obtained using CF1, CF2 and CF3 with the DNS RDF for $St_{\unicode[STIX]{x1D702}}=10$ and 80, respectively, at $Re_{\unicode[STIX]{x1D706}}=76$ . Figure 8(c,d) show the corresponding plots at $Re_{\unicode[STIX]{x1D706}}=131$ . For $St_{\unicode[STIX]{x1D702}}=10$ , the CF1 and CF3 RDFs show good qualitative and quantitative agreement with the DNS RDF at both Reynolds numbers. For $St_{\unicode[STIX]{x1D702}}=10$ and both $Re_{\unicode[STIX]{x1D706}}$ , CF2 overpredicts clustering as compared to DNS for both viscous and inertial separations. For $St_{\unicode[STIX]{x1D702}}=80$ , all the RDFs are close to unity, suggesting only a small amount of particle clustering. One also notices that the RDFs plateau, i.e. become essentially independent of $r$ , for separations in the inertial subrange. The plateauing is delayed, i.e. starts at smaller separations, for the $St_{\unicode[STIX]{x1D702}}=10$ pairs than for the $St_{\unicode[STIX]{x1D702}}=80$ pairs. The above RDF trends can be deduced by considering the governing equation for $\langle U_{i}\rangle$ , given by:

For isotropic turbulence, $\langle U_{i}\rangle =0$ , so that (4.20) becomes

where $\unicode[STIX]{x1D6FC}=1,2,3$ (repeated $\unicode[STIX]{x1D6FC}$ does not denote a summation). We can now write

which yields the rather elegant result for $St_{r}\gg 1$ pairs:

where $C(St)$ is an unknown coefficient that depends on the Stokes number. This dependence of the RDF may be contrasted with the power-law scaling of the RDF for the $St_{\unicode[STIX]{x1D702}}\ll 1$ particles at separations $r\lesssim \unicode[STIX]{x1D702}$ (Chun et al. Reference Chun, Koch, Rani, Ahluwalia and Collins2005):

where $C_{3}\sim St_{\unicode[STIX]{x1D702}}$ .

Figure 9.

$\langle U^{2}\rangle /u_{rms}^{2}$ as a function of $r/L$ for all Stokes numbers. (a) $Re_{\unicode[STIX]{x1D706}}=76$ and (b)  $Re_{\unicode[STIX]{x1D706}}=131$ . Lines denote CF1 and symbols denote CF2.

From (4.21), we can readily see that the flattening of the RDFs is related to the flattening of $\langle U^{2}\rangle$ . By reading figure 8(a,c) in conjunction with figure 9(a,b), one can see that the delayed plateauing of RDF for the $St_{\unicode[STIX]{x1D702}}=10$ particles is related to the correspondingly delayed plateauing of $\langle U^{2}\rangle$ at both Reynolds numbers. The overprediction of RDFs by CF2 for the viscous and inertial separations may also be inferred from (4.23) which shows that the RDF is inversely proportional to the relative-velocity variance. Since CF2 yields the lowest variances among the three closures, we see the associated overprediction of RDFs.

Figure 9(a,b) also suggest that $\langle U^{2}\rangle$ remains finite as the separation $r\rightarrow 0$ for all the $St_{\unicode[STIX]{x1D702}}$ considered in this study. Bec et al. (Reference Bec, Biferale, Cencini, Lanotte and Toschi2010) performed a DNS study of the low-order velocity structure functions of inertial particles in isotropic turbulence, and found that the structure functions were independent of separation in the dissipation range for $St_{\unicode[STIX]{x1D702}}>7$ . In Rani et al. (Reference Rani, Dhariwal and Koch2014), we had classified high-inertia particle pairs into ‘lingerers’ and ‘flyers’. Lingerers are low-relative-velocity particles that are highly correlated and remain correlated far longer than the time scales of fluid that influence their relative motion. Flyers are uncorrelated particles with large relative velocities, i.e. they undergo essentially ballistic motion so that their relative motion is unaffected by fluid eddies with sizes comparable to the pair separation. Flyers are responsible for maintaining a finite $\langle U^{2}\rangle$ as the separation $r\rightarrow 0$ .

Figure 10.

RDF versus $St_{\unicode[STIX]{x1D702}}$ at $Re_{\unicode[STIX]{x1D706}}=76$ and at specific pair separations: (a) $r/\unicode[STIX]{x1D702}=1$ , (b)  $r/\unicode[STIX]{x1D702}=11$ , (c) $r/\unicode[STIX]{x1D702}=22$ and (d) $r/\unicode[STIX]{x1D702}=44$ ( ${\approx}L$ ). CF1, CF2, CF3, and DNS are compared.

Figure 11.

RDF versus $St_{\unicode[STIX]{x1D702}}$ at $Re_{\unicode[STIX]{x1D706}}=131$ and at specific pair separations: (a) $r/\unicode[STIX]{x1D702}=1$ , (b)  $r/\unicode[STIX]{x1D702}=23$ , (c) $r/\unicode[STIX]{x1D702}=46$ and (d) $r/\unicode[STIX]{x1D702}=92$ ( ${\approx}L$ ). CF1, CF2, CF3, and DNS are compared.

In figure 10, the RDFs from CF1, CF2, CF3 and DNS are plotted as a function of $St_{\unicode[STIX]{x1D702}}$ at four separations for $Re_{\unicode[STIX]{x1D706}}=76$ . The corresponding plots for $Re_{\unicode[STIX]{x1D706}}=131$ are shown in figure 11. In general, the CF1 RDFs show the best agreement with the DNS RDFs. At larger separations ( $r=L/2,L$ ), one notices that the agreement of CF1 with DNS improves with Stokes number, while the agreement of CF2 and CF3 with DNS deteriorates at higher Stokes numbers. This may be attributed to the underprediction of $\unicode[STIX]{x1D60B}_{UU}$ at large separations by CF2 and CF3. For lower Stokes numbers at both $Re_{\unicode[STIX]{x1D706}}$ , CF3 shows better agreement with DNS than CF2. At higher Stokes numbers, CF2 and CF3 approach each other, as is to be expected.

Figure 12.

RDF versus pair separation $r/\unicode[STIX]{x1D702}$ for (a) $St_{\unicode[STIX]{x1D702}}=10$ and (b) $St_{\unicode[STIX]{x1D702}}=20$ . Black curves are for $Re_{\unicode[STIX]{x1D706}}=131$ , and grey curves are for $Re_{\unicode[STIX]{x1D706}}=76$ . CF1, CF3 and DNS are compared.

In figure 12, the effects of $Re_{\unicode[STIX]{x1D706}}$  on particle accumulation are shown by comparing the RDFs for $St_{\unicode[STIX]{x1D702}}=10$ , 20 at $Re_{\unicode[STIX]{x1D706}}=76$ and 131. The RDFs are shown for the two lower Stokes numbers (and for CF1 and CF3 only) so as to clearly illustrate the effects of $Re_{\unicode[STIX]{x1D706}}$  variation on clustering. It can be seen from figure 12(a,b) that for both Stokes numbers, the RDFs increase with $Re_{\unicode[STIX]{x1D706}}$ . This is because the response times of the particles considered are of the order of inertial time scales. An increase in $Re_{\unicode[STIX]{x1D706}}$ , with its concomitant broadening of the inertial subrange, would mean that the particles respond to a greater number of scales, thereby resulting in increased clustering at these separations. This observation is consistent with the findings in the study of Ireland et al. (Reference Ireland, Bragg and Collins2015).

4.3 Pair relative-velocity statistics

Figure 13.

$\langle U^{2}\rangle /u_{\unicode[STIX]{x1D702}}^{2}$ versus $St_{\unicode[STIX]{x1D702}}$ at $Re_{\unicode[STIX]{x1D706}}=76$ for various separations. (a) $r/L=1/20$ , (b)  $r/L=1/10$ , (c) $r/L=1/2$ and (d) $r/L=1$ . Dashed line in (d) corresponds to the analytical expression for the variance of uncorrelated pairs, $\langle U^{2}\rangle =2u_{rms}^{2}(T_{L}/(T_{L}+\unicode[STIX]{x1D70F}_{v}))$ , where $T_{L}$ is the Lagrangian integral time scale. $T_{L}$ is obtained using $T_{L}=T_{E}/1.1$ (Pan & Padoan Reference Pan and Padoan2013).

Figure 13 shows the pair relative-velocity variance as a function of Stokes number at four separations for $Re_{\unicode[STIX]{x1D706}}=76$ . The variances obtained from the Langevin simulations using the three diffusivities are compared with the DNS variances. It is seen that CF1 shows the best agreement with the DNS for all Stokes numbers and separations. CF3 agrees better with DNS than does CF2, especially at the smaller Stokes numbers considered. One also notices a slow change in the variances from $r\approx 4.5\unicode[STIX]{x1D702}$ in figure 13(b) to $r\approx 2.5\unicode[STIX]{x1D702}$ in figure 13(a). This may be attributed to the high inertia of particles because of which they retain memory of their relative velocities even after their separations have transitioned from the inertial range to the dissipative range. In figure 13(c,d), i.e. at $r/L=1/2$ and 1 respectively, one can see that CF1 overpredicts the DNS variances for $St_{\unicode[STIX]{x1D702}}=10$ , but the comparison improves significantly for $St_{\unicode[STIX]{x1D702}}>10$ . This is to be expected since CF1 involves modelling $\unicode[STIX]{x1D60B}_{UU}$ as the time integral of the two-time correlation of fluid relative velocities seen by nearly stationary pairs. Thus, CF1 becomes more accurate at high Stokes numbers where the modelled pair dynamics approaches the pair behaviour in DNS. In figure 13(d), corresponding to $r\approx L$ , we also show the analytical expression for the relative velocity variance of uncorrelated pairs, $\langle U^{2}\rangle =2u_{rms}^{2}T_{L}/(T_{L}+\unicode[STIX]{x1D70F}_{v})$ , where $T_{L}$ is the Lagrangian integral time scale of turbulence (Pan & Padoan Reference Pan and Padoan2013). Except at $St_{\unicode[STIX]{x1D702}}=10$ , CF1 shows excellent agreement with this expression.

In figure 13(d), the CF1 to CF2 and CF1 to CF3 variance ratios are 2.12 and 1.52 for $St_{\unicode[STIX]{x1D702}}=10$ , and 2.13 and 2.00 for $St_{\unicode[STIX]{x1D702}}=80$ . As already seen in figure 3, for $St_{\unicode[STIX]{x1D702}}=10$ and $r\gtrsim L$ , we see that $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[2]}\approx 2.05$ and $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[3]}\approx 1.50$ . For $St_{\unicode[STIX]{x1D702}}=80$ and $r\gtrsim L$ , figure 4 shows that $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[2]}$ and $\unicode[STIX]{x1D60B}_{UU}^{[1]}/\unicode[STIX]{x1D60B}_{UU}^{[3]}$ are 2.05 and 1.93, respectively. The correspondence between the variances and diffusivities at integral-scale separations becomes evident from these ratios. From (4.1) and (4.4), when $r\gtrsim L$ , the relative-velocity variance limits to $(2/\unicode[STIX]{x1D70F}_{v})u_{rms}^{2}T_{E}$ for CF1, and to $(0.53/\unicode[STIX]{x1D70F}_{v})u_{rms}^{2}T_{eddy}$ for CF2.

Figure 14.

$\langle U^{2}\rangle /u_{\unicode[STIX]{x1D702}}^{2}$ versus $St_{\unicode[STIX]{x1D702}}$ at $Re_{\unicode[STIX]{x1D706}}=131$ for various separations. (a) $r/L=1/20$ , (b)  $r/L=1/10$ , (c) $r/L=1/2$ and (d) $r/L=1$ . Dashed line in (d) corresponds to $\langle U^{2}\rangle =2u_{rms}^{2}(T_{L}/(T_{L}+\unicode[STIX]{x1D70F}_{v}))$ , where $T_{L}$ is the Lagrangian integral time scale. $T_{L}$ is obtained using $T_{L}=T_{E}/1.1$ (Pan & Padoan Reference Pan and Padoan2013).

In figure 14, the relative-velocity variances computed using CF1, CF2 and CF3 are compared with the DNS variances for $Re_{\unicode[STIX]{x1D706}}=131$ . Both CF2 and CF3 underpredict the DNS variances, although CF3 performs better for smaller Stokes numbers. In contrast to its behaviour at $Re_{\unicode[STIX]{x1D706}}=76$ , CF1 now overpredicts the DNS variances, although the comparison gets better as the Stokes number increases. The improved agreement at higher $St_{\unicode[STIX]{x1D702}}$ is expected, since the validity of the principal approximation in CF1 – pairs are essentially fixed during flow time scales – improves as the Stokes number increases. It will also be seen in subsequent discussion that the CF1 behaviour at $Re_{\unicode[STIX]{x1D706}}=131$ can be explained by considering the effects of $Re_{\unicode[STIX]{x1D706}}$ on the diffusivity $\unicode[STIX]{x1D60B}_{UU}$ . In figure 14(d), at $r/L=1$ , CF1 approaches the analytical limit, except for $St_{\unicode[STIX]{x1D702}}=10$ .

Figure 15.

$\langle U^{2}\rangle /u_{rms}^{2}$ versus $St_{\unicode[STIX]{x1D702}}$ . (a) $r/L=1/20$ and (b) $r/L=1$ . Open symbols denote $Re_{\unicode[STIX]{x1D706}}=131$ and filled symbols $Re_{\unicode[STIX]{x1D706}}=76$ .

The effects of $Re_{\unicode[STIX]{x1D706}}$ on the relative-velocity variances are illustrated in figure 15. In figure 15(a), the variances obtained from CF1, CF3 and DNS are compared for $Re_{\unicode[STIX]{x1D706}}=76$ , and 131 at $r=L/20$ . The corresponding comparison of variances for $r=L$ is shown in figure 15(b). At both separations, increase in $Re_{\unicode[STIX]{x1D706}}$  has only a marginal impact on the DNS variances. The variation of $Re_{\unicode[STIX]{x1D706}}$ , however, has a substantial effect on the CF1 variances at both separations. This can be attributed to the strong dependence of the CF1 diffusivity on $Re_{\unicode[STIX]{x1D706}}$ , as will be demonstrated in the following discussion. The CF3 variances also show a rather weak dependence on $Re_{\unicode[STIX]{x1D706}}$ . This behaviour of CF3, as compared to CF1, seems surprising, but will also be explained below.

We will now elucidate the trends in figures 14 and 15, specifically those concerning the effects of increase in $Re_{\unicode[STIX]{x1D706}}$ on the CF1 and CF3 variances. First, we present the ratios of dimensionless diffusivities at $Re_{\unicode[STIX]{x1D706}}=131$ and $Re_{\unicode[STIX]{x1D706}}=76$ , where the diffusivities are normalized using integral scale quantities. These ratios are (at $St_{\unicode[STIX]{x1D702}}=10$ and $r>L$ ):

where $\widetilde{\unicode[STIX]{x1D60B}}_{UU}=\unicode[STIX]{x1D60B}_{UU}/(u_{rms}^{2}\times T_{eddy})$ , and the values of $u_{rms}$ and $T_{eddy}=L/u_{rms}$ for the respective $Re_{\unicode[STIX]{x1D706}}$ are obtained from table 1. It is interesting to note that at large separations, the CF1 diffusivity shows a significant increase, while the CF3 diffusivity shows a marginal decrease with the Reynolds number.

The principal reason for the strong and weak dependence of CF1 and CF3, respectively, on $Re_{\unicode[STIX]{x1D706}}$ may be attributed to the relevant time scales for these closures. For CF1, the relevant time scale at large separations is the Eulerian integral time scale, $T_{E}$ , while for CF2 and CF3, the time scales are $L/u_{rms}$ and $L/W_{rms}$ respectively. It is clear from table 1 that $L$ and $u_{rms}$ , and thereby $T_{eddy}$ , do not change significantly between the two DNS runs. This is because the turbulent kinetic energy also changes slowly in the two DNS runs. Consequently, the integral length scale $L$ , which is determined by the mean dissipation rate and $u_{rms}$ , is also nearly the same in the two DNS runs. However, for CF1, we find that $T_{E}(Re_{\unicode[STIX]{x1D706}}=131)/T_{E}(Re_{\unicode[STIX]{x1D706}}=76)\approx 1.4$ , which is close to the ratio of diffusivities $\unicode[STIX]{x1D60B}_{UU}^{[1]}(Re_{\unicode[STIX]{x1D706}}=131)/\unicode[STIX]{x1D60B}_{UU}^{[1]}(Re_{\unicode[STIX]{x1D706}}=76)$ for $r>L$ . Since $\unicode[STIX]{x1D60B}_{UU}^{[1]}(r\gtrsim L)=\langle U^{2}\rangle (r\gtrsim L)/\unicode[STIX]{x1D70F}_{v}$ , an increase in the diffusivity directly results in higher relative-velocity variances at large separations. The ‘flyer’ pairs with high variances at large separations then result in increased variances at smaller separations through their ballistic motion. We believe that the increase in $T_{E}$ may be due to the forcing methodology used to achieve stationary isotropic turbulence. As in Ireland et al. (Reference Ireland, Vaithianathan, Sukheswalla, Ray and Collins2013), we use a deterministic forcing method that involves resupplying the energy dissipated during each simulation time step. The dissipated energy is added at low wavenumbers so that the medium and high wavenumbers are, hopefully, not significantly influenced by the forcing. From the current DNS runs, it seems that the deterministic forcing of small wavenumbers has the effect of increasing the temporal coherence of large-scale eddies as $Re_{\unicode[STIX]{x1D706}}$ is increased.

Figure 16.

Dimensionless $\unicode[STIX]{x1D60B}_{UU}\times St_{\unicode[STIX]{x1D702}}^{2}$ is plotted as a function of $r/\unicode[STIX]{x1D702}$ for $Re_{\unicode[STIX]{x1D706}}=76,131$ . Effects of $Re_{\unicode[STIX]{x1D706}}$ on the CF1 closure are shown. The longitudinal and transverse components of $\unicode[STIX]{x1D60B}_{UU}^{[1]}$ are compared at the two $Re_{\unicode[STIX]{x1D706}}$ . $\unicode[STIX]{x1D60B}_{UU}$ is made dimensionless with the Kolmogorov-scale quantities, and then multiplied with $St_{\unicode[STIX]{x1D702}}^{2}$ .

We now present further illustration of the effects of increasing $Re_{\unicode[STIX]{x1D706}}$ on the CF1 closure of $\unicode[STIX]{x1D60B}_{UU}$ . In figure 16, the dimensionless $\unicode[STIX]{x1D60B}_{UU}^{[1]}\times St_{\unicode[STIX]{x1D702}}^{2}$ is plotted as a function of $r/\unicode[STIX]{x1D702}$ at $Re_{\unicode[STIX]{x1D706}}=76$ and $131$ , where $\unicode[STIX]{x1D60B}_{UU}$ has been made dimensionless using the Kolmogorov-scale quantities.

In CF1, we have

The effects of $Re_{\unicode[STIX]{x1D706}}$ on CF1 can be understood by approximating the Eulerian two-time relative-velocity correlation on the right-hand side of (4.27) as the product of the Eulerian two-point structure function and an Eulerian autocorrelation of fluid relative velocities. This allows us to write:

where $\unicode[STIX]{x1D61A}(\boldsymbol{r})$ is the structure function, and $T_{r}$ is the Eulerian two-point time scale at separation $r$ . In both viscous and inertial ranges, it can be shown from (4.6)–(4.7) and (4.9)–(4.11) that $\unicode[STIX]{x1D61A}(\boldsymbol{r})T_{r}/(u_{\unicode[STIX]{x1D702}}^{2}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})$ is independent of $Re_{\unicode[STIX]{x1D706}}$ .

In the integral range, however, $\unicode[STIX]{x1D61A}_{||}/u_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D61A}_{\bot }/u_{\unicode[STIX]{x1D702}}^{2}\sim (u^{\prime }/u_{\unicode[STIX]{x1D702}})^{2}$ . Using $u^{\prime }/u_{\unicode[STIX]{x1D702}}$ values from Yeung & Pope (Reference Yeung and Pope1989), one arrives at the following scaling: $u^{\prime }/u_{\unicode[STIX]{x1D702}}=0.50505\sqrt{Re_{\unicode[STIX]{x1D706}}}-0.0061$ for $38\leqslant Re_{\unicode[STIX]{x1D706}}\leqslant 93$ . Using the DNS data of Ireland et al. (Reference Ireland, Bragg and Collins2015), we get the scaling: $u^{\prime }/u_{\unicode[STIX]{x1D702}}=0.50626\sqrt{Re_{\unicode[STIX]{x1D706}}}+0.018761$ for $88\leqslant Re_{\unicode[STIX]{x1D706}}\leqslant 597$ , which to leading order is nearly identical to the Young & Pope scaling. For the Eulerian integral time scale $T_{E}$ , we obtain the following scaling from Ireland et al. (Reference Ireland, Bragg and Collins2015): $T_{E}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=0.1039Re_{\unicode[STIX]{x1D706}}+2.8525$ . Therefore, at integral-scale separations, the $Re_{\unicode[STIX]{x1D706}}$ dependence of $\unicode[STIX]{x1D60B}_{UU}^{[1],non\text{-}dim}$ arises from both $\unicode[STIX]{x1D61A}$ and $T_{r}$ . It can be seen in figure 16 that in the dissipative and inertial ranges, the $Re_{\unicode[STIX]{x1D706}}$ effects on $\unicode[STIX]{x1D60B}_{UU}^{[1],non\text{-}dim}$ are weak, but in the integral range, increase in $Re_{\unicode[STIX]{x1D706}}$ dependence leads to the higher values of diffusivity for $Re_{\unicode[STIX]{x1D706}}=131$ .

Figure 17.

Relative-velocity PDF $\unicode[STIX]{x1D6FA}(U|r)$ normalized by $\langle U^{2}\rangle ^{1/2}$ for $Re_{\unicode[STIX]{x1D706}}=76$ and at $r/L=1/20$ . (a) $St_{\unicode[STIX]{x1D702}}=10$ and (b) $St_{\unicode[STIX]{x1D702}}=80$ . Grey line represents the normal distribution.

In figure 17, the PDF of relative velocity conditioned on separation $r$ , i.e.  $\unicode[STIX]{x1D6FA}(U|r)$ , normalized by $\langle U^{2}\rangle ^{1/2}$ is presented at $r=L/20$ for $St_{\unicode[STIX]{x1D702}}=10$ , 80 and $Re_{\unicode[STIX]{x1D706}}=76$ . The normalization of a PDF by the standard deviation sheds light on the shape of the PDF, e.g. its deviation from Gaussianity. In figure 17, the PDFs obtained from LS using CF1 and CF3 are compared with the DNS PDFs. It is known that for separations of the order of the integral length scale $L$ , particle pairs are effectively uncorrelated so that they behave like individual particles. In such a scenario, the relative-velocity PDF should be Gaussian. Indeed, this is what we see in the DNS, as well as LS (figure not shown here). As seen in figure 17, at smaller separations, the PDFs become non-Gaussian (increasingly so as the separation decreases). The non-Gaussianity is manifested in the form of PDFs with wider tails and sharper peaks than a Gaussian PDF. The wider tails are representative of uncorrelated pairs with high relative velocities, referred to as flyers. The higher peaks at low relative velocities are representative of pairs whose relative motion is strongly correlated, referred to as lingerers. Thus, the transition from a Gaussian PDF at large separations to non-Gaussian PDF at small separations is characterized by two distinct trends: flyers that become lingerers, and flyers that remain as flyers. At $r=L/20$ , the transition of pairs into lingerers is more prominent for the $St_{\unicode[STIX]{x1D702}}=10$ pairs than for the $St_{\unicode[STIX]{x1D702}}=80$ pairs, as manifested by the higher peak in figure 17(a). This is because the smaller $St_{\unicode[STIX]{x1D702}}$ particles relax to the local flow more effectively than do the higher $St_{\unicode[STIX]{x1D702}}$ particles that still retain some memory of their ballistic motion at larger separations. For both Stokes numbers, the PDFs of CF1 and CF3 have wider tails than the DNS PDF, suggesting that they overpredict the number of flyers. For $St_{\unicode[STIX]{x1D702}}=10$ , the inset of figure 17(a) shows that both CF1 and CF3 underpredict the occurrence of lingerers.

Figure 18.

Relative-velocity PDF $\unicode[STIX]{x1D6FA}(U|r)$ scaled by $u_{rms}$ for $Re_{\unicode[STIX]{x1D706}}=76$ and at $r/L=1/20$ . (a) $St_{\unicode[STIX]{x1D702}}=10$ , and (b) $St_{\unicode[STIX]{x1D702}}=80$ .

Figure 19.

Relative-velocity PDF $\unicode[STIX]{x1D6FA}(U|r)$ normalized by $\langle U^{2}\rangle ^{1/2}$ for $Re_{\unicode[STIX]{x1D706}}=131$ and at $r/L=1/20$ . (a) $St_{\unicode[STIX]{x1D702}}=10$ and (b) $St_{\unicode[STIX]{x1D702}}=80$ . Grey line represents the normal distribution.

Figure 20.

Relative-velocity PDF $\unicode[STIX]{x1D6FA}(U|r)$ scaled by $u_{rms}$ for $Re_{\unicode[STIX]{x1D706}}=131$ and at $r/L=1/20$ . (a) $St_{\unicode[STIX]{x1D702}}=10$ , and (b) $St_{\unicode[STIX]{x1D702}}=80$ .

The PDF $\unicode[STIX]{x1D6FA}(U|r)$ scaled by the turbulence intensity $u_{rms}$ is shown in figure 18. The PDFs obtained using CF1 and CF3 are compared with those computed from DNS. These PDFs enable us to understand the trends in relative-velocity statistics such as the variance $\langle U^{2}\rangle$ . Among the closures, the CF1 PDFs show the best agreement with the DNS PDFs. At $r=L/20$ , the CF1 PDFs for $St_{\unicode[STIX]{x1D702}}=10$ have wider tails than the corresponding DNS PDFs, which leads to an overprediction of variance by CF1, as seen in figure 13(a). For $St_{\unicode[STIX]{x1D702}}=80$ , however, we see that the CF1 PDF has narrower tails than the DNS PDF, which explains the lower CF1 variance compared to the DNS variance in figure 13(d). These trends suggest that CF1 overpredicts (underpredicts) the number of high-relative-velocity flyers at low (high) Stokes numbers. At both Stokes numbers, the CF3 PDFs are narrower than the DNS PDFs.

In figure 19, we present the normalized relative-velocity PDFs at $Re_{\unicode[STIX]{x1D706}}=131$ . For both $St_{\unicode[STIX]{x1D702}}=10$ and $St_{\unicode[STIX]{x1D702}}=80$ , CF1 shows good agreement with DNS for low to moderate relative velocities ( $-4\lesssim U/\langle U^{2}\rangle ^{1/2}\lesssim 4$ ), but has wider tails than DNS for higher relative velocities. For $St_{\unicode[STIX]{x1D702}}=10$ , when the relative velocity $U/\langle U^{2}\rangle ^{1/2}\sim \pm 2$ , one notices an inflection point in the DNS PDF. It is interesting to note that CF1 captures the inflection point, but not CF3. Further, CF3 predicts lower peaks than DNS for $St_{\unicode[STIX]{x1D702}}=10$ , and higher peaks than DNS for $St_{\unicode[STIX]{x1D702}}=80$ .

The PDF $\unicode[STIX]{x1D6FA}(U|r)$ scaled by $u_{rms}$ for $Re_{\unicode[STIX]{x1D706}}=131$ is shown in figure 20. We see that CF1 gives rise to PDFs with wider tails than does DNS, particularly for $St_{\unicode[STIX]{x1D702}}=10$ . This is consistent with the significant overprediction of variances by CF1 for low Stokes numbers at $Re_{\unicode[STIX]{x1D706}}=131$ (figure 14). However, for $St_{\unicode[STIX]{x1D702}}=80$ , the CF1 PDF shows reasonable agreement with the DNS PDF. For $St_{\unicode[STIX]{x1D702}}=80$ , the CF3 PDF is narrower than the DNS PDF. As a result, CF3 underpredicts the DNS variances.

Figure 21.

PDF of radial relative velocity $\unicode[STIX]{x1D6FA}(U_{r}|r)$ scaled by $u_{rms}$ for $St_{\unicode[STIX]{x1D702}}=10,80$ at $Re_{\unicode[STIX]{x1D706}}=76$ and at specific separations: (a) $r/L=1/20$ and (b) $r/L=1$ .

Next, we present in figure 21 the PDF of the radial component of relative velocity $U_{r}=\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{r}/r$ at $Re_{\unicode[STIX]{x1D706}}=76$ . The PDF $\unicode[STIX]{x1D6FA}(U_{r}|r)$ is of interest since it is a key input to the collision kernel. We compare CF1 and CF3 with the DNS for $St_{\unicode[STIX]{x1D702}}=10$ and 80 at two separations $r=L/20$ and $L$ . The most important property of these PDFs is the transition from a negatively skewed PDF at $r=L$ to a nearly symmetric PDF at $r=L/20$ . The transition in skewness suggests that the clustering of high-inertia particles at small separations is driven by the inward-migration bias occurring at much larger separations. At both $r=L/20$ and $L$ , the PDFs for the $St_{\unicode[STIX]{x1D702}}=10$ particles are more negatively skewed than those for the $St_{\unicode[STIX]{x1D702}}=80$ particles. This means that lower Stokes number particles tend to have higher radially inward relative velocities, which may cause the increased clustering of these particles. For $r=L/20$ and $St_{\unicode[STIX]{x1D702}}=10$ , the CF1 PDF is more negatively skewed than the DNS PDF, which explains the higher RDF for CF1 in figure 10. The CF3 PDF is less negatively skewed than the DNS PDF, and hence the lower RDF of CF3.

The transition from Gaussian to non-Gaussian relative velocities can also be demonstrated using the kurtosis of relative velocity, $\langle U^{4}\rangle /\langle U^{2}\rangle ^{2}$ . A kurtosis of 3 denotes a Gaussian distribution, and a deviation from this value is indicative of a non-Gaussian PDF. In figure 22(a,b), the kurtosis is plotted as a function of separation $r/L$ for the various $St_{\unicode[STIX]{x1D702}}$ at $Re_{\unicode[STIX]{x1D706}}=76$ and 131. In figure 22(a), at $Re_{\unicode[STIX]{x1D706}}=76$ , we see that CF1 and CF3 compare well with DNS at larger separations. For smaller pair separations ( $r/L\lesssim 0.5$ ), CF1 shows the best agreement with DNS. It is also evident that for $r\sim L$ , kurtosis approaches 3, indicating the Gaussianity of relative velocities. At smaller $r$ , the kurtosis for the $St_{\unicode[STIX]{x1D702}}=80$ pairs deviates more slowly from 3 when compared to the $St_{\unicode[STIX]{x1D702}}=10$ pairs. This suggests that the relative motion of the former pairs retains the ballistic nature for a wider range of separations as compared to the latter pairs. In figure 22(b), for $Re_{\unicode[STIX]{x1D706}}=131$ and $St_{\unicode[STIX]{x1D702}}=10$ , the CF1 kurtosis exceeds the CF3 kurtosis at all separations. As already seen, the increased $Re_{\unicode[STIX]{x1D706}}$  leads to a noticeably higher CF1 diffusivity at large separations, which in turn impacts the CF1 kurtosis. When we compare figure 22(a,b), it can be seen that the non-Gaussianity increases with Reynolds number.

Figure 22.

Kurtosis as a function of dimensionless pair separation $r/L$ at: (a) $Re_{\unicode[STIX]{x1D706}}=76$ , and (b) $Re_{\unicode[STIX]{x1D706}}=131$ . CF1, CF3 and DNS are compared for $St_{\unicode[STIX]{x1D702}}=10$ and 80.

4.4 Collision kernel

The collision kernel $K(\unicode[STIX]{x1D70E})$ for monodisperse particles is (Ray & Collins Reference Ray and Collins2011):

where $\unicode[STIX]{x1D70E}$ is the particle diameter, $g(\unicode[STIX]{x1D70E})$ the RDF of particle pairs at contact, $U_{r}=\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{r}/r$ is the radial component of relative velocity and $P(U_{r}|\unicode[STIX]{x1D70E})$ is the PDF of $U_{r}$ at contact. Equation (4.29) shows that the collision kernel depends on the RDF and the PDF of $U_{r}$ , the former a measure of particle spatial concentration, and the latter a measure of the rate of particle encounters. Figures 23 and 24 show the collision kernels for all $St_{\unicode[STIX]{x1D702}}$ and at $Re_{\unicode[STIX]{x1D706}}=76$ and 131, respectively. It is relevant to mention that the ‘collision kernels’ for CF1, CF3 and the current DNS are presented at separation $r=2.3\unicode[STIX]{x1D702}$ for $Re_{\unicode[STIX]{x1D706}}=76$ , and $r=4.7\unicode[STIX]{x1D702}$ for $Re_{\unicode[STIX]{x1D706}}=131$ , since these are the smallest separations for which a statistically stationary $P(U_{r}|r)$ could be computed in the respective Langevin simulations. At both $Re_{\unicode[STIX]{x1D706}}$ , CF1 shows the best agreement with DNS. At $Re_{\unicode[STIX]{x1D706}}=131$ , the comparison of CF1 with DNS improves with Stokes number.

Figure 23.

Collision kernel as a function of Stokes number at $Re_{\unicode[STIX]{x1D706}}=76$ . CF1, CF3 and DNS are compared. Curve 1 shows the collision kernel when the RDF $g(r)=1$ and the PDF $P(U_{r}|r)$ is Gaussian. Curve 2 represents the collision kernel computed using the Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) theory. Curve 3 represents the collision kernel computed using the Abrahamson theory (Abrahamson Reference Abrahamson1975; Zaichik & Alipchenkov Reference Zaichik and Alipchenkov2009). Curve 4 represents the collision kernel from the Mehlig, Uski & Wilkinson (Reference Mehlig, Uski and Wilkinson2007) theory. Curve 5 represents the collision kernel computed using equation (56) of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009). Curves 2 through 4 are at $r/\unicode[STIX]{x1D702}=1$ and $Re_{\unicode[STIX]{x1D706}}=75$ . Curve 5 is at $r/\unicode[STIX]{x1D702}=2.3$ and $Re_{\unicode[STIX]{x1D706}}=76$ .

Figure 24.

Collision kernel as a function of Stokes number at $Re_{\unicode[STIX]{x1D706}}=131$ . CF1, CF3 and DNS are compared.

In figure 23, curve 2 represents the collision kernel at $r=\unicode[STIX]{x1D702}$ computed by Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) from (Wang, Wexler & Zhou Reference Wang, Wexler and Zhou2000)

where the particle diameter $d=\unicode[STIX]{x1D702}$ and $\langle |U_{r}(d)|\rangle =\sqrt{2/\unicode[STIX]{x03C0}\langle U_{r}^{2}(d)\rangle }$ . Curve 3 represents the collision kernel of Abrahamson (Reference Abrahamson1975) (cf. Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009)). Curve 4 represents the collision kernel computed from the theory of Mehlig et al. (Reference Mehlig, Uski and Wilkinson2007). Finally, curve 5 is the kernel computed from (4.30), with $\langle |U_{r}(d)|\rangle$ and $g(d)$ values from the current DNS. It can be seen that the collision kernel of Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) overpredicts the DNS values for $St_{\unicode[STIX]{x1D702}}<20$ , and approaches the DNS for $St_{\unicode[STIX]{x1D702}}>40$ . One also notices that curve 5 obtained from (4.30) is in good agreement with the curve computed from (4.29).

The use of $\langle |U_{r}|\rangle =\sqrt{2/\unicode[STIX]{x03C0}\langle U_{r}^{2}\rangle }$ by Zaichik & Alipchenkov (Reference Zaichik and Alipchenkov2009) assumes that the pair relative velocities are normally distributed. They recognized that such an assumption was, at the best, reasonable only at large Stokes numbers. However, it was shown by Wang, Wexler & Zhou (Reference Wang, Wexler and Zhou1998) that even for zero Stokes number particles, the ratio $\langle |U_{r}|\rangle /\sqrt{\langle U_{r}^{2}\rangle }=0.77$ , which is quite close to $\sqrt{2/\unicode[STIX]{x03C0}}=0.798$ for normally distributed relative velocities.

Also indicated in figures 23 and 24 are the collision kernels when particle pairs are uncorrelated, corresponding to $g(r)=1$ and $P(U_{r}|r)$ being Gaussian. Interestingly, the collision kernels of high Stokes number pairs in isotropic turbulence are smaller than the collision rates of uncorrelated pairs. Since $g(r)>1$ when particles cluster, we can attribute the higher collision kernels for the uncorrelated pairs as arising from the integral over $U_{r}$ in the kernel equation (4.29). This scenario is confirmed in figure 25, where we compare the PDF $P(U_{r}|r)$ for the $St_{\unicode[STIX]{x1D702}}=10,80$ pairs at both $Re_{\unicode[STIX]{x1D706}}$ with the Gaussian PDF for uncorrelated velocities. It is clear that the wider tails of the Gaussian PDF result in the corresponding higher collision kernels. In addition, comparison of figures 23 and 24 shows that the collision kernel increases with $Re_{\unicode[STIX]{x1D706}}$ , due to an increase in both the RDF and the relative velocities (i.e. relative velocity PDFs with wider tails) with $Re_{\unicode[STIX]{x1D706}}$ . This is confirmed in figure 25 where we see that for a given $St_{\unicode[STIX]{x1D702}}$ , the PDF tails become wider as $Re_{\unicode[STIX]{x1D706}}$  is increased. The higher collision kernels for CF1 than CF3 at all $St_{\unicode[STIX]{x1D702}}$ and both $Re_{\unicode[STIX]{x1D706}}$ can also be explained by the PDFs in figure 25.

Figure 25.

Effects of Reynolds number on the PDF of radial relative velocity $\unicode[STIX]{x1D6FA}(U_{r}|r)$ for $St_{\unicode[STIX]{x1D702}}=10,80$ . (a) CF1 closure, and (b) CF3 closure. The PDFs for $Re_{\unicode[STIX]{x1D706}}=76$ are shown for $r=2.3\unicode[STIX]{x1D702}$ , and those for $Re_{\unicode[STIX]{x1D706}}=131$ are for $r=4.7\unicode[STIX]{x1D702}$ .