Turbulent pair dispersion in the presence of gravity

Let us define the particle pair separation $\Delta {\boldsymbol{r}} ={{{\boldsymbol{r}} }_{1}}-{{{\boldsymbol{r}} }_{2}}$, the separation increment ${\boldsymbol{R}} =\Delta {\boldsymbol{r}} (t)-\Delta {\boldsymbol{r}} (0)$, and the relative velocity $\Delta {\boldsymbol{v}} ={{{\boldsymbol{v}} }_{1}}-{{{\boldsymbol{v}} }_{2}}$. Following the work of Batchelor [1], the Taylor expansion of the particle pair dispersion $\langle {{{\boldsymbol{R}} }^{2}}(t)\rangle$ in the short-time limit is

$$\begin{eqnarray}&&\langle {{{\boldsymbol{R}} }^{2}}(t)\rangle =S({{r}_{0}}){{t}^{2}},\end{eqnarray} \tag{ 6 }$$

where $S(r)=\langle {{(\Delta {\boldsymbol{v}} (r))}^{2}}\rangle$ is the heavy particle Eulerian second order structure function and r₀ is the initial separation. The brackets $\langle \cdot \rangle$ indicate an ensemble average over all particle pairs along their trajectories. The monodisperse case has been considered by Bec et al [5], Gibert et al [6] and Salazar and Collins [26]. Here, we treat the bidisperse case and include gravity in our analysis. In the following section, we show that for small separations the effect of gravity on bidisperse particle pair dispersion is an additive term in the particle structure function.

Obtaining the structure function requires formulating the relative velocity $\Delta {\boldsymbol{v}}$ using equation (5) for bidisperse particles, which we obtain as follows

$$\begin{eqnarray}&&\Delta {\boldsymbol{v}} =\Delta {\boldsymbol{u}} -\gamma {{u}_{\eta }}\Delta {\rm St}{\boldsymbol{k}} -{{\tau }_{\eta }}({\rm S}{{{\rm t}}_{1}}{{{\boldsymbol{a}} }_{1}}-{\rm S}{{{\rm t}}_{2}}{{{\boldsymbol{a}} }_{2}}),\end{eqnarray} \tag{ 7 }$$

where $\Delta {\rm St}=|{\rm S}{{{\rm t}}_{1}}-{\rm S}{{{\rm t}}_{2}}|$. The appearance of $\Delta {\rm St}$ necessitates the set of transformations $\overline{{\rm St}}=\frac{1}{2}\;({\rm S}{{{\rm t}}_{1}}+{\rm S}{{{\rm t}}_{2}})$, $\Delta {\boldsymbol{a}} ={{{\boldsymbol{a}} }_{1}}-{{{\boldsymbol{a}} }_{2}}$, and ${\boldsymbol{\bar{a}}} =\frac{1}{2}\;({{{\boldsymbol{a}} }_{1}}+{{{\boldsymbol{a}} }_{2}})$, so that ${\rm S}{{{\rm t}}_{1}}\;{{{\boldsymbol{a}} }_{1}}-{\rm S}{{{\rm t}}_{2}}\;{{{\boldsymbol{a}} }_{2}}$ can be expressed as $\Delta {\rm St}\;{\boldsymbol{\bar{a}}} +\overline{{\rm St}}\;\Delta {\boldsymbol{a}}$. Upon squaring $\Delta {\boldsymbol{v}}$ and averaging it over an ensemble of particle pairs, we arrive at

$$\begin{eqnarray}\begin{array}{rcl} S(r) & = & \langle {{(\Delta {\boldsymbol{u}} )}^{2}}\rangle +{{\gamma }^{2}}\;u_{\eta }^{2}\;{{(\Delta {\rm St})}^{2}}+\tau _{\eta }^{2}\;{{(\Delta {\rm St})}^{2}}\;\langle {{{{\boldsymbol{\bar{a}}} }}^{2}}\rangle -2\;{{\tau }_{\eta }}\;\overline{{\rm St}}\;\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle +\tau _{\eta }^{2}\;{{\overline{{\rm St}}}^{2}}\;\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle \\ {} & {} & -2\;\gamma \;{{u}_{\eta }}\;\Delta {\rm St}\;\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{u}} \rangle -2\;{{\tau }_{\eta }}\;\Delta {\rm St}\;\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{u}} \rangle +2\;\gamma \;\eta \;{{(\Delta {\rm St})}^{2}}\;\langle {\boldsymbol{k}} \cdot {\boldsymbol{\bar{a}}} \rangle \\ {} & {} & +2\;\gamma \;\eta \;\overline{{\rm St}}\;\Delta {\rm St}\;\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{a}} \rangle -2\;\tau _{\eta }^{2}\;\overline{{\rm St}}\;\Delta {\rm St}\;\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{a}} \rangle . \\ \end{array}\end{eqnarray} \tag{ 8 }$$

We shall analyse S(r) in the dissipation range, $r\ll \eta$, because these length scales are most relevant to the cloud droplet collision problem. In this regime, we can apply the dissipation-range scaling for the fluid tracer structure function (see e.g. [19]), $\langle {{(\Delta {\boldsymbol{u}} )}^{2}}\rangle =\frac{1}{3}\;u_{\eta }^{2}\;{{(r/\eta )}^{2}}$.

Equation (8) contains the variance of the mean acceleration term, $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$, which for fluid tracers reduces to $3\;{{a}_{0}}\;a_{\eta }^{2}$, where a₀ is the Heisenberg–Yaglom constant [22, 23]. The case for bidisperse inertial particles does not permit simple analysis. We could, however, obtain some qualitative understanding from the monodisperse case. As noted by various authors experimentally and numerically, see e.g. [9, 10, 27], inertial particle acceleration variance decreases monotonically with Stokes number from fluid tracer acceleration variance. This behavior is attributed to two effects: preferential sampling of inertial particles in regions with high strains and low vorticity, and filtering of high-frequency fluctuations effected by the selective response of the inertial particles to temporal rate of change in the acceleration. In our problem, because the operation $\langle \cdot \rangle$ involves taking the average along inertial particle trajectories, preferential sampling must necessarily play a role in determining the value of $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$.

The term $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle$, which is known to asymptote to the value $-2\;\varepsilon$ in the inertial range (see e.g. [28–30]), has no known analytical expression in the dissipation range. By use of reasonings in the same vein as Kolmogorov's, we anticipate, however, $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle$ to scale as r² in the dissipative range of scales. We note that the variance of the acceleration increment is connected to the acceleration correlation by the relation $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle =\langle {\boldsymbol{a}} _{1}^{2}\rangle +\langle {\boldsymbol{a}} _{2}^{2}\rangle -2\;\langle {{{\boldsymbol{a}} }_{1}}\cdot {{{\boldsymbol{a}} }_{2}}\rangle$. Obukhov and Yaglom [31] have shown that $\langle {{{\boldsymbol{a}} }_{1}}\cdot {{{\boldsymbol{a}} }_{2}}\rangle$ approaches simple scaling laws r² in the dissipation range. In this connection, $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle$ may contain similar scaling laws. The remaining terms containing $\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{u}} \rangle$, $\langle {\boldsymbol{k}} \cdot {\boldsymbol{\bar{a}}} \rangle$, $\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{a}} \rangle$, $\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{u}} \rangle$ and $\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{a}} \rangle$ vanish for fluid tracers in isotropic and zero mean flow. Whether these approximations are reasonable is addressed in section 4.2.

We show in figure 1 the pair dispersion, $\langle {{{\boldsymbol{R}} }^{2}}\rangle$, of bidisperse heavy particles (St = 0.1 and 0.5) in the presence and absence of gravity, as well as that of fluid tracers. The statistics are collected the instant the particles are initialized. In all three cases, $\langle {{{\boldsymbol{R}} }^{2}}\rangle$ scales quadratically with t at short time, in agreement with the Batchelor prediction, as expressed in equation (6). The simulation also confirms that, in the dissipation range, heavy particles in a turbulent flow experiencing gravity separate faster than both fluid tracers and heavy particles without gravity. The essential features of the short-time behavior is captured by the particle structure function, which we discuss in the following section.

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In calculating the structure function, fluid tracer statistics are collected immediately after initialization, while heavy particle statistics are collected only after they have relaxed to a stationary state (typically two relaxation time of the heavier particle after initialization). We have numerically investigated the individual correlations appearing in equation (8) for fluid tracers, the details of which could be found in the supplementary material document. We briefly summarize the results of the our investigation as follows. We find that the most significant term is the acceleration variance, $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$, which we estimated, following the approach of Bec et al [9], by calculating the fluid tracer accelerations measured along inertial particle trajectories for separations in the range $1.4\leqslant r/\eta \leqslant 1.5$. The values of $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$ for various pairing of the Stokes numbers calculated with this approach are shown in table 2. All the variances are below the isotropic limit $3\;{{a}_{0}}$, whose value in our flow is approximately 11.6. These values of $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$ are in qualitative agreement with those estimated by taking the average between the variances for the monodisperse case calculated with the expressions derived by Ayyalasomayajula et al [27] and Zaichik and Alipchenkov [35]. Two additional terms contribute to S(r), namely $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle$ and $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle$, whose scaling in the dissipation range we have empirically determined to follow the power laws $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle =-0.034\;{{u}_{\eta }}\;{{a}_{\eta }}\;{{(r/\eta )}^{2}}$ and $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle =0.27\;a_{\eta }^{2}\;{{(r/\eta )}^{2}}$, respectively (see supplementary material document for further details). The remaining terms $\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{u}} \rangle$, $\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{u}} \rangle$, $\langle {\boldsymbol{k}} \cdot {\boldsymbol{\bar{a}}} \rangle$, $\langle {\boldsymbol{k}} \cdot \Delta {\boldsymbol{a}} \rangle$ and $\langle {\boldsymbol{\bar{a}}} \cdot \Delta {\boldsymbol{a}} \rangle$ are found to have negligible contributions. Using these empirical results, we can derive the following expression for the structure function

$$\begin{eqnarray}&&S(r)/{{u}_{\eta }}^{2}=\left( \frac{1}{3}+0.067\overline{{\rm St}}+0.27\;{{\overline{{\rm St}}}^{2}} \right){{(r/\eta )}^{2}}+{{(\Delta {\rm St})}^{2}}\;(\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle +{{g}^{2}})/{{a}_{\eta }}^{2},\end{eqnarray} \tag{ 9 }$$

valid only for separations in the range $r\ll \eta$. In the above expression, the linear $\overline{{\rm St}}$ term originates from the correlation $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle$, whereas the ${{\overline{{\rm St}}}^{2}}$ term can be traced back to the term $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle$. It is worthwhile to mention that both contributions from $\langle \Delta {\boldsymbol{u}} \cdot \Delta {\boldsymbol{a}} \rangle$ and $\langle {{(\Delta {\boldsymbol{a}} )}^{2}}\rangle$ are essentially negligible for ${\rm St}\ll 1$.

Table 2.  The acceleration variance $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$, in units of ${{a}_{\eta }}$, measured along the trajectories of bidisperse inertial particle pairs.

Stokes number pairing	0.1 and 0.5	0.2 and 0.4	0.4 and 0.5	0.8 and 0.9
$\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle /a_{\eta }^{2}$	4.3	3.1	3.3	2.9

We test equation (9) for fluid tracers, and for heavy particles in the absence and presence of gravity in figure 2. Although not visible on a logarithmic plot, the fluid tracer structure function is exactly zero at zero separation. The measured structure functions for heavy particles, however, maintain a positive offset at zero separation. In the insets, we show the data for the structure function in more detail by dividing out the prediction from equation (9). It is interesting to note that beyond the slight change in $\langle {{{\boldsymbol{\bar{a}}} }^{2}}\rangle$ the structure function is insensitive to $\overline{{\rm St}}$, as evidenced by the Stokes number pairs 0.4 with 0.5, and 0.8 with 0.9. The $\Delta {\rm St}$ in both cases are the same, but their $\overline{{\rm St}}$ are significantly different, 0.45 and 0.85, respectively. Remarkably, not only the numerical results are indistinguishable from each other, but they agree with the highly simplified expression, equation (9). At increasing separation distances, $(r\geqslant \eta )$, the simulation data start to deviate from the prediction. We believe the deviation is due to the dissipation and inertial range transition dynamics, which is not captured by our dissipation range model.

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We digress a bit by noting that the expression in (9) bears some resemblance to the gravitational settling collision rate of Saffman and Turner [36], and to the diffusivity tensor considered by Chun et al [37]. We can establish the link between the velocity structure function and the collision rate, ${\rm d}N/{\rm d}t$, starting with the canonical form for the collision rate

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}=\pi {{n}_{1}}{{n}_{2}}{{({{r}_{1}}+{{r}_{2}})}^{2}}\int |{\boldsymbol{w}} |P({\boldsymbol{w}} ){\rm d}{\boldsymbol{w}} ,\end{eqnarray} \tag{ 10 }$$

where n₁ and n₂ are the mean number concentrations of droplets with two distinct radii r₁ and r₂ respectively, ${\boldsymbol{w}}$ is the relative velocity of the pair just before they collide (which is $\Delta {\boldsymbol{v}}$ in our notation), and $P({\boldsymbol{w}} )$ is the probability distribution of the relative velocities. The integral involving the relative velocity is evaluated over the entire space. Under the assumption that $P({\boldsymbol{w}} )$ is normally distributed, Saffman and Turner showed that this integral can be integrated to yield a collision rate

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}={{n}_{1}}{{n}_{2}}{{({{r}_{1}}+{{r}_{2}})}^{2}}{{\left( \frac{8\pi S}{3} \right)}^{1/2}},\end{eqnarray} \tag{ 11 }$$

where S is the structure function given by equation (9), without the $\overline{{\rm St}}$ corrections, evaluated at $r={{r}_{1}}+{{r}_{2}}$.

The relationship between the structure function and the diffusivity, defined as

$$\begin{eqnarray}&&D=\int _{-\infty }^{t}\langle \Delta {\boldsymbol{v}} (t)\cdot \Delta {\boldsymbol{v}} ({{t}^{\prime }})\rangle {\rm d}{{t}^{\prime }},\end{eqnarray} \tag{ 12 }$$

is less apparent, but can be understood if we consider its short-time behavior. For time scale much shorter than the correlation time of the relative velocities, ${{\tau }_{c}}$, the two-time two-point correlation $\langle \Delta {\boldsymbol{v}} (t)\cdot \Delta {\boldsymbol{v}} ({{t}^{\prime }})\rangle$ is approximately equal to the single-time structure function, $\langle \Delta {\boldsymbol{v}} (t)\cdot \Delta {\boldsymbol{v}} ({{t}^{\prime }})\rangle \approx \langle {{(\Delta {\boldsymbol{v}} (0))}^{2}}\rangle$, and the diffusivity is approximately $D\approx \langle {{(\Delta {\boldsymbol{v}} (0))}^{2}}\rangle \;{{\tau }_{c}}$ (see e.g. Falkovich et al [38]). This implies that the spatial information of the diffusivity is fully contained within the structure function, whereas its temporal information is described by the relative velocity correlation time. Along the same line of analysis, Lu et al [39] extended the theory of Chun et al [37] to include the contribution of gravity and the diffusivity tensor they derived is, up to a multiplicative constant, identical to the expression shown in equation (9).

The foregoing analysis reveals three points about turbulent pair dispersion of bidisperse heavy particles, as summarized in equations (6) and (9). First, the separation of bidisperse particles in the presence of gravity at short time scales remains ballistic. In other words, the square of the separation increment is quadratic in time. Second, the particle differential response to gravity and fluid acceleration at short time scales is fully described by the particle structure function. Third, gravity contributes to the dispersion of heavy particles with the same order of the differential Stokes number as the fluid acceleration. It is perhaps illuminating to mention that in the parameter space we considered, gravity dominates over the fluid acceleration, so that the largest contribution to the dispersion of heavy particles is from gravity.

We note that this result for the structure function is not applicable to the monodisperse case, which has been treated by Gibert et al [6] and Salazar and Collins [26]. Gibert et al [6] measured the pair dispersion of monodisperse solid spherical particles in a laboratory experiment with fully developed turbulence in water at a Taylor Reynolds number of 442. They observed a Batchelor regime in the early stages of the relative motion of particle pairs. Their measurement shows that particle inertia increases the particle structure function but leaves the short-time ballistic scaling unchanged. Salazar and Collins [26] studied pair dispersion in a DNS of homogeneous isotropic turbulent flow laden with monodisperse heavy particles in the absence of gravity and at a Taylor Reynolds number as high as 120. For Stokes numbers 0.2 and higher, their simulations showed that the particle structure function consistently retains a constant positive offset from that of the fluid for separations in the dissipation range. The simple model presented here is unable to predict this offset. Whether it is therefore able to capture the early dispersion is open to question, and is addressed in the next section.

In a paper that preceded the work of Batchelor [1], Obukhov [40] applied Kolmogorov's dimensional argument to explain Richardson's empirical scaling [4] for turbulent pair dispersion, $\langle {{{\boldsymbol{R}} }^{2}}\rangle =g\varepsilon {{t}^{3}}$, valid only in the inertial range of scales. Fluid particles in turbulence are understood to separate ballistically at very short time and gradually transition to the Richardson regime at later time. This change in the separation rates influences, for example, the droplet collision rates and the chemical mixing rates. It is intriguing to investigate whether the presence or absence of gravity affects this transition. We present our measurement of the heavy particle pair dispersion, normalized by the Batchelor short-time solution given by equation (6), in figure 3.

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The figure shows that particles both with and without gravity separate ballistically for very short time but the Batchelor regime for heavy particles in the presence of gravity persists for a longer time than that for heavy particles in the absence of gravity or for tracers. At time scales of the order of the Kolmogorov time, the particles transition into a non-ballistic regime, whose scaling with time we are unable to ascertain accurately due to the limitation in the simulation time.

Qualitatively, we can understand the extension of the ballistic regime for particles experiencing differential sedimentation through the following heuristic argument. The Batchelor regime is defined by selecting the eddy turnover time corresponding to the initial particle separation distance r₀ (assuming the initial separation lies within the inertial range), such that ${{\tau }_{B}}={{(r_{0}^{2}/\varepsilon )}^{1/3}}$. Alternatively, by Kolmogorov reasoning, we could replace the initial separation with ${{r}_{0}}={{(\Delta {\boldsymbol{v}} )}^{3}}/\varepsilon$, where $\Delta {\boldsymbol{v}}$ is the initial differential velocity, to yield ${{\tau }_{B}}={{(\Delta {\boldsymbol{v}} )}^{2}}/\varepsilon$. We can define a differential settling velocity, $\Delta {{{\boldsymbol{v}} }_{g}}=g\;{{\tau }_{\eta }}\;\Delta {\rm St}$, to obtain a 'gravitational Batchelor time'

$$\begin{eqnarray}&&{{\tau }_{g}}=\frac{{{(g\;{{\tau }_{\eta }}\;\Delta {\rm St})}^{2}}}{\varepsilon }.\end{eqnarray} \tag{ 13 }$$

In effect, particles of different size and therefore different terminal speed, are expected to separate ballistically by differential sedimentation until they reach a separation distance at which the corresponding eddy velocity is equal to that sedimentation velocity. At larger scales the turbulent velocity becomes dominant in the pair dispersion. For the bidisperse particles in our simulation ${{\tau }_{B}}=1.26\;{{\tau }_{\eta }}$, whereas ${{\tau }_{g}}=16\;{{\tau }_{\eta }}$. A curve fit to the observed dispersion curves indeed shows that the transition from the linear to higher-order terms occurs at approximately $10\;{{\tau }_{\eta }}$ for the bidisperse particles in the presence of gravity, whereas for both tracers and bidisperse inertial particles without gravity the transition time is approximately $2\;{{\tau }_{\eta }}$. This lends at least semi-quantitative support to the notion of a gravitational Batchelor time for the transition from ballistic to super-ballistic dispersion.

An intriguing feature of figure 3 is the rather distinct change in the negative dip to sub-ballistic dispersion: the pronounced negative dip observed for the tracers nearly disappears for bidisperse inertial particles both with and without gravity. The initial transition to sub-ballistic dispersion for tracers has been observed in experiments by Ouellette et al [2], and they attributed it to the t³ term in the Taylor expansion of the tracer pair dispersion

$$\begin{eqnarray}&&\langle {{{\boldsymbol{R}} }^{2}}(t)\rangle =S({{r}_{0}})\;{{t}^{2}}+\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle \;{{t}^{3}}+\cdots .\end{eqnarray} \tag{ 14 }$$

Equation (14) contains the mixed structure function $\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle$. Mann et al [28, 29] and Hill [30] have shown that $\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle =-2\;\varepsilon$ within the inertial range, thereby explaining the negative behavior. The simulation results presented here show not only the temporary sub-ballistic excursion, but the eventual transition to super-ballistic behavior, presumably due to still higher-order terms in the Taylor expansion. How does the presence of particle inertia and gravity change this behavior?

We gain further insight into this transition by considering both mono and bidisperse particle pairs, in comparison with fluid tracer pairs. It is immediately evident from inspecting figure 3 that monodisperse particles with inertia or with gravity still show pronounced sub-ballistic regimes prior to entering the super-ballistic regime. In contrast, both with and without gravity, the bidisperse particles exhibit strongly suppressed sub-ballistic behavior. Especially for the no-gravity case, there is little difference in the dispersion curves for fluid tracers and for inertial particles; similar can be said for fluid tracers and St = 0.1 monodisperse particles. Dispersion of bidisperse particles shows a more rapid onset of super-ballistic behavior with and without gravity, as compared to fluid tracer and all monodisperse particles; as was observed in the figure 3, bidisperse particles without gravity are fastest to transition to super-ballistic dispersion.

To quantify the very different behavior for bidisperse particles we have computed the zero-time mixed structure function, and to allow for meaningful comparison we have scaled it by the second-order velocity structure function. Particle accelerations in the mixed structure function are obtained with the gaussian filtering and differentiation technique (e.g. [41]). It should be noted that the initial pair separation is not the same for all cases: it is $1.4\;\eta$ for tracer and bidisperse particles, and is $1.4\;\eta \;\sqrt{2}$ for monodisperse particles. The results for all particle-pair cases are displayed in the dimensionless form $-{{\tau }_{\eta }}\;\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle /S({{r}_{0}})$ in table 3. The largest magnitude is for fluid tracers, and both with and without gravity there is a decrease in magnitude with increasing ${\rm St}$. The values for monodisperse particle pairs of a given ${\rm St}$ are nearly the same with and without gravity, and for St = 0.5 the scaled mixed structure function is approximately a factor of 10 smaller than for tracers. In other words, with quite negligible influence of gravity, the mixed velocity–acceleration structure function decreases in magnitude relative to the second-order velocity structure function with increasing particle inertia. Bidisperse particles without gravity have a value that is essentially the mean of the two monodisperse values. This is somewhat surprising given the observation from figure 3(b) that bidisperse inertial particles without gravity exhibit very different transitions from the ballistic to super-ballistic dispersion regimes. Finally, in contrast to the no-gravity case, bidisperse particle pairs in the presence of gravity have the smallest magnitude observed for the scaled mixed structure function.

Table 3.  The normalized coefficients for the linear-t term in the expansion of $\langle {{R}^{2}}(t)\rangle /(S({{r}_{0}})\;{{t}^{2}})$ for fluid tracers, bidisperse (St = 0.1 and 0.5), and monodisperse inertial particles.

Gravity	Stokes number	$-{{\tau }_{\eta }}\;\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle /S({{r}_{0}})$	$\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle /({{\sigma }_{\Delta v}}\;{{\sigma }_{\Delta a}})$
0	0	0.132	−0.1131
0	0.1 and 0.5	0.0204	−0.0283
0	0.1	0.0404	−0.147
0	0.5	0.009 54	−0.153
$10\;{{a}_{\eta }}$	0.1 and 0.5	0.005 14	−0.0325
$10\;{{a}_{\eta }}$	0.1	0.0418	−0.150
$10\;{{a}_{\eta }}$	0.5	0.0122	−0.136

The observed behavior of the mixed structure function for the different cases could be the result of changes in the scaling factor $S({{r}_{0}})$, the magnitude of the acceleration increment $\Delta {\boldsymbol{a}}$, or changes in the correlation between velocity and acceleration increments $\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle$. The latter is perhaps most insightful, and we show the zero-time correlation coefficient $\langle \Delta {\boldsymbol{v}} \cdot \Delta {\boldsymbol{a}} \rangle /({{\sigma }_{\Delta v}}{{\sigma }_{\Delta a}})$ in table 3. Perhaps surprisingly, it is observed that monodisperse inertial particles exhibit stronger anticorrelation than do tracers. This is opposite to the observed trend for the scaled mixed structure function. Both with and without gravity, the correlation coefficient for velocity–acceleration increments diminishes significantly in magnitude (i.e., the anticorrelation becomes weaker) compared to tracers and to monodisperse inertial particles also with and without gravity. The strong decorrelation that results for bidisperse particles is very likely the cause of the striking difference in transition behavior observed in figure 3. We can speculate that the decorrelation is related to that observed in inertial particle clustering, for which it has been clearly observed in simulation and experiment that increasing the gap between the inertial response of particles of two different St induces the two populations to move independently from each other, thereby reducing the overlap between individual clusters (as measured through the radial distribution function) [15, 16]. Even when inertial particles of two different St exhibit considerable self-clustering, the spatial locations of the clusters are not collocated, and therefore the bidisperse correlations are diminished. In the dispersion problem this apparently results in a suppression of the sub-ballistic behavior and an earlier transition to the super-ballistic regime.