2.1 Relative velocities and particle accelerations

To gain insight into the effect of bidispersity and gravity on the particle-pair relative velocities, it is informative to consider the formal solution of (2.5), namely (in the rest of this section we drop $\widetilde{\cdot }$ for notational ease)

where $\unicode[STIX]{x0394}\boldsymbol{u}^{p}(s)\equiv \unicode[STIX]{x0394}\boldsymbol{u}(\boldsymbol{x}^{p}(s),\boldsymbol{r}^{p}(s),s)$ . In (2.6), and throughout, we assume $t\gg St_{2}$ such that initial conditions/effects can be ignored (valid since we are interested in the statistically stationary state of the system).

In the monodisperse case, $\unicode[STIX]{x0394}St=0$ , and only the first integral in (2.6) remains. This case has been studied in great detail using theoretical and numerical approaches (Bec et al. Reference Bec, Biferale, Cencini, Lanotte and Toschi2010; Gustavsson & Mehlig Reference Gustavsson and Mehlig2011; Salazar & Collins Reference Salazar and Collins2012b ; Bragg & Collins Reference Bragg and Collins2014b ; Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014; Ireland, Bragg & Collins Reference Ireland, Bragg and Collins2016a ; Ireland et al. Reference Ireland, Bragg and Collins2016b ). Here we summarize the findings and refer the reader to those papers for detailed discussions.

When $St_{r}$ is finite, the inertial particles preferentially sample the fluid velocity field, showing a tendency to avoid regions of strong rotation due to the centrifuge effect, and accumulate in regions of strain (Maxey Reference Maxey1987). At scales outside of the dissipation range, the same effect occurs and can be understood in terms of the particle interaction with the coarse-grained strain and rotation fields at those scales (Bragg et al. Reference Bragg, Ireland and Collins2015a ). When $St_{r}\geqslant O(1)$ , the dynamics becomes temporally non-local; $\boldsymbol{w}^{p}(t)$ depends upon $\unicode[STIX]{x0394}\boldsymbol{u}^{p}$ along the path history of the particle pair over the time span $t-s\leqslant O(\unicode[STIX]{x1D70F}_{p})$ . In the dissipation and inertial ranges where the statistics of $\unicode[STIX]{x0394}\boldsymbol{u}$ depend upon separation, this can have a dramatic effect. In particular, if the particle separation changes substantially over a time span $O(\unicode[STIX]{x1D70F}_{p})$ , then the particle pairs will remember their interaction with scales of motion of the turbulence that have very different properties from those at their current separation. This ‘path-history effect’ gives rise to ‘caustics’ (Wilkinson & Mehlig Reference Wilkinson and Mehlig2005), where $\Vert \boldsymbol{w}^{p}(t)\Vert \gg \Vert \unicode[STIX]{x0394}\boldsymbol{u}^{p}(t)\Vert$ at small separations. Finally, when $St_{r}\geqslant O(1)$ there also exists the ‘filtering effect’, which describes the fact that due to their inertia, the particles cannot fully respond to all of the turbulent fluctuations and so behave sluggishly.

In the dissipation range, the path-history effect dominates the behaviour of $\boldsymbol{w}^{p}(t)$ when $St_{r}\geqslant O(1)$ . As the separation increases, the path-history effect remains important when $St_{r}\geqslant O(1)$ , but its effect weakens because the separation dependence of $\unicode[STIX]{x0394}\boldsymbol{u}$ weakens. In this case, the preferential sampling and filtering effects play important roles. Finally, at the large scales of homogeneous turbulence, the path-history effect vanishes (since the statistics of $\unicode[STIX]{x0394}\boldsymbol{u}$ no longer depend upon separation), the preferential sampling plays a weak role, and the filtering effect dominates.

When $\unicode[STIX]{x0394}St=0$ , the effect of gravity on $\boldsymbol{w}^{p}(t)$ is only implicit; gravity affects $\unicode[STIX]{x0394}\boldsymbol{u}^{p}$ by changing the way the particles interact with the turbulent flow, and most importantly, by reducing the correlation time scale of $\unicode[STIX]{x0394}\boldsymbol{u}^{p}$ . This leads to a reduction of the preferential sampling and path-history effects (and therefore a reduction of caustics), and to an enhancement of the filtering effect (see Ireland et al. (Reference Ireland, Bragg and Collins2016b ) for a detailed discussion of these effects).

When $|\unicode[STIX]{x0394}St|>0$ and $Fr=\infty$ (i.e. no gravity), the main effect of the bidispersity is through the ‘acceleration term’ in (2.6), with an implicit effect upon the first integral in (2.6) through the modification to the properties of $\unicode[STIX]{x0394}\boldsymbol{u}^{p}$ . An essential difference between the two integrals in (2.6) is that the first depends upon the particle separation, whereas the second does not. Since $\unicode[STIX]{x0394}\boldsymbol{u}^{p}$ decreases, on average, with decreasing separation, then provided $|\unicode[STIX]{x0394}St|>0$ , there will be a separation below which the second integral is $\gg$ than the first. In this regime, the statistics of $\boldsymbol{w}^{p}(t)$ will not depend upon the particle separation. This acceleration contribution causes the relative velocities of bidisperse particles to exceed those of monodisperse particles at small separations, as observed in a number of previous theoretical and numerical studies (Zhou et al. Reference Zhou, Wexler and Wang2001; Zaichik, Simonin & Alipchenkov Reference Zaichik, Simonin and Alipchenkov2006; Zaichik et al. Reference Zaichik, Fede, Simonin and Alipchenkov2009; Pan & Padoan Reference Pan and Padoan2010; Pan et al. Reference Pan, Padoan and Scalo2014b ). In homogeneous turbulence, there is no path-history effect associated with the second integral in (2.6); the statistical properties of $\boldsymbol{a}^{p}(s)$ are equivalent along all primary particle trajectories $\boldsymbol{x}^{p}$ . The accelerations are, however, strongly affected by both the preferential sampling and filtering effects (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006; Ayyalasomayajula et al. Reference Ayyalasomayajula, Warhaft and Collins2008; Salazar & Collins Reference Salazar and Collins2012a ; Ireland et al. Reference Ireland, Bragg and Collins2016b ), with filtering causing the fluctuations of $\boldsymbol{a}^{p}$ to decrease rapidly with increasing $St$ .

When $|\unicode[STIX]{x0394}St|>0$ and $Fr<\infty$ , gravity can lead to a number of effects on the relative motion of bidisperse particles. First, we note that the role of gravity in (2.5) is a little subtle since $\boldsymbol{a}^{p}(t)-\boldsymbol{e}_{\mathfrak{g}}Fr^{-1}=St_{1}^{-1}(\boldsymbol{u}(\boldsymbol{x}^{p}(t),t)-\boldsymbol{v}^{p}(t))$ , i.e. the explicit role of gravity appears to be absent. However, the direct effect of gravity is hidden in $\boldsymbol{v}^{p}(t)$ , which may be brought out using the formal solution for $\boldsymbol{v}^{p}(t)$ to obtain

where we have again assumed $t\gg St_{1}$ . The important point then is that $\boldsymbol{a}^{p}$ is in fact only implicitly affected by gravity, despite the fact that gravity appears explicitly in its dynamical equation.

The effect of gravity on the first term on the right-hand side of (2.6) is qualitatively the same as in the monodisperse case, described earlier. The effect of gravity on the second term on the right-hand side of (2.6) is clear; it increases as $Fr$ decreases, and becomes increasingly important as $|\unicode[STIX]{x0394}St|$ increases. However, this term only acts in the direction of gravity, and plays no role in the plane normal to this. Therefore, in the plane normal to $\boldsymbol{e}_{\mathfrak{g}}$ the effect of gravity is only implicit. The third term on the right-hand side of (2.6) involves $\boldsymbol{a}^{p}$ , and in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) we showed that gravity can enhance $\boldsymbol{a}^{p}$ , the effect being dramatic when $Fr\ll 1$ . This enhancement of inertial particle accelerations due to gravity was also observed by Parishani et al. (Reference Parishani, Ayala, Rosa, Wang and Grabowski2015), who proposed that the enhancement is due to the preferential sweeping effect (Wang & Maxey Reference Wang and Maxey1993). However, in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) we argued that this cannot be the explanation since the acceleration enhancement due to gravity is strongest in a regime of $St$ and $Fr$ where the preferential sampling of the fluid velocity gradient field is absent. We argued, in contrast, that the acceleration enhancement occurs because when $St$ is finite, the inertial particles have large settling velocities when $Fr$ is small, causing the fluid velocity seen by the particles to change rapidly in time. That is, fast settling leads to large values of $\boldsymbol{u}^{p}(t)-\boldsymbol{u}^{p}(t^{\prime })$ even when $t-t^{\prime }\ll St$ , which in turn through (2.7) implies large fluctuations in $\boldsymbol{a}^{p}(t)$ . Further, we showed in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) that fluctuations in $\boldsymbol{a}^{p}(t)$ are weaker in the direction of gravity than in the plane normal to gravity, which we explained in terms of the difference in the longitudinal and transverse correlation length scales of $\boldsymbol{u}$ in isotropic turbulence.

In the plane normal to gravity, the third term, which always dominates over the first term at sufficiently small separations, can increase as $Fr$ is decreased due to the enhanced accelerations. In Ireland et al. (Reference Ireland, Bragg and Collins2016b ) we derived a theoretical model for the effect of strong gravity on the particle accelerations. For simplicity, let us define the gravitational force to act in the $x_{3}$ direction, so that $\boldsymbol{e}_{\mathfrak{g}}=(0,0,1)$ . This will be referred to as the ‘vertical’ direction, whereas $x_{1}$ and $x_{2}$ will be referred to as the ‘horizontal’ directions. (Note also that since we are considering isotropic turbulence, the statistics of the particle motion are axisymmetric about $\boldsymbol{e}_{\mathfrak{g}}$ when $Fr<\infty$ .) Using this notation, the result derived in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) for the inertial particle accelerations in the horizontal direction is

where $u^{\prime }$ is the fluid root-mean-square (r.m.s.) velocity, and $\ell$ is the integral length scale of the flow. Comparisons of (2.8) with DNS in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) showed excellent agreement in the regime $St_{1}Fr^{-1}\gg u^{\prime }u_{\unicode[STIX]{x1D702}}^{-1}$ . For fixed $R_{\unicode[STIX]{x1D706}}$ , (2.8) increases as $Fr$ decreases, resulting in strong accelerations in the plane normal to gravity. Since $w_{1}^{p}$ depends upon $a_{1}^{p}$ , gravity can therefore lead to an enhancement of the relative velocities in the plane normal to its action, and this enhancement could be strong when $Fr\ll 1$ .

Finally, consider the limit $Fr\rightarrow 0$ where the relative velocities in the vertical direction are deterministic and given by

which is simply the difference in the Stokes settling velocities of the two particles (i.e. the differential settling velocity), in dimensionless form. However, in the horizontal directions, if $R_{\unicode[STIX]{x1D706}}\gg 1$ the motion remains turbulent even in the limit $Fr\rightarrow 0$ since

The effect of gravity is implicit in (2.10), and in the limit $Fr\rightarrow 0$ the large settling velocities of the particles causes them to experience $\unicode[STIX]{x0394}u_{1}^{p}$ as a white noise signal (Bec et al. Reference Bec, Homann and Ray2014). As such, the limit $Fr\rightarrow 0$ is subtle in the bidisperse case. In the vertical direction, $w_{3}^{p}(t)=-\unicode[STIX]{x0394}StFr^{-1}$ when $Fr\rightarrow 0$ , which applies to quiescent and turbulent flows. However, for the horizontal direction, in a quiescent flow $w_{1}^{p}(t)=0\,\forall Fr$ , but in a turbulent flow, $w_{1}^{p}(t)\neq 0$ in the limit $Fr\rightarrow 0$ . As a consequence of this, $\lim _{Fr\rightarrow 0}\boldsymbol{w}^{p}(t)\not \rightarrow -\unicode[STIX]{x0394}StFr^{-1}\boldsymbol{e}_{\mathfrak{g}}$ in a turbulent flow.

Furthermore, provided $R_{\unicode[STIX]{x1D706}}<\infty$ , then in the limit $Fr\rightarrow 0$ (2.8) becomes

which is independent of $Fr$ . This predicts that even in the limit $Fr\rightarrow 0$ , turbulence can lead to horizontal accelerations for the inertial particles that are not only finite, but $\gg a_{\unicode[STIX]{x1D702}}$ .

In Chang, Malec & Shaw (Reference Chang, Malec and Shaw2015) a semi-empirical formula for the second-order structure function $\langle \Vert \boldsymbol{w}^{p}(t)\Vert ^{2}\rangle _{\boldsymbol{r}}$ was constructed for $St_{2}\ll 1$ and $|\unicode[STIX]{x0394}St|\ll 1$ , where $\langle \cdot \rangle _{\boldsymbol{r}}$ denotes an ensemble average conditioned on $\boldsymbol{r}^{p}(t)=\boldsymbol{r}$ . By comparing their results with DNS data for $Fr=0.1$ , they found good agreement between their prediction and the DNS data even when $St_{2}=O(1)$ (with $|\unicode[STIX]{x0394}St|\ll 1$ ). However, this is mainly because in their data with $Fr=0.1$ , $\langle \Vert \boldsymbol{w}^{p}(t)\Vert ^{2}\rangle _{\boldsymbol{r}}\approx (\unicode[STIX]{x0394}St)^{2}Fr^{-2}$ for small $r/\unicode[STIX]{x1D702}$ , i.e. the turbulent contribution is negligible, and this gravitational settling contribution is in exact, closed form for arbitrary $St_{2}$ , $\unicode[STIX]{x0394}St$ . Deriving closed form predictions for $\langle \Vert \boldsymbol{w}^{p}(t)\Vert ^{2}\rangle _{\boldsymbol{r}}$ , and more generally $\langle \Vert \boldsymbol{w}^{p}(t)\Vert ^{N}\rangle _{\boldsymbol{r}}$ , when $Fr\geqslant O(1)$ and $St_{2}=O(1)$ is much more difficult due to closure problems arising from the contribution of the turbulence along the particle trajectory history (Pan & Padoan Reference Pan and Padoan2010; Bragg & Collins Reference Bragg and Collins2014b ). Also, predicting the horizontal structure functions $\langle [w_{1}^{p}(t)]^{N}\rangle _{\boldsymbol{r}}$ is challenging, since as discussed earlier, $w_{1}^{p}(t)$ is always affected by turbulence for any $Fr$ .

In a recent paper (Bragg Reference Bragg2017), we advanced the model first proposed by Pan & Padoan (Reference Pan and Padoan2010) for $\langle \boldsymbol{w}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ , by improving one of their original closure assumptions by using our new theoretical predictions for the backward-in-time pair dispersion of inertial particles in turbulence (Bragg, Ireland & Collins Reference Bragg, Ireland and Collins2016). Comparisons with DNS data showed that this development of the model led to significant improvements in the predictions of $\langle \boldsymbol{w}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ when compared with the predictions from the original model. In future work, we intend to further extend the predictions to the case of settling, bidisperse particles, taking into account the theoretical considerations discussed in this section.

2.2 Spatial clustering

To consider the spatial clustering of settling, bidisperse particles, we begin by defining the probability density function (PDF) for $\boldsymbol{r}^{p}(t),\boldsymbol{w}^{p}(t)$ in the phase space $\boldsymbol{r},\boldsymbol{w}$ ,

for which the evolution equation is

where $\unicode[STIX]{x1D735}_{\boldsymbol{r}}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{r}$ and $\unicode[STIX]{x1D735}_{\boldsymbol{w}}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{w}$ .

The spatial clustering of the particles is described by the marginal distribution

By multiplying the stationary form of (2.13) by $\boldsymbol{w}$ and then integrating over all $\boldsymbol{w}$ , we can obtain an equation for $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D711}\langle \boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ , where $\unicode[STIX]{x1D711}\langle \boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}\equiv \int _{\mathbb{R}^{3}}\boldsymbol{w}{\mathcal{P}}\,\text{d}\boldsymbol{w}$ and $\langle \cdot \rangle _{\boldsymbol{r}}$ denotes an ensemble average conditioned on $\boldsymbol{r}^{p}(t)=\boldsymbol{r}$ . Then, inserting the equation of motion for $\dot{\boldsymbol{w}}^{p}(t)$ , and considering the horizontal component $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D711}\langle w_{1}^{p}(t)\rangle _{\boldsymbol{r}}$ , we obtain an equation governing the steady-state form of $\unicode[STIX]{x1D711}$ ,

where we have used $\langle w_{1}^{p}(t)\rangle _{\boldsymbol{r}}=0$ due to statistical stationarity and isotropy of the system in the horizontal plane. Note that although $a_{1}^{p}(t)$ is not a function of $\boldsymbol{r}^{p}(t)$ , $\langle a_{1}^{p}(t)\rangle _{\boldsymbol{r}}\neq \langle a_{1}^{p}(t)\rangle$ because $\boldsymbol{r}^{p}(t)$ is a functional of $a_{1}^{p}(t)$ .

In previous works, we have studied (2.15) in detail for the monodisperse case $\unicode[STIX]{x0394}St=0$ (Bragg & Collins Reference Bragg and Collins2014a ; Bragg et al. Reference Bragg, Ireland and Collins2015a ,Reference Bragg, Ireland and Collins b ). In that case we have demonstrated that clustering (i.e. non-uniform $\unicode[STIX]{x1D711}$ ) arises through $-St_{2}\unicode[STIX]{x1D711}\unicode[STIX]{x1D735}_{\boldsymbol{r}}\boldsymbol{\cdot }\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}<0$ , corresponding to a net drift of the particle pair towards each other. When $St_{2}\ll 1$ , this inward drift arises from the preferential sampling of strain over rotation along their trajectory (Chun et al. Reference Chun, Koch, Rani, Ahluwalia and Collins2005), associated with the centrifuge effect (Maxey Reference Maxey1987). However, when $St_{2}\geqslant O(1)$ , a fundamentally different clustering mechanism begins to dominate: particle pairs that are approaching carry a memory of more energetic turbulence along their path history than particles that are separating, and this asymmetry gives rise to a net inward drift leading to clustering.

The clustering of bidisperse particles for $Fr=\infty$ has been considered previously in theoretical studies, and quantitative predictions have been developed (Chun et al. Reference Chun, Koch, Rani, Ahluwalia and Collins2005; Zaichik et al. Reference Zaichik, Simonin and Alipchenkov2006). Here we will simply attempt to qualitatively consider how gravity affects the clustering of bidisperse particles, leaving a quantitative theoretical analysis for future work.

Under standard closure assumptions, the terms $\langle \unicode[STIX]{x0394}u_{1}^{p}(t)\rangle _{\boldsymbol{r}}$ and $\langle a_{1}^{p}(t)\rangle _{\boldsymbol{r}}$ can be modelled as diffusive fluxes of the form (Chun et al. Reference Chun, Koch, Rani, Ahluwalia and Collins2005; Zaichik et al. Reference Zaichik, Simonin and Alipchenkov2006)

where $\boldsymbol{{\mathcal{D}}}^{\unicode[STIX]{x0394}u}$ and $\boldsymbol{{\mathcal{D}}}^{a}$ are diffusion tensors that depend upon the properties of $\unicode[STIX]{x0394}u_{1}^{p}(t)$ and $a_{1}^{p}(t)$ , respectively. Their particular form is not important for this qualitative discussion; the important point is that $\boldsymbol{{\mathcal{D}}}^{\unicode[STIX]{x0394}u}$ decreases with decreasing $\Vert \boldsymbol{r}\Vert$ , whereas $\boldsymbol{{\mathcal{D}}}^{a}$ does not depend upon $\boldsymbol{r}$ . Using (2.16) and (2.17) in (2.15) we obtain

We first consider bidisperse clustering for $Fr=\infty$ . In this case, the main effect of bidispersity on the clustering is through the additional diffusion contribution $(\unicode[STIX]{x0394}St)^{2}\boldsymbol{{\mathcal{D}}}^{a}$ . In the limit of small separations, $\Vert \boldsymbol{{\mathcal{D}}}^{a}\Vert \gg \Vert \boldsymbol{{\mathcal{D}}}^{\unicode[STIX]{x0394}u}\Vert$ , and the enhanced diffusion due to the acceleration contribution leads to a reduction in the clustering as $|\unicode[STIX]{x0394}St|$ is increased, and to a plateau of $\unicode[STIX]{x1D711}$ at small separations. This reduced clustering due to the particle acceleration was captured in the theoretical analysis by Chun et al. (Reference Chun, Koch, Rani, Ahluwalia and Collins2005) for the regime $St_{2}\ll 1$ and $|\unicode[STIX]{x0394}St|\ll 1$ . This reduced clustering due to the particle accelerations carries over to the $|\unicode[STIX]{x0394}St|\geqslant O(1)$ regime. However, in addition to this, in the regime $|\unicode[STIX]{x0394}St|\geqslant O(1)$ the clustering will also be affected implicitly through the effect of $a_{1}^{p}(t)$ on $\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ and $\unicode[STIX]{x1D735}_{\boldsymbol{r}}\boldsymbol{\cdot }\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ . As discussed earlier, bidispersity leads both to an enhancement of $\boldsymbol{w}^{p}(t)$ , and also causes $\boldsymbol{w}^{p}(t)$ to depend weakly upon separation when $|\unicode[STIX]{x0394}St|\geqslant O(1)$ , i.e. where the acceleration effect dominates the behaviour of $\boldsymbol{w}^{p}(t)$ . These effects lead to an enhancement of $\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ and a reduction of $\unicode[STIX]{x1D735}_{\boldsymbol{r}}\boldsymbol{\cdot }\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ , and thus to further suppression of clustering when $|\unicode[STIX]{x0394}St|\geqslant O(1)$ . For these reasons, we expect the clustering of monodisperse particles to be stronger than that for bidisperse particles, something that has already been observed in previous DNS studies (e.g. Zhou et al. Reference Zhou, Wexler and Wang2001; Pan et al. Reference Pan, Padoan, Scalo, Kritsuk and Norman2011).

We now turn to consider the effect of gravity on the clustering. In the monodisperse case, the effect of gravity on the clustering was found to be non-trivial, with gravity leading to either reduced or enhanced clustering depending upon $St,Fr$ (Bec et al. Reference Bec, Homann and Ray2014; Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014; Ireland et al. Reference Ireland, Bragg and Collins2016b ). The theoretical explanation given for this in Ireland et al. (Reference Ireland, Bragg and Collins2016b ) can be summarized conceptually as follows. Gravity leads to a reduction in the correlation time of the flow seen by the inertial particles. This reduces the effectiveness of the centrifuge mechanism, and also reduces the path-history effect since it reduces the temporal span over which the particles are affected by their past interaction with the turbulence. Both of these effects lead to reduced clustering. Now, although the path-history effect generates strong clustering over a range of $St$ , when $St$ is large enough the path-history effect begins to destroy the clustering since the particle relative motion becomes increasingly ballistic. However, since gravity reduces the path-history effect, one has to go to larger $St$ before the path-history effect begins to destroy the clustering. As a result, at large enough $St$ , gravity has the effect of enhancing clustering relative to the $Fr=\infty$ case.

In the bidisperse case, for the regime $St_{2}\ll 1$ and $|\unicode[STIX]{x0394}St|\ll 1$ , Lu, Nordsiek & Shaw (Reference Lu, Nordsiek and Shaw2010) extended the theory by Chun et al. (Reference Chun, Koch, Rani, Ahluwalia and Collins2005) and considered the effect of gravity on the particle clustering. Their analysis showed that in this regime, gravity suppresses the clustering of bidisperse particles. Relating their analysis to (2.18), their argument is essentially that in the regime $St_{2}\ll 1$ and $|\unicode[STIX]{x0394}St|\ll 1$ , the leading-order effect of gravity is through its contribution to $\boldsymbol{{\mathcal{D}}}^{a}$ , leading to an overall enhancement of the diffusion, and hence to a reduction of the particle clustering. When $St_{2}\geqslant O(1)$ , the clustering enhancement due to gravity may occur when if $|\unicode[STIX]{x0394}St|$ is sufficiently small and $Fr$ is not too small. However, in general, we expect that gravity will reduce the clustering, and not enhance it. This is because, as explained earlier, as $Fr$ is decreased, $\boldsymbol{a}^{p}(t)$ increases, and $\boldsymbol{w}^{p}(t)$ both increases and becomes less dependent upon the particle separation. Thus, in terms of (2.18), decreasing $Fr$ both enhances the diffusion contributions $\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ and $\boldsymbol{{\mathcal{D}}}^{a}$ , and reduces the drift $\unicode[STIX]{x1D735}_{\boldsymbol{r}}\boldsymbol{\cdot }\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ , leading to a suppression of clustering.

4.1 Relative velocities

We begin by considering results for the relative velocities of the particles. The parallel (longitudinal) and perpendicular (transverse) projections of $\boldsymbol{w}^{p}(t)$ are defined as $w_{\Vert }^{p}(t)\equiv \boldsymbol{e}_{\Vert }^{p}(t)\boldsymbol{\cdot }\boldsymbol{w}^{p}(t)$ and $w_{\bot }^{p}(t)\equiv \boldsymbol{e}_{\bot }^{p}(t)\boldsymbol{\cdot }\boldsymbol{w}^{p}(t)$ , where $\boldsymbol{e}_{\Vert }^{p}(t),\boldsymbol{e}_{\bot }^{p}(t)$ are unit vectors parallel and perpendicular to $\boldsymbol{r}^{p}(t)$ , respectively.

Figure 1 shows the DNS results for the second-order, longitudinal velocity structure function $S_{2\Vert }(r)\equiv \langle [w_{\Vert }^{p}(t)]^{2}\rangle _{r}$ , where $\langle \cdot \rangle _{r}$ denotes an ensemble average conditioned on $\Vert \boldsymbol{r}^{p}(t)\Vert =r$ . We also include the monodisperse results with $Fr=\infty$ for comparison. It can be seen that the bidisperse values, with and without gravity, are bounded from below by the monodisperse values. This is in agreement with previous studies that considered bidisperse particle motion without gravitational effects (Zhou et al. Reference Zhou, Wexler and Wang2001; Zaichik et al. Reference Zaichik, Simonin and Alipchenkov2006, Reference Zaichik, Fede, Simonin and Alipchenkov2009; Pan & Padoan Reference Pan and Padoan2010; Pan et al. Reference Pan, Padoan and Scalo2014b ). We also observe in figure 1 that $S_{2\Vert }(r)$ becomes independent of $r$ at sufficiently small separations. Both of these effects arise because of the acceleration contribution to the bidisperse particle motion, as explained in § 2.

The enhancement of the relative velocities of bidisperse particles at small separations grows as $Fr$ is reduced and/or $|\unicode[STIX]{x0394}St|$ is increased. For sufficiently large $r$ , the effect of the bidispersity weakens, corresponding to the dominance of the first term on the right-hand side of (2.6) at these separations. The results for $S_{2\bot }(r)\equiv \langle [w_{\bot }^{p}(t)]^{2}\rangle _{r},$ are shown in figure 2, which are quantitatively similar to those for $S_{2\Vert }(r)$ .

Figure 1.

DNS results for $S_{2\Vert }/u_{\unicode[STIX]{x1D702}}^{2}$ as a function of $r/\unicode[STIX]{x1D702}$ for different $St_{1},St_{2},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ and solid lines correspond to the monodisperse case with $Fr=\infty$ . The numbers next to the dashed lines indicate the $St_{2}$ value that they correspond to.

In figure 3 we show results for ${\mathcal{P}}(|w_{\Vert }||r)$ , the PDF of the absolute value of the longitudinal relative velocities at separation $r$ . As observed by Pan et al. (Reference Pan, Padoan and Scalo2014b ), for $Fr=\infty$ , increasing $|\unicode[STIX]{x0394}St|$ leads to a broadening of the PDF, corresponding to the enhanced relative velocities through the acceleration effect. Decreasing $Fr$ leads to further broadening of the PDF. Most strikingly, when $\unicode[STIX]{x0394}St$ is sufficiently large and $Fr=0.052$ , the PDF is almost constant up to $|w_{\Vert }|\gg u_{\unicode[STIX]{x1D702}}$ and then suddenly reduces. In § 2 we showed that reducing $Fr$ can enhance $\boldsymbol{w}^{p}(t)$ for bidisperse particles in two ways: explicitly through the differential settling contribution $u_{\unicode[STIX]{x1D702}}\unicode[STIX]{x0394}StFr^{-1}\boldsymbol{e}_{\mathfrak{g}}$ , and implicitly through the way that gravity enhances the primary particle acceleration $\boldsymbol{a}^{p}$ . Since the orientation of $\boldsymbol{r}^{p}(t)$ is time dependent, the effect of gravity on the statistics of the parallel and perpendicular projections of $\boldsymbol{w}^{p}(t)$ is not entirely clear, and will involve a mixture of these explicit and implicit effects.

Figure 2.

DNS results for $S_{2\bot }/u_{\unicode[STIX]{x1D702}}^{2}$ as a function of $r/\unicode[STIX]{x1D702}$ for different $St_{1},St_{2},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ and solid lines correspond to the monodisperse case with $Fr=\infty$ . The numbers next to the dashed lines indicate the $St_{2}$ value that they correspond to.

In order to untangle the explicit and implicit effects of gravity on the statistics of $\boldsymbol{w}^{p}(t)$ and gain further insight, we consider the second-order structure functions based on the Cartesian components of $\boldsymbol{w}^{p}(t)$ . That is, we consider $\langle [w_{3}^{p}(t)]^{2}\rangle _{r}$ and $\langle [w_{1}^{p}(t)]^{2}\rangle _{r}$ , corresponding to the velocities in the vertical and horizontal directions, respectively (recall also that $\langle [w_{1}^{p}(t)]^{2}\rangle _{r}=\langle [w_{2}^{p}(t)]^{2}\rangle _{r}$ due to the axisymmetry of the statistics).

The results are shown in figures 4 and 5, where we also show the $Fr\rightarrow 0$ prediction

which follows from (2.9). When $Fr=0.3$ , $\langle [w_{3}^{p}(t)]^{2}\rangle _{r}$ is well approximated by (4.1), with some small discrepancies for certain $St_{1},St_{2}$ combinations. However, for $Fr=0.052$ , (4.1) is in almost perfect agreement with the data for each of the $St_{1},St_{2}$ combinations shown. When gravity is active, $\langle [w_{3}^{p}(t)]^{2}\rangle _{r}>\langle [w_{1}^{p}(t)]^{2}\rangle _{r}$ , and the difference grows with increasing $|\unicode[STIX]{x0394}St|$ and decreasing $Fr$ . The analysis in § 2 shows why: as $|\unicode[STIX]{x0394}St|$ increases and $Fr$ decreases, the contribution to $\langle [w_{3}^{p}(t)]^{2}\rangle _{r}$ from the differential settling velocity grows, yet this contribution is absent from $\langle [w_{1}^{p}(t)]^{2}\rangle _{r}$ . The results in figures 4 and 5 also show that the implicit effect of gravity can lead to strong enhancements of $\langle [w_{1}^{p}(t)]^{2}\rangle _{r}$ , compared with the $Fr=\infty$ case, and in some cases the increase corresponds to an order of magnitude. This strong implicit effect of gravity on the horizontal relative velocities could have important implications for understanding and modelling the small-scale mixing and dispersion of bidisperse particles that are settling under gravity.

Figure 3.

DNS results for ${\mathcal{P}}(|w_{\Vert }||r)$ as a function of $|w_{\Vert }|/u_{\unicode[STIX]{x1D702}}$ for different $St_{1},St_{2},Fr$ combinations, with $r=\unicode[STIX]{x1D702}$ . Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , and open symbols to $Fr=0.052$ .

Figure 4.

DNS results for $\langle [w_{1,3}^{p}(t)]^{2}\rangle _{r}/u_{\unicode[STIX]{x1D702}}^{2}$ as a function of $r/\unicode[STIX]{x1D702}$ for $St_{1}=0.1$ and different $St_{2},Fr$ combinations. Circles correspond to the vertical velocities and squares correspond to the horizontal velocities. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The dotted and solid horizontal lines correspond to (4.1) for $Fr=0.3$ and $Fr=0.052$ , respectively.

Figure 5.

DNS results for $\langle [w_{1,3}^{p}(t)]^{2}\rangle _{r}/u_{\unicode[STIX]{x1D702}}^{2}$ as a function of $r/\unicode[STIX]{x1D702}$ for $St_{1}=1$ and different $St_{2},Fr$ combinations. Circles correspond to the vertical velocities and squares correspond to the horizontal velocities. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The dotted and solid horizontal lines correspond to (4.1) for $Fr=0.3$ and $Fr=0.052$ , respectively.

Figure 6.

DNS results for ${\mathcal{P}}(|w_{1}||r)$ and ${\mathcal{P}}(|w_{3}||r)$ as a function of $|w_{1,3}|/u_{\unicode[STIX]{x1D702}}$ for $St_{1}=0.1$ , different $St_{2},Fr$ combinations, and with $r=\unicode[STIX]{x1D702}$ . Circles correspond to the vertical velocities and squares correspond to the horizontal velocities. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The vertical lines correspond to $u_{\unicode[STIX]{x1D702}}|\unicode[STIX]{x0394}St|Fr^{-1}$ for $Fr=0.3$ and $Fr=0.052$ .

Figure 7.

DNS results for ${\mathcal{P}}(|w_{1}||r)$ and ${\mathcal{P}}(|w_{3}||r)$ as a function of $|w_{1,3}|/u_{\unicode[STIX]{x1D702}}$ for $St_{1}=1$ , different $St_{2},Fr$ combinations, and with $r\in [0,2]\unicode[STIX]{x1D702}$ . Circles correspond to the vertical velocities and squares correspond to the horizontal velocities. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The vertical lines correspond to $u_{\unicode[STIX]{x1D702}}|\unicode[STIX]{x0394}St|Fr^{-1}$ for $Fr=0.3$ and $Fr=0.052$ .

In figures 6 and 7 we show results for ${\mathcal{P}}(|w_{1}||r)$ and ${\mathcal{P}}(|w_{3}||r)$ , the PDFs of the absolute value of the horizontal and vertical relative velocities at separation $r$ , respectively. As expected, the results show that gravity affects ${\mathcal{P}}(|w_{1}||r)$ and ${\mathcal{P}}(|w_{3}||r)$ in very different ways, since gravity only plays an explicit role in the vertical direction. Consistent with the results for the structure functions and the arguments of § 2, the PDFs for the horizontal velocities shift towards larger values, corresponding to the enhancement arising from the implicit effect of gravity. For the vertical velocities, the mode of the PDF is close to $u_{\unicode[STIX]{x1D702}}|\unicode[STIX]{x0394}St|Fr^{-1}$ , which is indicated by the vertical lines in figures 6 and 7. However, only for $|\unicode[STIX]{x0394}St|>2$ and $Fr=0.052$ does the PDF approach the asymptotic behaviour

When $Fr=0.052$ , the explicit effect of gravity was shown to dominate $\langle [w_{3}^{p}(t)]^{2}\rangle _{r}$ , even when $|\unicode[STIX]{x0394}St|<1$ . However, the results in figures 6 and 7 show that for the same parameter range, ${\mathcal{P}}(|w_{3}||r)$ is not dominated by the gravity asymptote, implying that $\langle [w_{3}^{p}(t)]^{n}\rangle _{r}=(u_{\unicode[STIX]{x1D702}}\unicode[STIX]{x0394}StFr^{-1})^{n}$ can fail for $n\neq 2$ . We would also expect the deviation from (4.2) to increase as $R_{\unicode[STIX]{x1D706}}$ is increased due to the enhanced intermittency in the turbulence, since the probability of the terms involving $\unicode[STIX]{x0394}u_{3}^{p}$ and $a_{3}^{p}$ becoming comparable to $u_{\unicode[STIX]{x1D702}}\unicode[STIX]{x0394}StFr^{-1}$ in the equation for $w_{3}^{p}(t)$ would increase. This implies that in applications such as clouds where $R_{\unicode[STIX]{x1D706}}=O(10^{4})$ (Grabowski & Wang Reference Grabowski and Wang2013), the asymptotic behaviour (4.2) may never be realized due to the extremely small values of $Fr$ that would be required. This could have important implications for understanding and modelling the vertical mixing of settling, bidisperse particles in a variety of turbulent flows.

4.2 Spatial clustering

We now turn to consider the spatial clustering of the particles, which may be quantified using the radial distribution function (RDF), defined as (McQuarrie Reference McQuarrie1976)

If we define the number of satellite particles (with Stokes number $St_{2}$ ) found at a distance $r$ from a given primary particle (with Stokes number $St_{1}$ ) to be $N_{i}^{\prime }$ , then $N_{i}$ is the sum of $N_{i}^{\prime }$ over all primary particles in the system. $\unicode[STIX]{x1D6FF}V_{i}$ is the volume of an elemental spherical shell with surface at a distance $r$ from the primary particle, $V$ is the total volume of the system and $N_{1}$ and $N_{2}$ are the total numbers of primary and satellite particles in the domain, respectively. Note that the RDF is the same (to within a constant) as $\Vert \boldsymbol{r}\Vert ^{-2}$ times the spherical average of the PDF $\unicode[STIX]{x1D711}(\boldsymbol{r})$ introduced in § 2.

Figure 8.

DNS results for $g(r)$ as a function of $r/\unicode[STIX]{x1D702}$ for different $St_{1},St_{2},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ and solid lines correspond to the monodisperse case with $Fr=\infty$ . The numbers next to the dashed lines indicate the $St_{2}$ value that they correspond to.

The results in figure 8 show that in general, increasing $|\unicode[STIX]{x0394}St|$ reduces the RDF, and that the clustering is significantly reduced for $r/\unicode[STIX]{x1D702}\ll 1$ even when $|\unicode[STIX]{x0394}St|\ll 1$ . These results for $Fr=\infty$ are consistent with previous investigations (Zhou et al. Reference Zhou, Wexler and Wang2001; Chun et al. Reference Chun, Koch, Rani, Ahluwalia and Collins2005; Pan et al. Reference Pan, Padoan, Scalo, Kritsuk and Norman2011). This reduction is consistent with the explanations in § 2, namely, that the acceleration contribution to the relative motion of bidisperse particles leads to an enhanced particle-pair diffusion and reduced inward drift, leading to reduced clustering, and to a plateau of the RDF at small separations.

We also note that although the RDF at $r/\unicode[STIX]{x1D702}\leqslant 1$ is always largest for $\unicode[STIX]{x0394}St=0$ , i.e. the monodisperse case, the same is not always true at larger separations (e.g. see figure 8 a). Over the $St_{1},St_{2}$ range we are here considering, $|\unicode[STIX]{x0394}St|>0$ monotonically increases the particle-pair diffusion tensor, and therefore, this behaviour must be caused by bidispersity leading to an enhancement of the inward drift term $St_{2}\unicode[STIX]{x1D711}\unicode[STIX]{x1D735}_{\boldsymbol{r}}\boldsymbol{\cdot }\langle w_{1}^{p}(t)\boldsymbol{w}^{p}(t)\rangle _{\boldsymbol{r}}$ in (2.18) in certain regimes of $St_{1},St_{2}$ .

In agreement with the arguments in § 2, the results in figure 8 also show that decreasing $Fr$ further suppresses the clustering of bidisperse particles. Indeed, even for $|\unicode[STIX]{x0394}St|=0.1$ , gravity is very effective at suppressing the clustering.

Figure 9.

DNS results for $g(r)$ as a function of $St_{2}$ , at specific values of $r/\unicode[STIX]{x1D702}$ and for different $St_{1},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ .

To see more clearly the effect of bidispersity and gravity on the RDF, in figure 9 we plot the RDF for a given $St_{1}$ as a function of $St_{2}$ at four different separations. As discussed earlier, the results show that the bidisperse RDFs generally peak when the difference between $St_{1}$ and $St_{2}$ is smallest. Figure 9 also shows more clearly how gravity systematically suppresses the clustering over the range of $St,\unicode[STIX]{x0394}St,Fr$ considered. This is in contrast to the monodisperse case for which it has been observed that gravity can actually enhance clustering in certain regimes of $St,Fr$ (Bec et al. Reference Bec, Homann and Ray2014; Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014; Ireland et al. Reference Ireland, Bragg and Collins2016b ). We expect that this enhancement could also be observed for bidisperse particles when $St_{2}>1$ if $|\unicode[STIX]{x0394}St|\ll 1$ . However, we do not have data for this regime, although it is something that should be investigated in future work. Also, the regime $Fr=O(1)$ should also be explored in future work to see if the clustering enhancement is observed for bidisperse particles in that regime.

We add the cautionary remark that when we used a box length of $2\unicode[STIX]{x03C0}$ for the $Fr=0.052$ case, we observed that the RDF was larger than for the $Fr=0.3$ case, giving the appearance of enhanced clustering due to gravity. However, this apparent enhancement disappeared for the much larger box length of $16\unicode[STIX]{x03C0}$ used in this study. This highlights the need to use large computational domains when $Fr\ll 1$ in order to ensure the dynamics of the settling particles is not artificially affected by the periodic boundary conditions used in the DNS, as discussed in detail in Ireland et al. (Reference Ireland, Bragg and Collins2016b ).

4.3 Collision kernel

Finally, we consider the collision kernel for the bidisperse particles. The collision kernel is given by (Sundaram & Collins Reference Sundaram and Collins1997)

where $d\equiv (d_{1}+d_{2})/2$ is the collision diameter of two spherical particles with diameters $d_{1}$ and $d_{2}$ , and $S_{-\Vert }(r)\equiv \langle w_{\Vert }^{p}(t)|<0\rangle _{r}$ is the average of $w_{\Vert }^{p}(t)$ conditioned on $w_{\Vert }^{p}(t)<0$ for the separation $r$ (i.e. the mean inward relative velocity).

In Ireland et al. (Reference Ireland, Bragg and Collins2016a ,Reference Ireland, Bragg and Collins b ), when plotting $K(d)$ we chose a fluid to particle density ratio, for which $d$ is then determined by $St$ . Since $d$ so determined is often smaller than the smallest separation for which we have data for $g$ and $S_{-\Vert }$ , we had to extrapolate our DNS data down to $d$ . For the monodisperse case considered in Ireland et al. (Reference Ireland, Bragg and Collins2016a ,Reference Ireland, Bragg and Collins b ), this was justified since the functional forms of $g$ and $S_{-\Vert }$ are known at small separations, i.e. they are power laws (Gustavsson & Mehlig Reference Gustavsson and Mehlig2011). However, in the bidisperse case, and with gravity, we do not know how $g$ and $S_{-\Vert }$ depend upon $r$ (they are not power laws), and therefore extrapolation would be unreliable. Therefore, we will simply consider $K(d=0.125\unicode[STIX]{x1D702})$ , corresponding to the smallest separation at which our DNS data are reliable.

In figure 10 we show results for the normalized collision kernel $\widehat{K}(d)\equiv K(d)/(d^{2}u_{\unicode[STIX]{x1D702}})$ . The results show that decreasing $Fr$ enhances $\widehat{K}(d)$ ; since clustering reduces as $Fr$ is decreased, this enhancement arises solely because of the enhanced relative velocities due to gravity. Also shown in the plot is the collision kernel for settling bidisperse particles in quiescent flow (Grabowski & Wang Reference Grabowski and Wang2013)

The results show that for $Fr=0.052$ , $K_{\mathfrak{g}}(d)$ is in excellent agreement with $K(d)$ . As we demonstrated earlier, turbulence plays a very significant role on the horizontal relative velocities even when $Fr\ll 1$ ; however, the relative velocities in the vertical direction are in general much larger, and so the vertical relative velocities dominate the behaviour of $S_{-\Vert }(d)$ . For $Fr=0.3$ , $K_{\mathfrak{g}}(d)$ is in reasonable agreement with $K(d)$ only if $|\unicode[STIX]{x0394}St|\gtrsim 1$ , and in general $K_{\mathfrak{g}}(d)$ significantly underpredicts $K(d)$ .

To determine the cause of this underprediction, in figure 11 we plot $\widehat{K}_{\mathfrak{g}}^{\prime }(d)=\unicode[STIX]{x03C0}g(d)|\unicode[STIX]{x0394}St|Fr^{-1}$ , which uses the DNS data for $g(d)$ , but the gravity estimate for $S_{-\Vert }(d)$ . The results show that for $Fr=0.052$ , $\widehat{K}_{\mathfrak{g}}^{\prime }(d)\approx \widehat{K}_{\mathfrak{g}}(d)\approx \widehat{K}(d)$ since the particle clustering is almost absent. However, for $Fr=0.3$ , $\widehat{K}_{\mathfrak{g}}^{\prime }(d)$ provides a significantly improved prediction for $\widehat{K}(d)$ compared with $\widehat{K}_{\mathfrak{g}}(d)$ , implying the importance of clustering on the collisions rates in this regime. However, for some combinations of $St_{1},St_{2}$ , $\widehat{K}_{\mathfrak{g}}^{\prime }(d)$ still underpredicts $K(d)$ which must be due to errors in the gravity estimate $S_{-\Vert }(r)=(1/4)u_{\unicode[STIX]{x1D702}}/|\unicode[STIX]{x0394}St|Fr^{-1}$ .

Figure 10.

DNS results for $\widehat{K}(d)$ as a function of $St_{2}$ , for $d=0.125\unicode[STIX]{x1D702}$ and for different $St_{1},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The dashed-dot and solid lines correspond to the predictions of (4.5) for $Fr=0.3$ and $Fr=0.052$ , respectively.

Figure 11.

DNS results for $\widehat{K}$ (circles) and $\widehat{K}_{\mathfrak{g}}^{\prime }$ (squares) as a function of $St_{2}$ , for $d=0.125\unicode[STIX]{x1D702}$ and for different $St_{1},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ .

In figure 12 we plot the DNS data for $S_{-\Vert }(r)$ and compare the results with the gravity estimate. As anticipated, the results show that whereas the gravity estimate

is in excellent agreement with the DNS data for $Fr=0.052$ and $r<\unicode[STIX]{x1D702}$ , it underpredicts the DNS data for $Fr=0.3$ for some combinations of $St_{1},St_{2}$ .

Figure 12.

DNS results for $S_{-\Vert }(r)$ as a function of $r/\unicode[STIX]{x1D702}$ , for different $St_{1},St_{2},Fr$ combinations. Dashed lines correspond to $Fr=\infty$ , filled symbols to $Fr=0.3$ , open symbols to $Fr=0.052$ . The numbers next to the dashed lines indicate the $St_{2}$ value that they correspond to. The dotted lines correspond to the predictions of (4.6) for $Fr=0.052$ , and the solid lines correspond to the predictions of (4.6) for $Fr=0.3$ , where the $St_{1},St_{2}$ combinations that the lines correspond to are shown in square brackets for clarity.