4.1. Volume distribution of inertial particle cells

To examine the effects of particle inertia and external forces, such as gravity, on how particles are clustered, we explore the volume distribution of modified Voronoi tessellations. Figure 2 shows the PDFs of the volume of modified Voronoi tessellations normalised by the mean $V_p/\overline {V_p}$ for different Stokes numbers, both (a) without and (b) with gravity. These visualisations provide insights into the spatial distribution and clustering behaviour of inertial particles. Similarly to Monchaux, Bourgoin & Cartellier (Reference Monchaux, Bourgoin and Cartellier2010), we define cluster cells as cells that have a volume smaller than the left intersection value $(V_p/\overline {V_p}\approx 0.4)$ of the PDF of the volume of inertial particles and the PDF of the volume of randomly distributed particles, and void cells as the cells that have a volume larger than the right intersection value $(V_p/\overline {V_p}\approx 3)$ . In the gravity-free case, it can be observed that, as the Stokes number increases and gets closer to 1, the number of cluster cells increases, and then decreases after exceeding $\textit{St}=1$ . This has also been observed in Dejoan & Monchaux (Reference Dejoan and Monchaux2013) and Baker et al. (Reference Baker, Frankel, Mani and Coletti2017), although here we show the volume of modified Voronoi cells and not that of the classical Voronoi. For void cells, a similar behaviour is observed, i.e. the number of void cells increases up to $\textit{St} = 1$ , and then slightly decreases. This illustrates that the clustering is the most pronounced for $\textit{St} \approx 1$ . In the case with gravity, the increasing trend in the number of cluster cells continues even for $\textit{St}\geqslant 1$ , in contrast to the gravity-free case. Similarly, for void cells, their number continues to increase slightly. Thus, the clustering is more pronounced persistently as the Stokes number increases even for Stokes numbers higher than 1.

Figure 2.

The PDFs of volume of modified Voronoi tessellation normalised by the mean $V_p/\overline {V_p}$ for different Stokes numbers (a) without and (b) with gravity. The black dashed line represents the volume distribution in the case of the same number of randomly distributed particles.

Figure 2. Long description

The image contains two line graphs labeled (a) and (b). Each graph shows the probability density function (PDF) of the volume of modified Voronoi tessellation normalized by the mean volume for different Stokes numbers. The x-axis represents the normalized volume (Vp/〈Vp〉) on a logarithmic scale ranging from 10^-4 to 10^3, and the y-axis represents the probability density function (p(Vp/〈Vp〉)) on a logarithmic scale ranging from 10^-10 to 10^2. The graphs include multiple colored lines representing different Stokes numbers (St): 0.05, 0.1, 0.2, 0.5, 1, 2, and 5. The black dashed line in both graphs represents the volume distribution in the case of the same number of randomly distributed particles. Graph (a) shows the data without gravity, while graph (b) shows the data with gravity. The trends indicate how the volume distribution changes with different Stokes numbers and the influence of gravity.

4.2. Divergence of particle velocity

In this subsection, we aim to understand how inertia and gravitational forces affect particle-velocity divergence, using tessellation-based divergence derived from modified Voronoi cell volume changes, as detailed in § 3.2. This understanding is crucial as the clustering formation and destruction of particles is inherently related to the particle-velocity divergence. By examining the particle-velocity divergence across different Stokes numbers, we assess the influence of gravity and inertia on particle motion in turbulent flows, providing insights into their dynamic behaviour.

Figure 3.

(a) Variance $\mathbb{V}$ and (b) flatness $\mathbb{F}$ of the particle-velocity divergence $\mathcal{D}(\boldsymbol{v}_p)$ normalised by the Kolmogorov time scale $\tau _\eta$ for different Stokes numbers without ( $\boldsymbol{f}_{\!g} = \boldsymbol{0}$ ) and with ( $\boldsymbol{f}_{\!g} = \boldsymbol{g}$ ) gravity.

Figure 3. Long description

The image contains two line graphs labeled (a) and (b). Graph (a) shows the variance of the particle-velocity divergence normalized by the Kolmogorov time scale for different Stokes numbers without gravity (blue line with circles) and with gravity (green line with squares). The x-axis represents Stokes numbers ranging from 0.05 to 5.00, and the y-axis represents the variance on a logarithmic scale from 10^-3 to 10^1. Graph (b) shows the flatness of the particle-velocity divergence normalized by the Kolmogorov time scale for different Stokes numbers without gravity (blue line with circles) and with gravity (green line with squares). The x-axis represents Stokes numbers ranging from 0.05 to 5.00, and the y-axis represents the flatness on a logarithmic scale from 10^1 to 10^3. Both graphs illustrate how the presence of gravity affects the variance and flatness of particle-velocity divergence across different Stokes numbers.

Figure 3 shows the variance and flatness of the particle-velocity divergence for different Stokes numbers with and without gravity. Here, the variance is defined as $\mathbb{V}(X) = \langle (X - \langle X \rangle )^2 \rangle$ , and the flatness by $\mathbb{F}(X) = \langle (X - \langle X \rangle )^4 \rangle /\langle (X - \langle X \rangle )^2 \rangle ^2$ , where $\langle \boldsymbol{\cdot }\rangle$ denotes the ensemble average. Variance offers insight into the average level of divergence fluctuation, while flatness, a normalised fourth-order statistical moment, provides an indication of the distribution’s tails. Specifically, high flatness values signify the presence of more statistically extreme events, characteristic of non-Gaussian and spatially intermittent behaviour. For a Gaussian distribution, the flatness value is equal to 3. For the corresponding PDFs see Appendix B.

For the case without gravity, we observe that the variance increases with the Stokes number. This is consistent with the expectation that particles exhibit a greater tendency to converge or to diverge due to the large particle relaxation time. The flatness also increases with the Stokes number for $\textit{St} \leqslant 0.2$ , but then decreases with the Stokes number for $\textit{St} \geqslant 0.2$ . This could suggest that the particle-velocity divergence is less intermittent for $\textit{St} \leqslant \mathcal{O}(1)$ .

We observe that the statistics for the particle-velocity divergence are similar in both scenarios without and with gravity for $\textit{St} \leqslant 0.2$ , i.e. $S_v \leqslant 1.36$ . For $\textit{St} \geqslant 0.5$ (i.e. $S_v \geqslant 3.39$ ), we can observe a discrepancy in both variance and flatness between the two cases for the absence and presence of gravity. These observations are expected, as notable changes are typically anticipated when $S_v\geqslant \mathcal{O}(1)$ (refer to table 1).

In the scenario with gravity for $\textit{St} \geqslant 0.5$ (i.e. $S_v \geqslant 3.39$ ), the variance is smaller than that in the scenario without gravity. It increases along with $\textit{St}$ up to $\textit{St} = 2$ (i.e. $S_v = 13.6$ ), although at a smaller rate compared with the gravity-free case, and decreases for $\textit{St}\gt 2$ . The flatness, on the other hand, shows a decrease from $\textit{St} = 0.5$ to $\textit{St} = 1$ although it increases with $\textit{St}$ for $\textit{St}\geqslant 1$ . The increase in the flatness for $\textit{St}\geqslant 1$ with gravity, suggests a transition to a different regime due to strong gravitational effects.

In Ireland et al. (Reference Ireland, Bragg and Collins2016b ), the variance of relative particle velocities, both parallel and perpendicular to the separation vector, is examined. It is shown that gravity reduces the variance of relative velocities at small separations, with a more significant reduction as the Stokes number increases. This reduction in the relative velocities is explained by the fact that gravity reduces the effect of the path-history interactions (Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014; Ireland et al. Reference Ireland, Bragg and Collins2016b ). This effect could also explain the observed reduction in the variance of the particle-velocity divergence in the presence of gravity, compared with the case without gravity.

Figure 4.

Two-dimensional slices of thickness $2\pi /1024$ in the $yx$ -plane of the inertial particle positions coloured by the particle-velocity divergence $\mathcal{D}(\boldsymbol{v}_p)$ (a,c,e) without and (b,d, f) with gravity for (a,b) $\textit{St} = 0.1$ , (c,d) 1 and (e, f) $\textit{St} = 5$ .

Figure 4. Long description

A heat map displays the particle-velocity divergence in two-dimensional slices of thickness in the -plane. The map is divided into six subplots, each representing different conditions. Subplots (a), (c), and (e) show data without gravity for different values, while subplots (b), (d), and (f) show data with gravity for the same values. The color scale ranges from blue to red, indicating the magnitude of particle-velocity divergence, with blue representing lower values and red representing higher values. The axes are labeled from 0 to 2π, and the color bar at the bottom indicates the range of values from -1.0 to 1.0. The heat map reveals distinct patterns and clusters of particle-velocity divergence under different conditions.

Figure 4 shows the spatial distribution of particles in two-dimensional slices of thickness $2\pi /1024$ , i.e. half of the grid spacing, of the domain. The particle positions are represented by colours that correspond to the particle-velocity divergence $\mathcal{D}(\boldsymbol{v}_p)$ . This visualisation is presented for three distinct Stokes numbers, both without and with gravity.

For the Stokes number of 0.1, as depicted in panels (a) and (b), we can see small void regions and a majority of the domain exhibits a nearly uniform distribution in cases both without and with gravity. The impact of gravity on the divergence values at the particle position appears significantly small. This observation is consistent with observations in the PDF of the divergence in figure 14.

For the Stokes number of 1, shown in panels (c) and (d), there is noticeable particle clustering. Here, we observe fine clusters and distinct void regions. Moreover, differences between the scenarios without and with gravity are apparent. In the absence of gravity, the distribution of clusters is isotropic, and we can see different types of clusters: clusters where all particles strictly converge or diverge, and clusters where some particles converge and others diverge within the same cluster. In contrast, with gravity, particles tend to align somewhat more with the direction of gravity, and clusters often exhibit more uniform divergence signs, i.e. clusters primarily composed of either converging or diverging particles.

At an even higher Stokes number of 5, as illustrated in panels (e) and (f), the influence of gravity is significant. In the absence of gravity, clusters are more dispersed, and void regions are less distinct compared with $\textit{St}=1$ . Moreover, in the absence of gravity, at a given position, neighbouring particles can take significantly different divergence values, which is a local signature of caustics where the particle velocity becomes multivalued. These large positive and negative divergence values and the more diffuse clusters (i.e. less filamentary structures with reduced void contrast) observed at large Stokes numbers are consistent with model studies showing that caustics can weaken spatial fractal clustering in the sense that the spatial correlation dimension increases, potentially up to three (the dimension of physical space), as caustics become more prevalent (Meibohm et al. Reference Meibohm, Gustavsson, Bec and Mehlig2020). In contrast, gravity introduces noticeable anisotropy by significantly elongating clusters in the direction of gravity, similar to the observations reported in Ireland et al. (Reference Ireland, Bragg and Collins2016b ). The divergence varies continuously within the domain, even though some locations of extreme variation can be observed. In this regime, there is a noticeable reduction in overall particle divergence compared with scenarios without gravity. This indicates that gravitational forces significantly influence particle trajectories. This could potentially reduce the turbulent interactions that typically drive convergence and divergence of particles. Overall, the observed patterns confirm previous studies (Bec et al. Reference Bec, Homann and Ray2014; Gustavsson et al. Reference Gustavsson, Vajedi and Mehlig2014) and provide the physical context for the interpretation of the multiresolution analyses introduced below.

For high Stokes numbers without gravity in figure 4, we can observe cluster regions exhibiting both positive and negative divergence. The behaviour could explain the increase in variance for $\textit{St} \geqslant 0.5$ as the Stokes number increases in the absence of gravity. This occurs when their inertial properties lead to more frequent crossing trajectories, i.e. the particle velocity becomes multivalued, which is referred to as caustics (Wilkinson & Mehlig Reference Wilkinson and Mehlig2005; Bec et al. Reference Bec, Biferale, Cencini, Lanotte and Toschi2010). Consequently, particles, on the whole, are more likely to converge or diverge strongly.

However, as this behaviour is more generalised, these events are statistically more frequent, resulting in a reduced flatness as observed in figure 3. In contrast, in the presence of gravity for $\textit{St}\gt 1$ , caustics are rarely observed (as illustrated in figure 4). When caustics do occur, they represent extreme deviations from typical fluctuations (i.e. the variance), thereby resulting in an increased flatness. It may be that, despite the overall damping effect of gravity, intermittent strong vortices occasionally cause particle clusters with distinct time histories to converge, leading to caustics.

These extreme divergence values originate from caustics. In phase space the particle dynamics evolves on a folded manifold, so that several velocity branches coexist at the same position. When this structure is projected onto configuration space ( $\boldsymbol{x}$ -space), particles on different branches approach the same location and cross with large relative velocities (Bec et al. Reference Bec, Gustavsson and Mehlig2024). In the idealised continuum description this collapse corresponds to the divergence of the particle-velocity field tending to $-\infty$ at the caustic (Meibohm et al. Reference Meibohm, Gustavsson, Bec and Mehlig2020) and then resuming from $+\infty$ immediately after crossing. In practice, such infinite values occur statistically only on sets of zero measure in configuration space, so that in a discrete particle system with a finite number of particles one observes very large but finite divergence values instead. This is analogous to the fact that, in one, two or three spatial dimensions, two points sampled from a continuous distribution coincide with probability zero. In a continuous-time description each particle trajectory is a one-dimensional curve in space–time, and in one spatial dimension (two-dimensional space–time) such curves can intersect with non-zero probability, whereas in two or three spatial dimensions their intersections occur only on sets of zero measure so that exact coincidences are not expected in a finite particle sample. Within particle clusters this process can occur almost simultaneously for many neighbouring particles, which produces the observed pattern of divergence alternating between large negative and large positive values at small scales.

We now aim to quantify observations made in figure 4 and assess the particle-velocity divergence at various scales. To this end, we apply the multiresolution analysis detailed in § 3.3, to the previously computed divergence. Figure 5 shows the wavelet energy spectrum of the particle-velocity divergence for different Stokes numbers, both in the absence and presence of gravity. The spectra are normalised by using the Kolmogorov scale $\eta$ , the kinematic viscosity $\nu$ and the energy dissipation rate $\epsilon$ . The advantage of this method over the Fourier spectrum lies in its spatial locality, which is essential for capturing transient and localised events. Furthermore, the method operates on graphs, which represent the native structure of the data we work with, thereby eliminating the need to project the data onto a Cartesian grid for classical Fourier and wavelet analyses.

Figure 5.

Wavelet energy spectrum of the particle-velocity divergence $\mathcal{E}_{\mathcal{D}}(k_V)$ for (a) small and (b) large Stokes number without (solid lines) and with (dashed lines) gravity.

Figure 5. Long description

The image contains two line graphs labeled (a) and (b). Graph (a) shows the wavelet energy spectrum of the particle-velocity divergence for small Stokes numbers, while graph (b) shows the same for large Stokes numbers. Each graph includes solid lines representing data without gravity and dashed lines representing data with gravity. The x-axis of both graphs is labeled k subscript v times eta, and the y-axis is labeled nu to the power of 1/4 times epsilon to the power of negative 3/4 times E subscript D of k subscript v. The graphs compare how the energy spectrum varies with different Stokes numbers and the influence of gravity. The data points are marked with different symbols for each Stokes number, and the trends show how the energy spectrum changes across different conditions. All values are approximated.

For $0.1 \leqslant St \leqslant 0.5$ , the spectra show a peak around $k_V\eta \approx 0.5$ . This means that the scale of cluster and void formation is almost the same for these Stokes numbers. Additionally, an increase in the Stokes number leads to an increase in the spectrum amplitude. This phenomenon could be linked to the increased particle relaxation time, as already discussed for figure 3. In contrast, for $\textit{St} = 0.05$ , the magnitude is much lower and no clear peak is observed at $k_V\eta \approx 0.5$ . For such low inertia, the particle divergence is expected to be nearly zero, so the results at small scales are dominated by correlated numerical noise, as similarly observed when computing the divergence for $\textit{St} = 0$ (Oujia et al. Reference Oujia, Matsuda and Schneider2020). The decay of the spectrum at large wavenumbers is consistent with the fact that, for small Stokes numbers, relative velocities at small spatial separations remain weak (Bhatnagar, Gustavsson & Mitra Reference Bhatnagar, Gustavsson and Mitra2018). As a result the particle-velocity field is smooth at the smallest scales and the divergence tends to vanish as the inter-particle separation goes to zero, so that little spectral energy is present at the largest wavenumbers.

In the cases of $\textit{St} \geqslant 1$ without gravity, we can see that, as the Stokes number increases, the overall amplitude of the spectrum increases, and the wavenumber with the spectrum peak shifts to higher values, i.e. smaller scales. This behaviour can be attributed to the formation of caustics. When the velocity becomes multivalued, the divergence calculated using the tessellation-based method becomes noisy at small scales. As the range of values that the multivalued particle velocity can take expands, the amplitude of the spectrum at high wavenumbers also increases, due to the sampling, which creates discontinuities in the particle velocity. For the case $\textit{St}=5$ without gravity, although a decay is not apparent over the wavenumber range accessible in our data, this does not imply that the spectrum keeps increasing indefinitely toward smaller scales. Joint statistics of separations and relative velocities at comparable Stokes numbers show that large relative velocities at very small separations are possible but remain much less probable than small relative velocities (Bhatnagar et al. Reference Bhatnagar, Gustavsson and Mitra2018). In the context of phase-space models (Bec et al. Reference Bec, Gustavsson and Mehlig2024), this corresponds to a regime where the phase-space correlation dimension $D_2$ remains below three, so that the second-order structure function of particle relative velocity, denoted $S_2(R)$ , still tends to zero as the separation $R$ tends to zero. Typical relative velocities therefore vanish at vanishing separation, even though rare caustic events produce large fluctuations of the relative velocity. In this situation, we expect the divergence spectrum to reach a maximum and then decrease at the largest wavenumbers, and the absence of a clear decay at $\textit{St}=5$ is most likely due to the finite number of particles and the limited range of resolved $k_V$ . For even larger Stokes numbers, when $D_2$ exceeds three and the spatial correlation dimension ( $\hat D_2 = \min \{D_2,3\}$ ) saturates at three, theory and model studies indicate that $S_2(R)$ approaches a constant as $R$ tends to zero (Simonin et al. Reference Simonin, Zaichik, Alipchenkov and Février2006; Bec et al. Reference Bec, Gustavsson and Mehlig2024). This regime of random uncorrelated motion corresponds to a divergence field that behaves like small-scale spatial noise in three dimensions, and in such a case the associated isotropic spectrum is not expected to decay at the largest wavenumbers, with its asymptotic scaling determined by the statistical properties of the underlying random field and one may conjecture a $k_V^2$ scaling at large wavenumbers.

In contrast, for $\textit{St} \geqslant 1$ with gravity (i.e. $S_v \geqslant 6.79$ ), the general trend is a decrease in the overall amplitude of the spectrum when the Stokes number increases, and the wavenumber with peak shifts to lower values. This behaviour could be explained by the influence of gravity, which is conjectured to reduce the residence time of particles with large Stokes numbers within small vortices (Matsuda et al. Reference Matsuda, Onishi and Takahashi2017), thereby leading to fewer interactions with these structures. Thus, this weaker interaction with small-scale eddies reduces their contribution to the particle-velocity divergence, resulting in the peak shift toward larger scales.

4.3. Rotation of particle velocity

In this subsection, we investigate the curl of the particle velocity to understand how inertia and gravity affect the rotational behaviour of inertial particles. The curl of the particle velocity is computed using the method described in § 3.2. Studying the curl of the particle velocity provides insights into the vortical motion and the shearing behaviour in particle motion. In the gravity-free case, the behaviour of inertial particles is isotropic since we consider isotropic turbulence. Hence, we do not need to distinguish between coordinates in the gravity-free case. However, in the presence of gravity, the particle behaviour can be anisotropic. Therefore, we examine statistics separately for the curl component in the $x$ -direction, i.e. in the same direction as gravity, and the curl components in both the $y$ - and $z$ -directions, i.e. perpendicular to the direction of gravity.

Figure 6 provides a quantitative measure of the variance and flatness of the curl of particle velocity ${\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_p)$ normalised by the Kolmogorov time scale $\tau _\eta$ for different Stokes numbers. The corresponding PDFs are shown in Appendix B. In all cases, the variance and flatness are computed over the union of the sets of values corresponding to the individual curl components. In the absence of gravity, this union is formed by combining the $x$ -, $y$ - and $z$ -component values. In the presence of gravity, however, only the horizontal components (i.e. the $y$ - and $z$ -components, which are statistically homogeneous) are merged, while the vertical $x$ -component is treated separately. We observe that for $\textit{St} \leqslant 0.2$ , indicating a negligible impact of gravity, the variance slightly decreases while the flatness is almost constant. However, as the Stokes number increases further, we note that the variance increases in the absence of gravity, which is explainable by a more frequent occurrence of particle trajectory crossings. In contrast, in the presence of gravity, it increases from $\textit{St} = 0.2$ to $\textit{St} = 2$ (i.e. from $S_v = 1.36$ to $S_v = 13.6$ ) and then decreases, illustrating the impact of the gravity on the particle dynamics. This decay in variance for $\textit{St}\gt 2$ is steep compared with the increase for $0.2 \leqslant St \leqslant 2$ , and the variance value for $\textit{St} = 5$ is significantly lower than that for all other considered Stokes numbers, even smaller ones like $\textit{St} = 0.05$ . Concerning the flatness, in the absence of gravity, flatness values increase from $\textit{St} = 0.2$ to $\textit{St} = 1$ , and then decrease for higher values of $\textit{St}$ . Conversely, in the presence of gravity, flatness continues to increase for $\textit{St} \geqslant 0.2$ (i.e. $S_v = 1.36$ ). The observed dependence of the flatness on the Stokes number can also be attributed to particle trajectory crossings. Indeed, the flatness of the curl appears to effectively quantify the intermittency associated with such crossing events. Furthermore, the variance and flatness are slightly lower for the curl along the $x$ -direction compared with those along the combined $y$ - and $z$ -directions for almost all considered Stokes numbers although the behaviour of these quantities is similar across the different directions.

Figure 6.

Variance $\mathbb{V}$ (a) and flatness $\mathbb{F}$ (b) of the curl the particle velocity ${\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_p)$ normalised by the Kolmogorov time scale $\tau _\eta$ for different Stokes numbers without ( $\boldsymbol{f}_{\!g} = \boldsymbol{0}$ ) and with ( $\boldsymbol{f}_{\!g} = \boldsymbol{g}$ ) gravity in the $x$ -direction ( $\mathcal{C}_{x}(\boldsymbol{v}_p)$ ), and the $y$ - and $z$ -directions ( $\mathcal{C}_{y,z}(\boldsymbol{v}_p)$ ).

Figure 6. Long description

Two line graphs compare variance and flatness of particle velocity curl for different Stokes numbers with and without gravity. The x-axis represents Stokes numbers ranging from 0.05 to 5.00. The y-axis on the left graph measures variance on a logarithmic scale from 10^-2 to 10^1, while the y-axis on the right graph measures flatness on a logarithmic scale from 10^0 to 10^3. Three data series are plotted: one without gravity, one with gravity in the x-direction, and one with gravity in the y and z directions. The blue line represents data without gravity, the green line represents data with gravity in the x-direction, and the red line represents data with gravity in the y and z directions. The graphs show how variance and flatness change with different Stokes numbers and the influence of gravity on these parameters.

Figure 7.

Two-dimensional slices of thickness $2\pi /1024$ in the $yx$ -plane of the inertial particle positions coloured by the enstrophy of the particle velocity $\tau _\eta ^2 \mathcal{Z}(\boldsymbol{v}_p)$ (a,c,e) without and (b,d, f) with gravity for (a,b) $\textit{St} = 0.1$ , (c,d) $\textit{St} = 1$ and (e, f) $\textit{St} = 5$ .

Figure 7. Long description

The image consists of six heat maps arranged in a 3x2 grid, each representing two-dimensional slices of thickness in the x-y plane of the inertial particle positions colored by the enstrophy of the particle velocity. The maps are labeled (a) through (f). The first row (a, b) corresponds to a Stokes number of 0.5, the second row (c, d) to a Stokes number of 1, and the third row (e, f) to a Stokes number of 2. The left column (a, c, e) shows the data without gravity, while the right column (b, d, f) shows the data with gravity. The color scale at the bottom ranges from blue to red, indicating the magnitude of enstrophy, with blue representing lower values and red representing higher values. The heat maps show varying patterns of particle distribution and enstrophy intensity, with higher Stokes numbers generally exhibiting more pronounced structures and higher enstrophy values. The presence of gravity appears to influence the distribution and intensity of enstrophy, particularly at higher Stokes numbers.

Figure 7 presents the spatial distribution of particles in two-dimensional slices of the domain, coloured by particle enstrophy $\mathcal{Z}(\boldsymbol{v}_p)$ , i.e. the squared norm of particle vorticity $ (\mathcal{ Z}(\boldsymbol{v}_p) = ||{\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_p)||_2^2 )$ , for three distinct Stokes numbers, for both cases, i.e. without and with gravity. For $\textit{St} = 0.1$ , enstrophy exhibits local fluctuations with significant variations between regions with strong rotation and those where particles are only transported. In regions where the particles are present, the enstrophy of inertial particles is similar to the enstrophy of fluid flow (e.g. see Maurel–Oujia et al. Reference Maurel–Oujia, Matsuda and Schneider2024), while the particles are not present in high enstrophy regions due to the centrifugal effect. This observation is consistent with the fact that particles continue to closely follow the fluid for this Stokes number. As expected, no significant difference can be observed between the cases without and with gravity. For $\textit{St} = 1$ , aside from the elongated clusters in the direction of gravity that we previously observed, difference in the enstrophy for the cases with or without gravity is not obvious in the visualisations. Conversely, for $\textit{St} = 5$ , a distinct difference can be observed in the magnitude of enstrophy between the cases without and with gravity. In particular, the value of enstrophy is significantly lower than that for the other considered Stokes numbers. In the gravity-free case, a similarity can be seen between the regions where the value of enstrophy is highest and the regions where the values of divergence oscillate, as shown in figure 4. This is due to the caustics, which are interpreted by the computational method as extreme values.

Figure 8.

Wavelet energy spectrum of the curl of the particle velocity for different Stokes numbers: (a) the spectra $\mathcal{E}_{\mathcal{C}}(k_V)$ for the case without gravity and (b) the spectra for the case with gravity in the $x$ -direction (solid lines) and in the $y$ - and $z$ -directions (dashed lines), $\mathcal{E}_{\mathcal{C}_x}(k_V)$ and $\mathcal{E}_{\mathcal{C}_{y,z}}(k_V)$ , respectively.

Figure 8. Long description

Two line graphs depict wavelet energy spectrum of the curl of the particle velocity for different Stokes numbers. The first graph (a) shows the spectra for the case without gravity, while the second graph (b) shows the spectra for the case with gravity in the z-direction (solid lines) and in the x- and y-directions (dashed lines). The x-axis represents the non-dimensional wavenumber, and the y-axis represents the normalized wavelet energy spectrum. Different colored lines represent different Stokes numbers, ranging from 0.05 to 5. The graphs illustrate how the energy spectrum varies with the Stokes number and the presence of gravity.

We aim to assess the curl of particle velocity across various scales of motion by applying the multiresolution analysis framework to the curl, aiming to quantify our observations from the preceding figure. Figure 8 presents the wavelet energy spectrum of the curl of the particle velocity for various Stokes numbers, illustrating particle behaviour both without and with gravity. Panel (a) depicts the scenario without gravitational influence, while panel (b) demonstrates the effects of gravity, analysed separately along the $x$ -direction, and the $y$ - and $z$ -directions. Each curve represents an average of the various quantities considered. Specifically, in panel (a) we display $\mathcal{E}_{\mathcal{ C}}(k_V) = (\mathcal{E}_{\mathcal{C}_x}(k_V)+\mathcal{E}_{\mathcal{C}_y}(k_V)+\mathcal{E}_{\mathcal{C}_z}(k_V))/3$ , while in panel (b) the solid line corresponds to $\mathcal{E}_{\mathcal{C}_x}(k_V)$ , and the dashed lines represent $\mathcal{E}_{\mathcal{ C}_{y,z}}(k_V) = (\mathcal{E}_{\mathcal{C}_y}(k_V)+\mathcal{E}_{\mathcal{C}_z}(k_V))/2$ . For small Stokes numbers, we observe a slight shift of the spectrum towards larger scales as inertia increases. Then, for $0.2 \leqslant St \leqslant 2$ , there is a decrease in the peak value of the spectrum along with the increase in $\textit{St}$ . This decrease in the peak value of the spectrum is more significant in the presence of gravity up to $\textit{St}=5$ . We can observe that, in the case with gravity, the peak value is slightly larger in the $x$ -direction than in the $y$ - and $z$ -directions, while the opposite can be observed for small wavenumbers. In the scenario without gravity, there is also a notable increase in the higher wavenumbers as the Stokes number increases for $\textit{St} \geqslant 0.5$ , which is due to particle crossings and random behaviour of the particles. This is particularly visible for $\textit{St} = 5$ , as we had already observed in figure 7. Thus, in the case without gravity, we observe that rotational motion at scales corresponding to $k_V\eta \approx 0.5$ diminishes as the Stokes number increases. For $\textit{St} = 5$ , it is noted that the majority of the contribution to the PDF of the curl of the particle velocity in figure 15(a) is due to small scales, which are attributable to caustics. In the case with gravity, we observe a decrease in the amplitude of the wavelet energy spectrum. Particles tend to be advected mainly in the direction of gravity at a relatively spatially uniform speed across different regions of the domain. This results in shear reduction of the particle velocity, and thus in reduced enstrophy values at all scales.

4.4. Helicity of particle velocity

In this subsection, we analyse the relative helicity of the particle velocity to discern the underlying swirling motion of inertial particle-laden turbulent flow and to understand the nature of particle motion influenced by forces such as inertia and gravity. The study of helicity provides insights into coherent or incoherent particle motion and is considered a topological statistics. Large helicity values, i.e. strong alignment or anti-alignment between velocity and vorticity, tend to suppress chaotic mixing and lead to swirling three-dimensional motion. Thus helicity can be seen as a measure of flow organisation. In this work, we refer to coherent particle motion as the coordinated, non-random motion of a group of particles (a particle and its neighbours, or clusters) that forms structures persisting in time and space. In such regions, particle related quantities, such as particle velocity, vorticity, divergence or helicity, exhibit weak fluctuations within the group, in contrast to the strongly irregular variations expected for random particle motion.

Rewriting the incompressible Euler equations in terms of the Lamb vector reveals that nonlinearity decreases when the velocity aligns with the vorticity, corresponding to a relative helicity of $-1$ or $1$ . This alignment between these two quantities simplifies the flow motion. Moreover, in incompressible flows it has been observed that vortex tubes often maximise helicity (Tsinober & Levich Reference Tsinober and Levich1983; Levich Reference Levich1987). In Farge, Pellegrino & Schneider (Reference Farge, Pellegrino and Schneider2001), helicity is utilised to identify vortex tubes and to distinguish between coherent structures and incoherent motion within the flow. It is observed that the PDF of the relative helicity of the coherent flow exhibits a convex shape with two pronounced maxima at $-1$ and $1$ , while the incoherent flow has a concave shape with a maximum at $0$ . In this section, we will adopt a similar approach for helicity of the particle flow. The relative helicity plays the same role even for coherent particle flows when the flows are close to continuous since the particle momentum equation obtained from the Boltzmann equation has a similar form to the Navier–Stokes equation (Kasbaoui et al. Reference Kasbaoui, Koch and Desjardins2019). However, the same idea may not be applicable when the effect of random particle motions exceeds the effect of the nonlinear transport relevant to the Lamb vector. Therefore, the relative helicity may not be an appropriate indicator of coherent structure in the caustic regions.

Unlike divergence or curl, helicity is not Galilean invariant. When a uniform mean velocity is applied to the entire flow field, the helicity value changes. We can express the particle velocity as $\boldsymbol{v}_{p} = \boldsymbol{v}_{p}' + \overline {\boldsymbol{v}}_{p}$ , where $\boldsymbol{ v}_{p}'$ represents the fluctuating component of the particle velocity and $\overline {\boldsymbol{v}}_{p}$ is the mean particle-velocity vector. For the case without gravity, $\overline {\boldsymbol{v}}_{p}$ is statistically zero vector since the particle motion is affected only by the HIT, whereas for the case without gravity, $\overline {\boldsymbol{v}}_{p}$ deviates from zero vector due to the gravitational settling velocity. When $\overline {\boldsymbol{v}}_{p}\neq \boldsymbol{0}$ , we have

(4.1) \begin{align} {\mathcal{H}}(\boldsymbol{v}_{p}) &= \frac {\boldsymbol{v}_{p}\boldsymbol{\cdot }{\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_{p})}{||\boldsymbol{v}_{p}||_{2} \, ||{\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_{p})||_{2}} \\[-10pt] \nonumber \end{align}

(4.2) \begin{align} &= \frac {\boldsymbol{v}_{p}'\boldsymbol{\cdot }{\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_{p}') + \overline {\boldsymbol{v}}_{p}\boldsymbol{\cdot }{\boldsymbol{\mathcal{C}}}(\boldsymbol{ v}_{p}')}{||\boldsymbol{v}_{p}' + \overline {\boldsymbol{v}}_{p}||_{2} \, ||{\boldsymbol{\mathcal{C}}}(\boldsymbol{v}_{p}')||_{2}} \neq {\mathcal{H}}(\boldsymbol{v}_{p}'). \end{align}

For this reason, in the case with gravity, we have also computed the relative helicity ${\mathcal{H}}(\boldsymbol{v}_{p}')$ after subtracting the mean particle velocity in the $x$ -direction from the particle velocity, in addition to ${\mathcal{H}}(\boldsymbol{v}_{p})$ .

Figure 9.

The PDFs of the relative helicity $\mathcal{H}(\boldsymbol{v}_p)$ of the particle velocity for different Stokes numbers (a) without and (b) with gravity and (c) the relative helicity $\mathcal{H}(\boldsymbol{v}_p^\prime )$ of the particle-velocity fluctuation for the case with gravity.

Figure 9. Long description

The image contains three line graphs labeled (a), (b), and (c). Each graph displays the probability density function of the relative helicity of particle velocity for different Stokes numbers. Graph (a) shows data without gravity, while graph (b) includes gravity. Graph (c) presents the relative helicity of the particle-velocity fluctuation for the case with gravity. The x-axis represents the relative helicity values ranging from -1.0 to 1.0, and the y-axis represents the probability density function values ranging from 0.40 to 0.70. Different colored lines represent various Stokes numbers, including 0.05, 0.1, 0.2, 0.5, 1, 2, and 5. The graphs illustrate how the relative helicity changes with different Stokes numbers under varying conditions.

Figure 9 shows the PDFs of the relative helicity of particle velocity for various Stokes numbers, both without and with gravity. As the impact of gravity is very low, no significant change is expected. In the gravity-free scenario, as the Stokes number increases, the PDF shape becomes more convex, exhibiting two distinct maxima a $\pm 1$ . In Farge et al. (Reference Farge, Pellegrino and Schneider2001), this pattern was shown to be characteristic of coherent structures in fluid turbulence. This suggests that similar conclusions can be drawn for the motion of inertial particles, i.e. this pattern indicates coherent structures in the motion of inertial particles. A transition occurs at $\textit{St} = 2$ , where the PDF shape shifts from convex to concave shape. At $\textit{St} = 5$ , the PDF becomes concave, showcasing a maximum at 0 and two minima at $\pm 1$ . This distribution implies that the vorticity is more likely to be orthogonal to the particle velocity at higher Stokes numbers, suggesting a predominantly two-dimensional particle motion, i.e. the flow is only weakly swirling. This pattern is indicative of incoherent motion in fluid turbulence (Farge et al. Reference Farge, Pellegrino and Schneider2001), and by analogy, similar conclusions can be applied to the motion of inertial particles.

In figure 9(b), we can observe that particle motion is strongly helical for the larger Stokes numbers. This is consistent with the results shown in figure 8, where the magnitude of $x$ -component of the curl is higher compared with the other directions. Given that the mean particle velocity is also oriented along the $x$ -direction due to gravity, it is expected that we observe a greater alignment between velocity and vorticity vectors, resulting in higher helicity values.

This behaviour is reduced in figure 9(c) where the mean particle velocity has been subtracted, although the motion remains helical. While in the gravity-free case, the PDF tended to become concave and the motion more two-dimensional for the larger Stokes numbers studied, in the presence of gravity, the relative helicity exhibits increasingly helical behaviour as the Stokes numbers increase, even for $\textit{St} = 5$ . This suggests that particles tend to exhibit swirling motion as they fall. The influence of gravity, therefore, appears to enhance the coherence in the particle motion, leading to more helical flow patterns, particularly at higher Stokes numbers.

Figure 10.

Average of the second derivative of the PDF of the relative helicity $\mathcal{H}(\boldsymbol{v}_p)$ for different Stokes numbers without ( $\boldsymbol{0}$ , $\mathcal{H}(\boldsymbol{v}_p)$ ) and with gravity without modification ( $\boldsymbol{g}$ , $\mathcal{ H}(\boldsymbol{v}_p)$ ) and by removing the mean particle velocity in the $x$ -direction ( $\boldsymbol{g}$ , $\mathcal{ H}(\boldsymbol{v}'_p)$ ).

Figure 10. Long description

The line graph presents the average of the second derivative of the probability density function (PDF) of the relative helicity for different Stokes numbers. The x-axis represents the Stokes number (St) ranging from 0.1 to 5.0. The y-axis represents the average of the second derivative of the PDF of the relative helicity (p double prime (H)) ranging from -0.1 to 0.7. Three data lines are plotted: one in blue circles representing (0, H(v_p)), one in green squares representing (g, H(v_p)), and one in red triangles representing (g, H(v’_p)). The blue line starts at approximately 0.05 at St 0.1, slightly increases, and then decreases to approximately -0.05 at St 5.0. The green line starts at approximately 0.05 at St 0.1, increases steadily, and reaches approximately 0.65 at St 5.0. The red line starts at approximately 0.05 at St 0.1, increases gradually, and reaches approximately 0.35 at St 5.0. All values are approximated.

These various observations can be quantified in figure 10, which presents the average values of the second derivative of the PDF of relative helicity, i.e.

(4.3) \begin{align} \overline {p''(\mathcal{H})} = \int _{-1}^1 \frac {d^2 p(\mathcal{H})}{{\rm d}\mathcal{H}^2} {\rm d}\mathcal{H}. \end{align}

This quantity permits assessment of the convexity of the PDF curves, thereby enabling a quantitative evaluation of the coherent or incoherent motion of the particles. Curve smoothing was applied using the Savitzky–Golay algorithm (Savitzky & Golay Reference Savitzky and Golay1964) before computing the second derivative of each PDF curve. A value close to zero indicates a curve with a linear shape, a positive value denotes a convex curve and a negative value signifies a concave curve. For $\textit{St} = 0.1$ , the value of convexity is positive but low. In the case without gravity, we can observe an increase in convexity with a maximum value at $\textit{St} = 0.5$ , followed by a decay until $\textit{St} = 5$ where the value becomes negative, with the highest amplitude among all Stokes numbers considered in the absence of gravity. In the case with gravity, there is no observed decrease in convexity, reaching a maximum at $\textit{St}=5$ . In the case with gravity, with the mean particle velocity subtracted, the convexity also increases but not as strongly compared with not subtracting the mean particle velocity. We can also note that the convexity values for $\textit{St} = 0.2$ and $\textit{St}=0.5$ are similar, which is further reflected by the fact that the PDFs are almost superimposed.

Figure 11.

Two-dimensional slices of thickness $2\pi /1024$ in the $yx$ -plane of the inertial particle positions coloured by the relative helicity of the fluctuating particle velocity (a,c) without and (b,d) with gravity for (a,b) $\textit{St} = 1$ and (c,d) $\textit{St} = 5$ .

Figure 11. Long description

A heat map displays two-dimensional slices of inertial particle positions in the -plane. The particles are colored according to the relative helicity of their fluctuating velocity. The heat map is divided into four sections: (a) and (c) show data without gravity, while (b) and (d) show data with gravity. The color scale ranges from blue to red, indicating varying levels of relative helicity, with blue representing lower values and red representing higher values. The axes are labeled from 0 to 2π, and the color bar at the bottom indicates the range of relative helicity values from -1.0 to 1.0.

Figure 12.

Conditional probabilities of observing a low relative helicity ${|\mathcal{H}|}\lt 0.1$ (blue lines) and a high relative helicity ${|\mathcal{H}|}\gt 0.9$ (red lines) as a function of the local variance of the particle-velocity divergence (used here as a proxy to identify caustic regions), $\mathbb{V}_{{bin}} (\tau _\eta \mathcal{D}(\boldsymbol{v}_p))$ , evaluated in spatial bins on the $yx$ -plane using a $512\times 512$ grid for particles belonging to a slice of thickness $2\pi /1024$ . The probabilities are estimated by grouping particles into $256$ variance quantile bins, and the curves are smoothed for readability using a Gaussian filter with $\sigma =3.0$ . Results are shown (a,c) without and (b,d) with gravity for (a,b) $\textit{St}=1$ and (c,d) $\textit{St}=5$ .

Figure 12. Long description

The image contains four line graphs labeled (a), (b), (c), and (d). Each graph shows conditional probabilities of observing low relative helicity (blue lines) and high relative helicity (red lines) as a function of the local variance of the particle-velocity divergence. The x-axis represents the local variance of the particle-velocity divergence, while the y-axis represents the probability. Graphs (a) and (b) are for a specific parameter value without and with gravity, respectively. Graphs (c) and (d) are for another parameter value without and with gravity, respectively. The probabilities are estimated by grouping particles into variance quantile bins, and the curves are smoothed for readability using a Gaussian filter. The graphs illustrate how the presence of gravity affects the distribution of relative helicity in different scenarios.

To further assess the coherence of particle motion and to clarify the role of caustics in shaping the relative helicity statistics, we complement the PDF based analysis with spatial visualisations and conditional probabilities of helicity. Figure 11 shows the spatial distribution of particles in two-dimensional slices of the domain, coloured by the relative helicity associated with the fluctuating particle velocity. Unlike the divergence and enstrophy visualisations, we do not show the low-inertia case (e.g. $\textit{St}=0.1$ ) since we focus on regimes where a difference between the cases without and with gravity becomes visible. We can observe that, except for the case $\textit{St}=5$ without gravity, helicity exhibits spatially continuous fluctuations, with progressive variations from one particle to another in regions where particles are present. We also find that, except for $\textit{St}=5$ without gravity, the majority of particle clusters has large helicity magnitude values which means that they exhibit swirling motion. This is particularly interesting for the case $\textit{St}=5$ with gravity where strong helicity magnitude indicates that the settling of heavy particles takes place in a swirling motion. Conversely, for $\textit{St}=5$ without gravity, we observe a strong intermittency of helicity, with abrupt variations, including changes in sign and amplitude, for particles that are close to each other. This intermittency is nevertheless not uniform throughout the domain, and some regions instead show smoother variations, comparable to those observed for the other parameters. By comparing the helicity distribution for $\textit{St}=5$ without gravity (figure 11 c) with the corresponding divergence visualisation (figure 4 e) we observe that the regions where helicity is the most intermittent coincide with regions where divergence exhibits intense oscillations that are characteristic of caustics. Conversely, the continuous transitions of helicity correspond to regions where these caustic signatures are absent or weak. A similar correspondence, although less pronounced, can also be observed for the other cases by comparing with the corresponding divergence figure. We predict that, when helicity is intermittent in caustic dominated regions, it should also exhibit strong temporal fluctuations. In particular, during trajectory crossing events associated with caustics, the local helicity carried by the particles is expected to vary rapidly, indicating that it is not temporally stable in those regions. However, the fact that helicity becomes intermittent in caustic regions does not provide information on the impact of caustics on the helicity distribution.

To quantify the impact of caustics on helicity, we relate the relative particle helicity to local divergence fluctuations, using the variance of $\tau _\eta \mathcal{D}(\boldsymbol{v}_p)$ as a proxy to localise regions dominated by caustics. We discretise the $yx$ -plane on a regular $512\times 512$ grid and within each bin we compute the variance $\mathbb{V}_{{bin}} (\tau _\eta \mathcal{D}(\boldsymbol{v}_p) )$ from the values carried by the particles located in that bin. Each particle is then assigned the variance of the bin that contains it. Figure 12 shows, for $\textit{St}=1$ and $\textit{St}=5$ and for the cases without and with gravity, the conditional probabilities of observing a low relative helicity $P(|\mathcal{H}|\lt 0.1 | \mathbb{V}_{{bin}} (\tau _\eta \mathcal{D}(\boldsymbol{v}_p) ))$ and a high relative helicity $P(|\mathcal{H}|\gt 0.9| \mathbb{V}_{{bin}} (\tau _\eta \mathcal{D}(\boldsymbol{v}_p) ))$ as a function of $\mathbb{V}_{{bin}} (\tau _\eta \mathcal{D}(\boldsymbol{v}_p) )$ . These probabilities are estimated by grouping particles into $256$ classes defined by variance quantiles and then computing within each class the fraction of particles satisfying each threshold. The curves are smoothed using a Gaussian filter with parameter $\sigma =3.0$ to improve readability. If caustics had no impact on helicity, we would expect to observe a constant curve, up to statistical noise. We can observe that, for $\textit{St}=1$ without and with gravity, indeed no clear trend emerges, as the statistics are dominated by noise due to the limited number of samples. Conversely, for $\textit{St}=5$ , a trend appears in the evolution of both probabilities. This trend is clear in the case without gravity. More precisely, the probability of observing a low relative helicity increases with the divergence variance, and thus with caustic intensity, whereas the probability of observing a high relative helicity decreases as the divergence variance increases.