Clustering in laboratory and numerical turbulent swirling flows

Nulls of all vector fields were computed in the DNSs using the methods described in Haynes & Parnell (2007) and Mora et al. (2021). Using the positions of the zeros, and the particles’ positions in the experiment and DNSs, 3D Voronoï tesselations were computed for all datasets. While the DNSs use $10^{6}$ particles, in the experiment we have a maximum of $\approx 29$ particles in each frame, resulting in a significantly smaller number of closed 3D Voronoï cells: there are an average of 3 closed cells per snapshot. When considering all snapshots this results in a total number of closed cells of $1.9\times 10^{5}$. Also, in DNSs, the tessellations were performed with periodic boundary conditions. These two observations will be relevant to interpret some of the results.

Figure 2 shows the PDFs of the normalized volumes of the Voronoï cells, $\altmathcal{V}=V/\langle V\rangle$, for STPS, ZAPS, and ZWEROs in TG and HIT, and the PDFs of the Voronoï cells for the particles in the DNSs and the experiment. For the experimental data, since the total number of particles fluctuates in time (as particles can enter or exit the observation volume $V_{obs}^{\text{EXP}}$, and as already mentioned, can explore regions behind the disks), $\langle V\rangle_{\text{EXP}}$ is defined as the volume occupied by the fluid between the propellers divided by the mean number of particles per image, calculated from all of the available images. This choice is robust to variations in the number of particles (Fiabane et al., 2013), and to the presence of spurious volumes near the boundaries of the images as will be discussed later.

It has been hypothesized that the forcing mechanism or the large-scale flow can introduce changes in the clustering of nulls of the fluid vector fields (Goto & Vassilicos, 2009). This figure confirms this is indeed the case: HIT displays stronger clustering of all fields than TG (i.e., larger probabilities for small volumes). In the figures, the PDF resulting from a random Poisson process (herein RPP, corresponding to homogeneous distribution of nulls or particles) is shown as a reference. The larger standard deviation of the PDFs (compared to the RPP, whose standard deviation is $\sigma_{\text{RPP}}\approx 0.42$) and the stronger tails (of clusters to the left, and of voids to the right) indicate enhanced clustering. However, the Voronoï cell volumes of the particles is similar for the three flows, and notably, the level of clustering as quantified by $\sigma{V}$ is similar for TG and EXP (see table 1). As will be discussed next, differences between EXP and TG can be associated with finite sampling and with boundary effects.

In Fig. 2 we also show an exponential and a -5/3 power law to serve only as references for the comparison. Note STPS in HIT are more similar to the $\altmathcal{V}^{-5/3}$ power law in the vicinity of $\altmathcal{V}\approx 1$ (i.e., for intermediate volumes, their distribution is closer to a power law than to an exponential decay), which is a consequence of the power-law behavior of the velocity auto-correlation function (Davila & Vassilicos, 2003; Smith et al., 2008). TG STPS display a less clear scaling, which is compatible with contamination of the scaling of the velocity by the mean flow (Angriman et al., 2020). ZAPS and WZERO do not exhibit a power law behavior, and are closer to an almost exponential decay $\sim e^{-\altmathcal{V}}$ in all DNSsfor large volumes (i.e., for $\altmathcal{V}>1$), with the aforementioned differences between HIT and TG for very small values of $\altmathcal{V}$. Remarkably, the PDFs of Voronoï cells of all particles are very similar in spite of the differences in the distribution of zeros, and for intermediate and large volumes ($\altmathcal{V}\gtrsim 1$) are closer to an exponential decay, indicating the clustering of particles for those volumes is more akin to the distribution of either ZAPS or WZERO than to the distribution of STPS. It is worth recalling that some studies of finite-size, neutrally buoyant particles in HIT laboratory flows indicate that small particles do not cluster and behave as tracers (Fiabane et al., 2012), at least from a Voronoï tessellation point of view. Fiabane et al. (2012) considered neutrally buoyant particles whose radii ranged from $2.25\eta$ to $8.5\eta$ ($0.0875\lambda$ to $0.23\lambda$). The particles employed here are well outside this range, with a size comparable to the Taylor microscale, a scale at which viscous effects from the turbulent flow become less relevant. Our results suggest that Taylor-scale neutrally buoyant particles cluster similarly to inertial particles. The numerical particle data thus serve the purpose of gauging the level of clustering the experimental particles experience when compared to the minimal model in the simulations. As particles in the DNS cluster akin to different flow nulls, the comparison allows to further relate the clustering in the experiment with specific properties of the background flow.

Measurement of 3D clustering in experiments poses multiple challenges. Particle seeding must be sufficiently small to remain in the regime of a dilute suspension (Elghobashi, 1994), which results in a few particles per measurement. When computing 3D Voronoï tessellations, this results in a small number of closed cells, roughly proportional to the cube root of the total number of detected particles. Thus, the number of closed Voronoï cells scales slowly with the number of particles (see Tagawa et al. 2012 for another study of 3D clustering in experiments). Lastly, boundaries have a drastic effect in the statistics. In this section we compare DNSs, synthetically post-processed DNSs, and laboratory data, to quantify the effect of these ever-present limitations. We therefore evaluate possible sources of biases in the 3D Voronoï tessellation calculated from the inertial particles’ instantaneous positions by processing DNS data so as to mimic experimental conditions.

We first focus on the effect of boundaries. Particles near the boundaries of the observation region in the experiment occupy either open cells (which are always discarded in the analyses) or artificially large cells, as particles closing their cells (or the domain walls) are not visible in the cameras field of view. The result is depicted in Fig. 3(a) (inset), which shows the Voronoï tessellation on the raw EXP data without imposing any restriction on the cells near boundaries (labeled “no BC” in the figure); note the power-law tail for very large $\altmathcal{V}$. We reproduced this situation in the TG DNS by computing the tessellation imposing periodic boundary conditions (“PBC”, considered as ground truth), by imposing no condition on the boundaries (“no BC”, which results in a power-law dependence), by removing all cells close to boundaries (labeled “FC” in the figure), and by dropping the largest cells that resulted in a total volume larger than the observation volume (that is, keeping only “physical” volumes, “PV”), see Fig. 3(a). The latter method, when applied to EXP data, results in the PDF shown in the inset of Fig. 3(a) (the strategy also used in Fig. 2(f)). In each case, $\langle V\rangle$ is the total volume occupied by the fluid divided by the mean number of particles, to remove biases (especially when large spurious volumes are kept). Boundaries mainly affect the statistics of large cells, but the left tails of the PDFs remain the same regardless of the strategy used for treating the boundaries. This suggests boundaries do not affect significantly the determination of clusters (as clusters are formed by small volumes $\altmathcal{V}$ below some threshold, as will be discussed in the following section). Furthermore, we verified that when analysing DNS data in the same way as EXP data (i.e., by considering the non-periodic VK cells in the TG DNS and keeping only “physical” Voronoï cells), the distribution of cluster volumes reported in the following section remains unchanged.

The relatively low number of particles in experiments also has an effect in the PDFs. Figure 3(b) shows EXP data together with the original TG data (tessellated using periodic boundary conditions), and the TG data sub-sampled to a maximum of 250 and 5000 particles. These sub-sampled sets were built so as to follow the same distribution of number of particles per frame as in the experiments. This was done in the following way. The maximum number of particles $N_{\text{max}}$ considered (either 250 or 5000) represents the DNS-equivalent of the total number of particles inside the experimental vessel (but not necessarily within the observation volume at once); e.g. $N_{\text{max}}=250$ corresponds to roughly 31 particles available in each of the 8 individual VK cells the DNS comprises. The experimental variability in the actual number of particles that simultaneously occupy the observation volume at any time instant was mimicked in these numerical datasets based on the occupation probability distribution (i.e., the probability to observe a certain number of particles at each time instant), which was calculated empirically from the EXP data. As a result, for $N_{\text{max}}=250$, the mean number of particles per frame in the complete numerical domain of volume $(2\pi)^{3}$ is $\approx$ 111, leading to approximately 13 particles per VK cell of the TG flow, comparable to what is attained in the experiments. These sub-sampled datasets were then processed in a manner identical to what was done with the experimental data; namely, employing a tessellation scheme without considering periodic boundary conditions and keeping only physical volumes in the entire $(2\pi)^{3}$ domain. To further mimic the experimental conditions, the sub-sampling of the TG data was also performed in one VK cell, i.e., in just the octant comprised by the subdomain $[0,\pi)\times[0,\pi)\times[0,\pi)$, thus keeping an average of only $13$ particles per snapshot. The tessellation was then carried out as in the case of $N_{\text{max}}=250$ and $N_{\text{max}}=5000$ but in the smaller subdomain; the resulting PDF is shown in Fig. 3(b) as well.

Decreasing the number of particles in the DNS in these ways -in both the ($2\pi)^{3}$ domain and in the VK cell- as shown in Fig. 3(b), results in PDFs whose left tails slowly move towards the RPP, partially explaining the slight defect of small volumes $\altmathcal{V}$ seen in the EXP data. The reduction in the number of particles, and the treatment of boundaries discussed above, also has a small effect in the concavity of the right tail of the observed PDF. For a detailed study of other experimental biases in 2D Voronoï tessellation, including sub-sampling effects, see Monchaux (2012), and for an investigation of finite-size effects on Voronoï analysis of randomly placed spheres, see Uhlmann (2020).

While small (larger than the Kolmogorov scale) particles do not cluster (Fiabane et al., 2012), neutrally buoyant particles with sizes of the order of the flow integral scale have been reported to cluster in von Kármán experiments (Machicoane et al., 2014). The clusters in that case were found to be associated to large-scale flow inhomogeneities (i.e., preferential sampling by the particles) and to confinement effects (Machicoane et al., 2016), rather than to inertial effects. It is then worth analyzing if the clustering behavior observed in our finite-size particles is the result of, or at least affected by, a global preferential sampling as the one reported by Machicoane et al. (2014). To this end we computed the 3D spatial concentration map $C({\bf{x}})$ of the particles in the experiment, using all available snapshots and normalized so that the average in the entire observation volume is $\langle C({\bf{x}})\rangle_{V_{obs}^{\text{EXP}}}=1$. Figure 4(a) and (b) show two-dimensional probability density maps of the particles’ positions (proportional to the 2D concentration maps) in the $x-z$ and $y-z$ planes respectively, where $\hat{z}$ is chosen parallel to the rotation axis of the propellers and to the direction of gravity. Our particles present a more uniform distribution than the larger particles considered in Machicoane et al. (2014) and Machicoane et al. (2016). This is consistent with previous observations: in Machicoane et al. (2014) the larger particles are the ones that display the stronger inhomogeneous preferential sampling, while their smaller particles (with $R/\lambda\approx 2.6$) sample the flow more uniformly than their larger particles. In spite of this, in Fig. 4 a very small tendency for our particles to be in the lower half of the cell ($z<0$) can still be observed, specially near the corners. This may be partially due to asymmetries in the position of the shear layer in the von Kármán flow (Ravelet et al., 2004; Huck et al., 2017), to a small effect of gravity, and to the particles entering and exiting the region behind the propellers; the latter effect is particularly visible in the lower corners of the concentration maps.

To quantify the effect that the particles preferential sampling has on the observed clustering, we computed the Voronoï cells compensating each cell volume $\altmathcal{V}$ by its local particle concentration. The purpose of doing this is to correct any underestimation (resp. overestimation) of the cell volumes resulting from the cells being located in regions of space with higher (resp. lower) particle concentration. This can be done in two ways. For a given particle located at ${\bf{x}_{0}}$, the volume of its Voronoï cell $\altmathcal{V}$ can be multiplied by the concentration at the particle position $C({\bf{x}_{0}})$, i.e., $\altmathcal{V}C(\bf{x}_{0})$, or by the average concentration in the entire cell that contains that particle $\langle C({\bf x})\rangle{V}$, i.e., $\altmathcal{V}\langle C\rangle{V}$. We thus computed PDFs using these two compensated cell volumes, which are shown in Fig. 4(c) along with the PDF of $\altmathcal{V}$ without any compensation. The overall shape of the PDFs remains similar, with the heavy tails that deviate from the RPP being preserved. The second compensation, using the mean concentration in each cell, does not change significantly the distribution of large volumes, but does decrease slightly the probability of finding small volumes. Even so, the compensated PDFs indicate that large-scale sampling effects as reported in Machicoane et al. (2014) play a smaller role in our case, and confirm that the clustering observed in the experiment is at least partially associated to inertial effects. As a reference, the standard deviation of the Voronoï volumes compensated by $\langle C\rangle{V}$ is $\sigma_{\altmathcal{V}_{\langle C\rangle}}/\sigma_{\text{RPP}}\approx 2.73$, which is slightly larger than $\sigma{V}/\sigma_{\text{RPP}}\approx 2.33$ for the uncompensated Voronoï volumes in the experiment (see table 1), and closer to the reference value for the minimal model of inertial particles in the TG DNSs.

Figure 5 shows the PDFs of the volumes of the clusters $V_{c}$ (formed by at least two adjacent cells with $\altmathcal{V}<\altmathcal{V}_{c}$, where $\altmathcal{V}_{c}$ is the first crossing between the PDF of $\altmathcal{V}$ with the RPP, see Monchaux et al. (2010)), for all vector fields nulls and for the particles, in the DNSs and EXP. From Mora et al. (2021) we expect a $V_{c}^{-5/3}$ scaling for STPS, and $V_{c}^{-2}$ for ZAPS and WZEROs, resulting from the fractal nature of the spatial distribution of these zeros. Hence, the presence or not of these power laws provides information on the clusters geometry and their fractality (see Obligado et al. (2014) and Mora et al. (2021) for simple models and a review of the origin of these power laws). For all nulls, HIT shows a slight excess of smaller volume clusters when compared with TG, but overall the PDFs and scalings are similar.

Figure 5(f) also shows the PDFs of $V_{c}$ for particles in the DNSs and EXP (in the latter case, with and without corrections by the mean average concentration per volume), which are expected to follow a $-2$ power-law scaling, with a shorter range as a result of the large St considered. Clusters of particles in the experiment have different PDFs than in TG or HIT. The right tails of the PDFs in the experiment, associated with volumes $V_{c}>\langle V\rangle$, present a sharp drop as the cluster size cannot be larger than the observation volume ($V_{obs}^{\text{EXP}}/\langle V\rangle$ is indicated by the arrow in the figure). On the other hand, in the PDF of uncompensated EXP volumes, the probability of having small clusters (with $V_{c}<\langle V\rangle$) is larger than in the DNSs. We verified that this is not an effect of sub-sampling or boundaries, so this difference underlines limitations in the scope of the toy model considered in Eq. (4) and, ultimately, in the effect of the mean large-scale flow as captured by the model, either from missing physical effects or from preferential sampling. Neglected interactions between particles is not the probable cause, as the volumetric loading ratio is $\lesssim 2\times 10^{-4}$. However, particles in the DNSs, while subjected to Stokes drag, are point particles that can only sense pressure gradients indirectly through the carrier velocity field. On the other hand, finite-size particles are expected to sense the pressure gradient directly as a force acting on their surface, as well as velocity gradients (and thus, shear stresses) that the point particles in the DNSs cannot sense. Pressure gradients have already been pointed out as important in the understanding of the formation of clusters of finite-size particles (Fiabane et al., 2012). Finally, when compensating each EXP particle volume by the mean concentration, the probability of finding smaller clusters decreases and becomes closer to that of the model, suggesting that the small preferential concentration induced by the large-scale flow geometry plays a role here. This is confirmed by the fact that small and large clusters in TG and EXP flows occur at slightly different places, which could also explain why their PDFs look differently.

In table 1 a comparison of the mean linear cluster size $\langle V_{c}\rangle^{1/3}$ for all quantities is given, normalized by $\eta$. The mean linear cluster sizes of WZERO and ZAPS are similar for TG and HIT, but STPS clusters are slightly larger for HIT, a difference that may be related to the fact that the HIT forcing scale is larger than in TG, and thus larger clusters may appear. The mean linear cluster size of particles in EXP is much larger than in TG or HIT. Even when conditioning the data as discussed in Sec. 3.2 (that is, sub-sampling the number of particles in the DNS, and considering boundary effects), or as in Sec. 3.3 (i.e., correcting by the particles concentration), we observe that clusters in the experiment are still 3 times larger, on average, than in TG. This difference is associated to the finite volume of the particles in EXP. Note that finite-size particles can only form clusters with a minimum linear size of order $4R$, which for EXP data is greater than $120\eta$. In other words, some clusters in TG cannot be realized in the experiment without the actual particles overlapping.

The PDFs of the number of clusters $M_{c}$ with $N_{PC}$ points inside each cluster are shown for the DNSs in Fig. 6. These PDFs follow a power-law behavior with an exponent close to $-16/9$ for STPS, and $-2$ for ZAPS and WZERO, as expected from Mora et al. (2021). Figure 6(c) indicates that the probability of having clusters of STPS with a larger number of nulls is larger for HIT than for TG, consistently with the larger mean linear size of clusters of STPS in HIT. Inertial particles also follow a power law with an exponent close to $-2$ for clusters with $N_{PC}\lesssim 10$, a behavior consistent with the one reported by Yoshimoto & Goto (2007). Overall, the observed scaling is similar for ZAPS, WZERO, and inertial particles, which is compatible with the sweep-stick mechanism (Goto & Vassilicos, 2008; Coleman & Vassilicos, 2009; Oka & Goto, 2021). The deviation from the power law observed for particles in Fig. 6(f) might relate to the fact that particles with large Stokes number filter out small flow scales. Indeed, these kinds of particles are expected to be less responsive to small scale motions in the fluid, so clusters with larger number of particles around very dense nulls in the carrier flow are less likely to form. Finally, the inset in Fig. 6(f) shows the linear cluster size normalized by $L_{F}$, as a function of the number of particles per cluster for the inertial particles. Note the cluster size grows with the number of particles with a $1/3$ scaling.