2.1. Planar, two-colour PIV in the log-layer

The two-colour PIV and PTV measurements were performed in the $200\ {\rm mm} \times 200\ {\rm mm} \times 2000\ {\rm mm}$ test section of the Technion High Speed Water Tunnel Facility. The experimental set-up for the two-colour PIV is illustrated in figure 1, where the streamwise ($x$), wall-normal ($y$) and spanwise ($z$) velocity components are denoted by $u$, $v$ and $w$, respectively. The wall-parallel ($x$–$z$) plane was illuminated with a 527 nm wavelength laser sheet produced by a dual-pulse laser (Litron LD30-527) and imaged by two high-speed cameras (Phantom VEO-340L and VEO-440L, $2560 \times 1600$ pixels) positioned above and below the test section, oriented perpendicular to the measurement plane. The field of view was centred on a streamwise position 847 mm downstream of the entrance of test section where the boundary layer was tripped via a small step in the wall surface. The laser sheet was approximately 0.8 mm thick and was positioned with its centre location, $h$, 2.9 mm above the floor of the test section. Previous streamwise/wall-normal measurements were used to determine the friction velocity of the flow by a modified Clauser method. Details of the flow, including the acceleration parameter, $K= ({\nu }/{u_\tau ^2})({{\rm d}U_\infty }/{{\rm d}\kern0.7pt x})$, are shown in table 1. The mean velocity profile, turbulence intensity profile and streamwise map of spectral energy density appear in Appendix A. The wall-normal range of the laser sheet corresponded to $h^+ = 229\unicode{x2013}302$, which falls within the upper range of the log layer described by Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) as $3 {Re}_\tau ^{1/2} < y^+ < 0.15 {Re}_\tau$ or, in this case, $139 < y^+ < 323$. The total measurement area was $5.9\delta$ in the streamwise direction and $3.7 \delta$ in the spanwise direction, centred within the test section.

Figure 1. An illustration of the experimental set-up, showing the green laser illumination of the wall-parallel plane ($x$–$z$) near the bottom wall of the test section. Green and red dots represent the glass tracer particles and fluorescent inertial particles, respectively. Two cameras were positioned above and below the test section, each with colour filters to record the two types of particles simultaneously. The resulting filtered fields are illustrated above and below the test section, in place of the cameras.

Table 1. Parameters for the underlying boundary layer flow and the location of the laser sheet at wall normal height, $h$. The boundary layer thickness, $\delta$, represents $\delta _{99}$. Mean flow profiles and streamwise energy spectra are provided in Appendix A.

The flow was seeded with small, glass tracers (Sigma-Aldrich 440345-500G, $1.1\ {\rm g}\ {\rm ml}^{-1}$, diameter $9\unicode{x2013}13\ \mathrm{\mu}{\rm m}$) to measure the velocity field, and large polyethylene fluorescent particles (Cospheric UVPMS-BR-1.20 $75\unicode{x2013}90\ \mathrm {\mu }{\rm m}$, $1.2\ {\rm g}\ {\rm ml}^{-1}$, diameter $75\unicode{x2013}90\ \mathrm {\mu } {\rm m}$) to identify particle clustering. The volume fractions for the tracers and particles were $6.7 \times 10^{-5}$ and $9.2 \times 10^{-6}$, respectively.

The key parameter describing the relative inertia of the particles compared with that of the carrier flow, and thus the extent of coupling, is the Stokes number, $St$, which is the ratio of the viscous time scale of the particle, $\tau _p$, to an appropriate time scale of the flow, $\tau _f$. The viscous time scale, in the limit of infinitesimal particle Reynolds number, ${Re}_p \equiv {d_p^3 \rho _f g (\rho _p-\rho _f)}/{18 \mu ^2} \ll 1$, is given in terms of the particle density, $\rho _p$, the fluid density, $\rho _f$, the particle diameter, $d_p$, and the dynamic viscosity, $\mu$, as $\tau _p= {(\rho _p-\rho _f)d_p^2}/{18\mu }$ (Eaton & Fessler Reference Eaton and Fessler1994). For heavy particles, $\rho _p \gg \rho _f$, this viscous time scale is approximately $\tau _p \approx {\rho _pd_p^2}/{18\mu }$ (Stokes Reference Stokes1851).

The flow time scale $\tau _f$ for turbulent flows can be defined in terms of the Kolmogorov scales, as $\tau _\eta = (\nu /\epsilon )^{1/2}$ (via the dissipation rate, $\epsilon$, and kinematic viscosity, $\nu$) following Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010) and Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002); or in terms of the friction velocity as $\tau _\nu = {\nu }/{u_\tau ^2}$, following Yamamoto et al. (Reference Yamamoto, Potthoff, Tanaka, Kajishima and Tsuji2001) and Zhao, Andersson & Gillissen (Reference Zhao, Andersson and Gillissen2013). The dissipation rate in the log layer was estimated from the planar PIV data, assuming that it is in balance with the production rate (see Brouwers (Reference Brouwers2007) for discussion). The production itself was approximated as $\mathcal {P} \approx -\overline {uv}({\partial U}/{\partial y}) -\overline {vv}({\partial V}/{\partial y})$, where the remaining contributions to the production are assumed negligible, following Blackman et al. (Reference Blackman, Perret, Calmet and Rivet2017), yielding a dissipation rate of $\epsilon \approx 0.6\ {\rm m}^2\ {\rm s}^{-3}$.

Table 2 summaries the Stokes numbers and mass/volume fractions for the particles used in this experiment, along with a selection of relevant past studies performed in a channel flow, boundary layer and in experiments with HIT. The current volume fraction for inertial particles was consistent with previous studies, although the mass fraction is much lower, due to the smaller density of the current particles. Nevertheless, the Stokes number of the inertial particles is nearly two orders of magnitude larger than the velocity tracers due to their size. According to Brandt & Coletti (Reference Brandt and Coletti2021), the low volume fraction in all of these experiments is indicative of the one-way coupling regime between the fluid and inertial particles, which is better suited to observe the influence of large-scale coherent motions of the canonical boundary layer on the particle clustering.

Table 2. Range of particle parameters for current experiment (top) and past clustering studies (bottom). The Kolmogorov time scale, $\tau _\eta$, was calculated using the dissipation rate obtained via the balance of production and dissipation in the log layer. Here DNS is direct numerical simulation.

2.2. Particle and velocity field processing

The fluorescent, inertial particles had an excitation peak at 575 nm and an emission peak at 607 nm. Therefore, the fluorescent light of the particles was isolated from the laser light by using a notch filter (Chroma, ZET532nf) blocking the laser wavelength, $527$ nm. The reflected light of the tracer particles was isolated from the fluorescent light by using a bandpass filter (Chroma, ET525/30m) overlapping the same wavelength. Because the fluorescent particles also reflected light at the laser wavelength, albeit at much weaker intensity than the glass tracers, the slight signature of the fluorescent particles was removed from the tracer image using a digital masking procedure, which also eliminated noise from the particle image. The fluorescent particles were masked using a fixed intensity threshold ($1000$ out of $4096$ grey levels). The resulting binary mask was then used for measuring the particle clustering behaviour, described in § 3, as well as for masking out the low-intensity reflections of the fluorescent particles from the velocity tracer images, prior to PIV processing of the velocity field. The size distribution of the fluorescent signature of the particles is discussed in the next section, § 2.3.

The masking and PIV processing was performed using commercial software (Davis 10.1.1). Five independent recordings were performed at 800 Hz with $\Delta t = 130\ \mathrm {\mu }{\rm s}$ ($\Delta t^+= 1.1$) between image pairs. The duration of each recording was approximately 188 eddy turnover times ($\delta / U_\infty$), yielding a total temporal record length of around 940 eddy turnover times. The convergence of all statistical quantities was verified by random sampling of the snapshots across different eddy turnover times. Perspective correction was applied to the image pairs according to a planar calibration image, and a sliding-average background of size 8 pixels was subtracted. The multipass vector calculation included an initial pass of a square $32\times 32$ pixel window, followed by a second pass with a $16\times 16$ pixel circular window. To avoid spatial aliasing, interrogation windows with $50\,\%$ overlap were applied for both passes. The correlation value was also calculated for the PIV algorithm and vectors with a correlation value lower than 0.5 were deleted. Vectors with greater deviation than two times the standard deviation were removed and replaced with interpolated velocity data. The spatial resolution of the underlying images was $0.054\ {\rm mm}\ {\rm pixel}^{-1}$, resulting in a spatial resolution of the velocity field, after PIV processing, of $\Delta x^+ = 39.5$. The spatial resolution for the particle field was $\Delta x^+ = 4.9$.

The particle velocities themselves were obtained by PTV applied to the pre-binarized fluorescent particle image pairs, where the time delay between pairs was $130\ \mathrm {\mu }{\rm s}$. A ‘Laplace of Gaussian’ filter was applied to the particle images first, and the particle centres were obtained with subpixel accuracy after a Gaussian interpolation, following Heyman (Reference Heyman2019). The velocities of the particles were then calculated by matching particle pairs between two pulses, using the numerical approach described in Janke, Schwarze & Bauer (Reference Janke, Schwarze and Bauer2020). More than $95\,\%$ of particles in the two pulses were matched with high confidence; the remaining $5\,\%$ particle losses were due to out-of-plane particle motions.

2.3. Particle cross-section homogenization

The threshold binarization of the fluorescent particle field had the effect of exaggerating the size of some particles due to their greater scattering cross-section. The variations in scattering cross-section can result from both variations in the physical size of the particle (which, in this case is relatively small for highly monodisperse particles) and variations in the intensity of incident light, which varies with the location of the particles relative to the peak intensity of the laser sheet as noted by Raffel et al. (Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018). In order to identify particle clusters purely by their spatial distribution, without bias due to the scattering cross-section of individual particles, the binarized particle images were homogenized to generate a field in which all particles were represented with a uniform number of pixels (i.e. perfectly monodisperse).

The homogenization process first eliminated all binary pixel clusters with fewer than two pixels. The pixel clusters were then fitted by an ellipse where the major ($d_{maj}$) and minor ($d_{min}$) axes of the ellipse were defined by the second moments of the underlying cluster area, as described in Appendix A of Haralock & Shapiro (Reference Haralock and Shapiro1991). The image processing was performed using MATLAB. Prior to homogenization, the mode of the major axes was approximately $3.7$ pixels, and the distribution was clearly not singular, as desired. Thus, all of the irregular shaped (polydisperse) pixel clusters were replaced by square clusters at their original centroid locations with side length 3 pixels, corresponding to a major axis of approximately $3.5$, in order for the homogenized particle field to retain particles of similar size as in the original particle field, but perfectly homogeneous in shape and pixel count.

3.1. Voronoi tessellation

The Voronoi technique was implemented following the approach of Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010) and Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2012). We first analysed each homogenized particle field to obtain the unique Voronoi tessellation of cells describing that field, as shown in figure 2(a). Each cell represents the region of the field closer to the particle it encloses than to any other particle. A cluster of particles is defined as a region of contiguous cells, each of area $A$, where the area of each cell normalized by the average area of all cells, $\langle A \rangle$, $\bar {A} = A/\langle A \rangle$, is smaller than some normalized threshold area, $\bar {A}_c$, that would be expected were the particles distributed spatially by a uniform random distribution with the total number of particles specified by a Poisson point process, with expected value, $\lambda$, set equal to the average number of particles per field detected in the real experiments. (An alternative method for generating synthetic particle fields with fixed numbers of particles per field will be described in § 3.2.2.)

Figure 2. (a) A sample particle field with particles denoted by blue points surrounded by the black edges of the Voronoi cells. Cells with area $\bar {A} < \bar {A}_c$ are shaded grey. (b) Probability density ($p)$ of normalized Voronoi cell areas ($\bar {A}$) for: $10^5$ synthetic fields generated by a Poisson process (dashed, red) with shape parameter equal to the mean number of particles in the experimental particle fields; and $4425$ experimental particles fields (solid, black). The vertical dashed (black) line represents the crossing point between the two p.d.f.s ($\bar {A}_c= 0.68$) which distinguishes between clustered and non-clustered cells.

Figure 2(b) shows the distribution of the cell areas from the synthetic Poisson fields (dashed) and measured particle fields (solid). Cells bordering the edge of the measurement region were eliminated. The intersection point between the two distributions is used as the threshold area value, $\bar {A}_c$. Cells with area smaller than this threshold represent regions of unusually dense particle concentration and are marked grey in figure 2(a).

The contiguous regions of grey cells in figure 2(a) were treated as particle clusters. Note that here, contiguity includes cells that share a common edge and even cells connected by only a single vertex. To quantify the size and orientation of these very irregular clusters of cells, each cell cluster was fitted to an ellipse by least squares (following the same procedure described in § 2.3 for the pixel clusters) and the major and minor axes, denoted $L_{maj}$ and $L_{min}$, respectively, and the orientation angle, $\theta$, measured with respect to the streamwise direction, were extracted. The characteristic ellipses associated with the Voronoi clusters are illustrated in figure 3(a). This ellipse fitting was done for consistency with the wavelet analysis presented below in § 3.2, in which the wavelets are naturally characterized by the geometric parameters of an ellipse. But the characteristic scale of the Voronoi clusters can also be calculated directly by their total area and perimeter, as done by Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010).

Figure 3. (a) Ellipses identifying the cell clusters previously identified in the example particle field from figure 2(a), marked with red solid lines. (b) Joint p.d.f. ($p$) of the major and minor axis lengths of ellipses bounding the Voronoi clusters, including all orientation angles. The domain of the map represents the physical measurement limits of the present experiment, including the definitional requirement for all ellipses that $L_{maj} > L_{min}$.

The parameters describing the cluster ellipses provide a convenient statistical perspective on both the size and orientation of particle clusters in the flow. Figure 3(b) shows the joint p.d.f. of the cluster major axes, $L_{maj}$, with respect to the minor axes, $L_{min}$, averaged over all orientation angles. The vast majority of particle clusters detected by the Voronoi technique is smaller than $\delta$ and there are almost no large clusters detected with a size of $2\delta$ or greater. The most probable clusters are slightly anisotropic, with $(L_{maj}/\delta,L_{min}/\delta ) \approx (0.29,0.18)$. In terms of the actual area of the Voronoi cluster, the mode was approximately $\sqrt {A_v}/\delta \approx 0.2$. These clusters are a bit larger than the typical size detected in previous studies; for instance, Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021) reported the most probable Voronoi area as $\sqrt {A_v}/\delta \approx 0.04$. The difference is due to the particle volume fraction: in Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021), the volume fractions are $7 \times 10^{-5}$ and $11\times 10^{-5}$, approximately one order of magnitude higher than the volume fraction in present experiment, $9.2 \times 10^{-6}$.

The tendency of the Voronoi approach to detect small-scale features is also apparent in reports of clustering in HIT by Monchaux et al. (Reference Monchaux, Bourgoin and Cartellier2010). There, they reported an inverse power law distribution (with an exponent of $-2$) of particle cluster areas, without any particular characteristic length scale, and all of the clusters identified were nearly an order of magnitude smaller than the integral length scale of the flow (granting that their field of view in the streamwise direction extended only approximately two integral length scales). Using a box-counting technique, Aliseda et al. (Reference Aliseda, Cartellier, Hainaux and Lasheras2002) were able to obtain a typical cluster size for HIT of approximately 10 Kolmogorov length scales, again about an order of magnitude smaller than the integral scales of their flow.

Based on Voronoi analysis alone, we might conclude that the significant organization of particles is confined to the small scales and does not extend in space as far as the coherent momentum structures, associated with LSMs. However, the Voronoi analysis is inherently limited in its ability to detect such extended features, because even a single non-contiguous cell can interrupt a cluster. In other words, multiple contiguous cells are needed to instantiate a large-scale cluster, but only a single large cell can prevent one from being detected, and thus we expect a bias towards the detection of small-scale features. In particular, if large-scale features are the result of a superposition of small-scale features, as is hypothesized in the case of momentum structures, then any separation between the constituent small-scale features will prevent detection of the resultant large scales by the Voronoi technique. We therefore turn to an alternative technique for spatial cluster identification in order to identify large-scale particle clustering behaviour without this bias: the wavelet.

3.2. Wavelet transformation

In order to identify large-scale particle clusters, the homogenized binary particle map was transformed via a wavelet transform. The wavelet transform represents a convolution of a wavelet kernal, $\psi (x,z)$, with the underlying wall-parallel ($x$–$z$) particle field in order to detect clusters of particles that correspond to the wavelet geometry. Farge (Reference Farge1992) noted that the continuous wavelet transform is better suited for tracking coherent structures, although worse for efficient modal representation of data. Because of this, the astrophysics studies referenced in § 1.3 generally employed the continuous wavelet transform to identify galaxy clusters, whereas previous turbulence studies utilized the discrete transform to produce efficient, low-order representations of turbulent flow fields.

3.2.1. Anisotropic wavelets

The wavelet employed for the cluster detection was the anisotropic (real-valued) Mexican hat wavelet as defined in Antoine et al. (Reference Antoine, Murenzi, Vandergheynst and Ali2004), which allows for independent variations in the minor axis length of the wavelet, $L_{min}$, the aspect ratio of the major and minor axes, $\varepsilon = L_{max}/L_{min}$, and the major axis orientation angle $\theta$, and is particularly well suited for feature detection applications, as noted by Hou & Qin (Reference Hou and Qin2012). The wavelet parameters are illustrated in figure 4(a). The major and minor axes describe the spatial domain of an ellipse that coincides with the positive annulus of the wavelet. (Note that not all anisotropic Mexican hat wavelet definitions are the same – see the discussion in Appendix B for details.)

Figure 4. (a) A representative wavelet amplitude map, illustrating the signed magnitude of the wavelet (red for the positive annulus, blue for the negative), showing its major and minor axes, $L_{maj}$ and $L_{min}$, and its orientation angle, $\theta$. The inset illustrates the cross-sectional slice of the wavelet across its major axis, showing the magnitudes of the positive (red) and negative (blue) annuli. (b) A grid of the wavelet dimensions and aspect ratios, where each intersection indicates a particular combination of wavelet parameters. The dashed horizontal line indicates the streamwise domain size.

As the Mexican hat wavelet is convolved with a particle field, the resulting wavelet coefficient field represents the magnitude of the spatial particle clustering within the positive annulus of the wavelet, discounting those particles that appear in the negative annulus, and ignoring particles situated beyond that region. Therefore, the spatial parameters which define the positive annulus of the wavelet also characterize the geometric clustering of the particles detected by that wavelet. The axes are thus conceptually analogous to the elliptical axes that were used to bound the grey cells of the Voronoi clusters in the previous section. By varying the wavelet parameters, $(L_{min},\varepsilon,\theta )$, different shapes and orientations of particle clusters were detected.

The wavelet transform was calculated over 10 linearly spaced minor axis scales, $L_{min}$, starting from the smallest scale of interest, $0.1 \delta$ or $215 \nu /u_\tau$, chosen to comfortably exceed the typical spanwise length scale of the near wall streaks (see Smith & Metzler Reference Smith and Metzler1983), and extending to $0.73 \delta$. This resulted in the largest major axis within the measurement domain, $L_{maj} \approx 5.9$, obtained when the minor axis is multiplied by the largest aspect ratio considered. The aspect ratios were linearly spaced between $\varepsilon = 1$ and $10$. The resulting set of wavelet dimensions is illustrated in the map in figure 4(b). Every combination of wavelet dimensions was deployed at all orientation angles, $\theta = 0^\circ, 15^\circ, 30^\circ,\ldots, 180^\circ$, resulting in a total number, $n$, of unique wavelets, $(L_{min},\varepsilon,\theta )_j$.

Each particle field, $i = 1,\ldots,N$, was transformed into the wavelet domain for each unique wavelet, $j = 1,\ldots,n$, yielding maps of wavelet coefficients $C_{ij}(x,z)$. The continuous wavelet was implemented via a Fourier transform technique (‘cwtft2’ in MATLAB), where finite boundary effects were avoided by assuming a periodic particle domain.

Following the approach of Escalera & Mazure (Reference Escalera and Mazure1992), the locations, $(x^*,z^*)_{ij}$, of the relative maxima, $C^*_{ij}(x^*,z^*)$, of the wavelet transform field were identified using a sparse peak finder. These maxima represent strong regions of local particle clustering at the scale (and orientation) of a specific wavelet. However, the presence of a relative maximum alone does not necessarily indicate a statistically significant spatial density of particles. In order to ascertain whether the maximum is significant, it was compared with a threshold value of the wavelet coefficient maximum, $\hat {C}_{95}$, at the 95th percentile level, determined via the wavelet transform of a uniform, spatial, random process with the same number of particles as the field of interest. To accomplish this comparison, synthetic particle fields were generated with a uniform random process in order to establish the statistical threshold of significance.

3.2.2. Significance testing via synthetic particle fields

The threshold for significant wavelet maxima, $\hat {C}_{95}$, was determined as a function of the wavelet geometric properties $L_{min}$ and $\varepsilon$ (excluding $\theta$ by symmetry arguments). However, the wavelet coefficient field was also found to depend on the size of the particles used in the homogenization procedure, $d_h$ (measured in pixels), as well as the number density of particles in a particular field, $\phi _p$, where number density is defined as the number of particles per field area.

The particle number density dependence also influenced the Voronoi technique described above, and was taken into account via the generation of synthetic particle fields. There, the random fields were based on a Poisson point process, in which the number of particles per field was selected from a Poisson distribution with the same mean number of particles as the measured fields. In this way, the dependence on number density was swept into the threshold generation implicitly. An alternative approach, adopted in the wavelet studies of Slezak et al. (Reference Slezak, Bijaoui and Mars1990) and Escalera & Mazure (Reference Escalera and Mazure1992), involved generating random fields with a fixed number of particles, identical to the experimental field of interest. In this way, the confounding effect of number density was eliminated from the threshold completely, not just in an average sense. The difficulty with using a fixed number of particles is that every new experimental particle field would require a corresponding set of synthetic fields with that same particle density to generate a new statistical threshold. Therefore, in this study, we chose the middle ground between these approaches: explicitly modelling the wavelet field dependence on the number density, $\phi _p$, to obtain higher accuracy than the Poisson assumption (which had previously been criticized by Escalera et al. (Reference Escalera, Slezak and Mazure1992)) without the computational cost of calculating separate synthetic fields for each new measured field.

The threshold wavelet maximum, $\hat {C}_{95}(L_{min},\varepsilon,d_h,\phi _c)$, was determined by (1) generating large sets of random particle fields for each parameter value; (2) finding the converged distribution of wavelet maxima $C^*$ for each set; (3) selecting the 95th percentile value of $C^*$ to define the threshold, $\hat {C}_{95}$, for each particular wavelet; and (4) fitting an empirical function to describe parametric dependence of $\hat {C}_{95}$ on these individual wavelet parameters. The functional representation of the threshold reduced the computational burden of calculating new synthetic fields for every possible combination of wavelet parameters. To obtain a statistically converged threshold, each set of wavelet parameters required a large number of synthetic fields (resulting in more than $2.5 \times 10^6$ maxima detected per wavelet). The convergence behaviour and accuracy of the threshold, $\hat {C}_{95}$, is discussed in Appendix C.

Figure 5 shows the variation of the threshold, $\hat {C}_{95}$, as a function of the particle size, $d_h$ in pixels (figure 5a), the particle number density (per frame), $\phi _p$ (figure 5b), the wavelet minor axis, $L_{min}$ (figure 5c), and the aspect ratio, $\varepsilon$ (figure 5d). These different relations were collapsed into the approximate threshold function,

(3.1)\begin{align} \hat{C}_{95}(d_h,L_{min},\varepsilon,\phi_p) = \hat{C}_{95,0} \left(\frac{d_h}{d_{h,0}}\right)^{\alpha_d} \left(\frac{\phi_p}{\phi_{p,0}}\right)^{\alpha_\phi} \left(\frac{L_{min}}{L_{{min},0}}\right)^{-\alpha_L\left({\phi_p}/{\phi_{p,0}}\right)^{\beta_L}} \!\!\left(\frac{\varepsilon}{\varepsilon_0}\right)^{-\alpha_\varepsilon \left({\phi_p}/{\phi_{p,0}}\right)^{\beta_\varepsilon}}, \end{align}

where each parameter, $A$, is normalized by a characteristic value, $A_0$, and then raised to a power that itself can depend on number density, in the form: $\alpha _A \phi _p^{\beta _A}$. The fit was performed by nonlinear least square regression, and the range of parameters and their resulting exponents and corresponding standard errors, are listed in table 3. The quality of the fit was assessed by comparing the actual statistical percentile of the synthetic data corresponding with the model threshold (i.e. the effective threshold percentile) versus the design percentile of $95$ from which the model was formulated. It was found that more than $87\,\%$ of the effective threshold percentiles were within the percentile of $95\pm 2.5$, such that using the model threshold was likely to produce nearly the same effective significance level as direct examination of the distribution of maxima from synthetic fields.

Figure 5. The variation of $\hat {C}_{95}$ with respect to (a) homogeneous particle size, $d_h$, holding the other parameters fixed; (b) particle number density, $\phi _p$; (c) wavelet size, $L_{min}$; (d) wavelet aspect ratio, $\varepsilon$. The individual points (red circles) are chosen from within the range given in the second column of table 3. The solid lines are fitted power laws with exponents given in the last two columns of table 3.

Table 3. Nonlinear, least-squares, best-fit parameters for the wavelet maxima threshold model described in (3.1). The uncertainty is listed as standard error, where $n=132$ unique wavelets were used for fitting the power-law surface.

The threshold relation in (3.1) was applied to the experimental particle fields to determine which wavelet maxima represented statistically significant spatial particle clusters. Unlike the threshold used previously in the Voronoi analysis, the percentile-based approach adopted here allows for the choice of significance level, and provides for an intuitive interpretation of the particle clusters. Significant particle clusters are defined in terms of the likelihood of those clusters appearing in a field of randomly distributed particles. Clusters of particles that result in a wavelet maximum in the top 5 % of all wavelet maxima from randomly distributed particle fields are defined as statistically significant. Other threshold values, from 10 % to 1 %, were also examined and result in similar trends as presented below, with some slight variation in the prominence of less likely (i.e. more extreme) cluster sizes.

3.2.3. Wavelet-identified clusters

Figure 6(a) illustrates the same example particle field used above for the Voronoi analysis, with all the significant wavelet maxima circumscribed by their corresponding positive annulus ellipses, showing the regions of significant, coherent clustering. The wavelet ellipses in the example appear to encompass all of the particles previously identified by the Voronoi analysis, along with a substantial number of additional particles not identified by the Voronoi. Over all the particle fields, $94\,\%$ of the particles associated with Voronoi clusters were also associated with wavelet clusters. And while the Voronoi analysis placed approximately $32\,\%$ of all particles inside clusters, the wavelet analysis placed nearly $72\,\%$ of all particles inside clusters. Of course, this number could be reduced by increasing the statistical significance threshold. But more important than the difference between these techniques in the number of particles identified within clusters is the difference in appearance of the cluster-bounding ellipses. Unlike the Voronoi ellipses, the wavelet ellipses overlap each other, i.e. multiple different wavelets detect the same underlying clusters of particles, but at different scales.

Figure 6. (a) Ellipses identifying the wavelet clusters, marked with red solid lines, overlayed on the Voronoi clusters from figure 3(a). (b) Joint p.d.f. of the two ellipse axes of the wavelet clusters for all orientation angles.

The overlapping of ellipses is indicative of the hierarchical nature of the wavelet detection technique, which identifies small clusters whose superposition creates the appearance of larger clusters, and also those large clusters themselves. This overlap reveals a hierarchy of particle clustering, similar to the momentum hierarchies familiar from wall-bounded turbulence. However, this redundancy also creates an interpretative difficulty, because the distribution of wavelet clusters inherently conveys a cumulative picture, potentially double-counting ‘child’ clusters within ‘parent’ clusters. Thus special care is required when analysing the distribution of the geometric cluster parameters obtained by the wavelet technique.

Figure 6(b) shows the joint p.d.f. of the cluster major axes, $L_{maj}$, with respect to the minor axes, $L_{min}$, averaged over all orientation angles, comparable to the joint p.d.f. of the Voronoi ellipses shown earlier in figure 3(b). The Voronoi ellipse parameters were continuously distributed and thus the joint p.d.f. could be constructed by a standard histogram technique. However, the wavelet ellipse parameters constitute a finite set of discrete values, illustrated by the grid in figure 4(b), and thus linear interpolation was used to obtain a continuous probability density surface, i.e. filling in between the grid locations. The outer boundary of the grid is shown in solid black outline. The final Voronoi and wavelet p.d.f.s were linearly interpolated to the same, underlying grid density for comparison.

The wavelet cluster joint p.d.f. shows a high concentration of isotropic clusters, ranging from the smallest wavelets up to the largest (along the bottom edge of the graph), with the highest probability clusters being small, nearly isotropic clusters consistent with the Voronoi distribution. However, unlike the Voronoi distribution, the wavelet p.d.f. also shows a substantial number of large-scale clusters with major axis larger than $1 \delta$ in length, including both highly anisotropic clusters as well as nearly isotropic clusters. These large scales are consistent with the overlapping, large-ellipses noted above, and thus are assumed to be associated with the hierarchical superposition of smaller particle clusters.

5.1. Particle velocity distribution

To study the velocity of the particles within the clusters, we first present the overall probability distribution of the fluid velocity, $u$, and the particle velocity, $u_p$, irrespective of the clusters, in figure 8(a). The fluid velocity distribution (in blue) is bimodal reflecting the signature of the high and low speed streaks in the wall-parallel plane at the height of the log layer. The particle velocity distribution is unimodal with negative skew, where most particles are measured as travelling faster than the mean velocity. In fact, the mode of the particle velocity distribution coincided roughly with the velocity of the near-wall, high-speed streaks, although the particle velocities are broadly distributed.

Figure 8. (a) Distribution of overall fluid velocity, $u$ (blue) and particle velocities, $u_p$ (red). (b) Distribution of particle velocities in-clusters (solid) and out-of-clusters (dashed) for the Voronoi method. (c) In-cluster and out-of-cluster particle velocities for the wavelet method.

In the hairpin paradigm, Tomkins & Adrian (Reference Tomkins and Adrian2003) argued that the high-speed streaks are associated with the forward flow induction produced by aligned hairpin legs. Dennis & Nickels (Reference Dennis and Nickels2011) provided additional experimental evidence in support of this view, via conditional averaging, showing that the high-speed regions appear adjacent to the hairpin legs, due to the forward flow induction. However, this does not necessarily mean that the particles are dynamically associated with the high-speed streaks themselves. They could just as easily be associated with the sweeping events induced along the backbone of the hairpin packet, as argued by Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021), and still obtain a similar velocity distribution due to the same induction process.

The overall particle velocity distribution was then conditioned on whether the particles in question were part of a cluster or not. For the Voronoi technique, shown in figure 8(b), the difference between the in-cluster and out-of-cluster particle velocity distributions was very subtle, with in-cluster particles exhibiting a slightly higher average velocity. For the wavelet technique, shown in figure 8(c), the difference was a bit more pronounced, again with in-cluster particles travelling faster. In both cases, the differences were not significant enough to drawn any inference about the relative position of the in-cluster particles with respect to coherent motions in the momentum field.

5.2. Particle clusters convection scaling

In order to establish the physical orientation of the particle clusters in the turbulent boundary layer with respect to the coherent structures, conditional averaging of the momentum field was attempted. However, as Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021) previously reported, such averaging is impractical when dealing with very low particle concentrations in the flow. Therefore, they employed other inferential techniques, including quadrant analysis in the vicinity of particles, as well as conditional spectral techniques, in order to argue that particles tended to collect along the sweep and ejection events that demarcate the outer envelope of hairpin packets (or, equivalently, uniform momentum zones). Here, we take a different approach towards inferring the particle orientation by exploiting the trend of particle velocities with respect to their cluster size. The convection velocity of coherent structures varies with their size, as has been known since the early work of Wills (Reference Wills1964), and thus so too should the velocity of particle clusters being convected by those structures.

In order to establish a relationship between cluster size and particle convection velocity, ideally we seek a unique convection velocity for each cluster size, ignoring the superposition effect. Thus, in this section, we focus on only the unique clusters identified by the Voronoi analysis. Then, to obtain the relationship between the particle velocities and the large-scale clustering, we averaged the particle velocities conditioned on the size of their associated Voronoi clusters in order to construct a map of the average particle velocities as a function of the cluster axis lengths. However, because the variation in particle velocities across different clusters was high, the averaging was performed using the ‘objective mapping interpolation’ technique, as described by McIntosh (Reference McIntosh1990), to obtain a smooth representation of the particle velocity trends. The maps of average particle velocities are shown in figure 9 for the Voronoi technique. Overlapping the particle velocities are the contour lines associated with the cluster size distribution from figure 3(b). A general trend of increasing particle velocity with cluster size is visible in both maps, with greater sensitivity to the variation in the ellipse minor axes compared with the major axes.

Figure 9. (a) Contour map of mean particle velocity for each Voronoi cluster as a function of $L_{maj}$ and $L_{min}$ calculated via objective mapping interpolation. (b) Grey scatter points correspond to the mean particle velocity for each Voronoi cluster; the red line is a $0.06 \delta$-wide, rectangular moving-average of the scattered clusters; the black line is the average trend calculated from the contour map. The blue triangle represents the slope of coherent structure convection velocities, $u_c$, with respect to their streamwise wavelength, $\lambda _x$.

To observe the trends in more details, we plot the average variation in particle velocity with respect to the major and minor axes separately (averaging over the respective axes) in figure 9(b). These plots indicate the high variability in particle velocities per cluster by the cloud of grey points in the background (each point representing the average velocity of particles in a specific cluster), and then show the moving-average smoothing of the cluster velocities in red, and the average trends obtained by the ‘objective mapping interpolation’ technique in black. Both smoothing techniques resulted in a robust trend of increasing average particle velocity with cluster size.

Adrian et al. (Reference Adrian, Meinhart and Tomkins2000) provided a conceptual model for the convection velocity of different sized hairpin packets. Smaller packets exist closer to the wall and induce more intense backward velocity than larger packets, which reach farther from the wall, meaning that larger packets advect downstream faster. Del Alamo & Jimenez (Reference Del Alamo and Jimenez2009) quantified how the advection velocity of eddies increases with size as the attached eddies reach farther into the high-speed outer flow. They employed a ‘correlation height’ for the large-scale coherent structures and deduced the corresponding convection velocity from a convolution of the mean velocity profile of the flow in the vicinity of the correlation height.

The approximate convection velocity of different scale coherent motions was calculated following the approach of Del Alamo & Jimenez (Reference Del Alamo and Jimenez2009) and is shown in figure 10 as a function of streamwise and spanwise wavelengths of the motions, denoted $\lambda _x$ and $\lambda _z$, respectively. The mean velocity profile used in the calculation was a channel flow reported by Hoyas & Jiménez (Reference Hoyas and Jiménez2008). The convection velocity averaged over spanwise wavelength is shown in figure 10(b), where the slope of the convection velocity with respect to the streamwise wavelength, $\lambda _x$ (for motions with $\lambda _x/\delta \gtrapprox 3.5$) is illustrated by a blue triangle. This slope triangle has also been inset in figure 9(b) for comparison with the slope of the particle advection velocity with respect to particle cluster size.

Figure 10. (a) Normalized convection velocity varying with streamwise ($\lambda _x$) and spanwise wavelength ($\lambda _z$) in the log layer. (b) The convection velocity profile averaged over $\lambda _z/\delta = 0\unicode{x2013}2$. The triangle displayed here and in figure 9(b) was calculated from the range of coherent motions $\lambda _x/\delta = 4\unicode{x2013}6$. Here $\delta$ corresponds to the half-channel height in the calculation of Del Alamo & Jimenez (Reference Del Alamo and Jimenez2009).

In figure 9(b), particle velocities and coherent motion convective velocities exhibit a similar trend, providing further evidence to support the notion that particle clusters collect on the back of coherent motions like hairpin packets, as proposed by Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021). However, in this slope comparison, the scale of the coherent motions was assumed much larger than that of the particle clusters. This discrepancy between the sizes of momentum and particle clusters may be the result of slicing the three-dimensional particle clusters in the wall-parallel measurement plane. The discrepancy might also be attributed to the hairpin thickness/height aspect ratio (see the attached eddy modelling of Perry & Chong (Reference Perry and Chong1982) and Marusic & Monty (Reference Marusic and Monty2019)) under the assumption that the particles are localized narrowly in the region of highest intensity straining flow adjacent to the vortex cores. Additional evidence that the particle clusters are associated with self-similar attached eddies can be found in the joint-distribution of the cluster shapes themselves, shown above in figure 3(b). Tracing along the peaks of the joint p.d.f. reveals that the most probable clusters have a roughly constant aspect ratio, independent of their size, which indicates wall-parallel slices through self-similar attached eddies that are involved in the transport of the particle clusters.

The particle cluster advection process, combining the previous report of Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021) with the convective velocity scaling of Del Alamo & Jimenez (Reference Del Alamo and Jimenez2009), and the current experiments, is illustrated schematically in figure 11. Figure 11(a) indicates the wall-parallel view of two particle clusters of different streamwise dimensions. Figure 11(b) shows the inference of the streamwise/wall-normal view of the particle clusters with respect to a hairpin packet. The particle clusters are shown in the yellow region indicating the outer interface of the hairpin packet which is illustrated as discrete, purple hairpins. The slice of the clusters in the measurement plane is shown in green. The sweep ($Q4$) and ejection ($Q2$) events are marked about the hairpin heads. Beneath the hairpin legs is a region of low-momentum fluid caused by the backward vortex induction described by Adrian et al. (Reference Adrian, Meinhart and Tomkins2000). Above the hairpin backbone is a region of higher momentum fluid being swept in towards the wall. The average particle velocity is denoted with a vector $\bar {u}_p$ exceeding that of the local mean flow in the measurement plane, $\bar {u}$, consistent with figure 9.

Figure 11. (a) Wall-parallel illustration of the particle field with two particle clusters labelled in green. (b) The streamwise/wall-normal inference of the particle clusters illustrated within the outer envelope of a hairpin packet (yellow), as proposed originally by Zhu et al. (Reference Zhu, Pan, Wang, Liang, Ji and Wang2021), but now taking into account the differential convection velocities of the particles due to two different size hairpin packets. The hairpins are illustrated in purple, the induced low-speed region beneath their legs in blue, and the high-speed fluid being swept in is shown in red. The vector, $\bar {u}_p$ reflects the relative convection velocity of the particles and coherent structures compared with the mean flow, $\bar {u}$, as a function of the two hairpin packet sizes.