Recent measurements of inertial particles in isotropic turbulence (Hammond & Meng, J. Fluid Mech., vol. 921, 2021, A16) revealed surprising extreme clustering of particles at near-contact separations $(r)$, whereby the radial distribution function, $g(r)$, grows from $O(10)$ to $O(10^3)$ with a $(r/a)^{-6}$ scaling (where $a$ is the particle radius), and a surprising upturn of the mean inward particle-pair relative velocity (MIRV). Hydrodynamic interactions (HIs) were proposed to explain the extreme clustering, but despite predicting the correct scaling $(r/a)^{-6}$, the HI theory underpredicted $g(r)$ by at least two orders of magnitude (Bragg et al., J. Fluid Mech., vol. 933, 2022, A31). To further understand the extreme clustering phenomenon and the relevance of HI, we characterize $g(r)$ and particle-pair kinematics for Stokes numbers $0.07 \leq St \leq 3.68$ in a homogeneous isotropic turbulence chamber using three-dimensional (3-D) particle tracking resolved to near–contact. A drift–diffusion equation governing $g(r)$ is presented to investigate the kinematic mechanisms of particle pairs. Measurements in all 24 conditions show that when $r/a\lessapprox 20$, extreme clustering consistently occurs, scaling as $g(r) \sim (r/a)^{-k}$ with $4.5 \leq k \leq 7.6$, which increases with $St$. Here $g(r)$ varies with $St$, particle size, density and polydispersity in ways that HI cannot explain. The extreme clustering region features an inward drift contributed by particle-pair turbophoresis and an inward radial relative acceleration. The latter indicates an interparticle attractive force at these separations that HI also cannot explain. The MIRV turns upward when approaching the extreme clustering region, opposite to direct numerical simulation predictions. These observations further support our previous assessment that extreme clustering arises from particle–particle interactions, but HI is not the main mechanism.

Expanding recent observations by Hammond & Meng (J. Fluid Mech., vol. 921, 2021, A16), we present a range of detailed experimental data of the radial distribution function (r.d.f.) of inertial particles in isotropic turbulence for different Stokes number, $St$, showing that the r.d.f. grows explosively with decreasing separation r, exhibiting $r^{-6}$ scaling as the collision radius is approached, regardless of $St$ or particle radius $a$. To understand such explosive clustering, we correct a number of errors in the theory by Yavuz et al. (Phys. Rev. Lett., vol. 120, 2018, 244504) based on hydrodynamic interactions between pairs of small, weakly inertial particles. A comparison between the corrected theory and the experiment shows that the theory by Yavuz et al. underpredicts the r.d.f. by orders of magnitude. To explain this discrepancy, we explore several alternative mechanisms for this discrepancy that were not included in the theory and show that none of them are likely the explanation. This suggests new, yet-to-be-identified physical mechanisms are at play, requiring further investigation and new theories.

Particle collisions in turbulent flow are critical to particle agglomeration and droplet coalescence. The collision kernel can be evaluated by radial distribution function (RDF) and radial relative velocity (RV) between particles at small separations $r$. Previously, the smallest $r$ was limited to roughly the Kolmogorov length $\eta$ due to particle position uncertainty and image overlap. We report a new approach to measuring RDF and RV near contact ($r/a \approx 2.07$, where $a$ is particle radius). Three-dimensional particle tracking velocimetry using the four-pulse shake-the-box algorithm recorded short tracks with the interpolated midpoints registered as particle positions, avoiding image overlap and track mismatch. We measured RDF and RV of inertial particles in a one metre diameter isotropic air turbulence chamber with Taylor Reynolds number $Re_\lambda =324$, $a=12 - 16\ \mathrm {\mu }\textrm {m}$ $({\approx }0.12\eta )$ and Stokes number ${\approx }0.7$. At large $r$ the measured RV agrees with the literature, but when $r<20\eta$ the first moment of negative RV starts to increase, reaching 10 times higher values than direct numerical simulations of non-interacting particles. Likewise, RDF scales as $r^{-0.39}$ when $r>\eta$, reflecting the well-known scaling for polydisperse particles, but when $r\lessapprox \eta$, RDF scales as $r^{-6}$, yielding 1000 times higher near-contact RDF than simulations. Such RV enhancement and extreme clustering at small $r$ can be attributed to particle–particle interactions including hydrodynamic interactions, which are not well-understood. Uncertainty analysis substantiates the observed trends. This first-ever simultaneous RDF and RV measurement at small separations provides a clear glimpse into the clustering and relative velocities of particles in turbulence near-contact.

The effects of Reynolds number ($R_{\unicode[STIX]{x1D706}}$) and Stokes number ($St$) on particle-pair relative velocity (RV) are investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber. We compare the measured results with direct numerical simulation (DNS), verifying whether the conclusions of the DNS for simplified conditions and limited $R_{\unicode[STIX]{x1D706}}$ are still valid in reality. Two experiments are performed: varying $R_{\unicode[STIX]{x1D706}}$ between 246 and 357 at six $St$ values, and varying $St$ between 0.02 and 4.63 at five $R_{\unicode[STIX]{x1D706}}$ values. The measured mean inward particle-pair RV $\langle w_{r}^{-}\rangle$ as a function of separation distance $r$ is compared with the DNS under closely matched conditions. At all experimental conditions, an excellent agreement is achieved, except when the particle separation distance $r\lesssim 10\unicode[STIX]{x1D702}$ ($\unicode[STIX]{x1D702}$ is the Kolmogorov length scale), where the experimental $\langle w_{r}^{-}\rangle$ is consistently higher, possibly due to particle polydispersity and finite laser thickness in the experiments (Dou et al., arXiv:1712.07506, 2017). At any fixed $St,\langle w_{r}^{-}\rangle$ is essentially independent of $R_{\unicode[STIX]{x1D706}}$, echoing the DNS finding of Ireland et al. (J. Fluid Mech., vol. 796, 2016, pp. 617–658). At any fixed $R_{\unicode[STIX]{x1D706}}$, $\langle w_{r}^{-}\rangle$ increases with $St$ at small $r$, showing dominance of the path-history effect in the dissipation range when $St\gtrsim O(1)$, but decreases with $St$ at large $r$, indicating dominance of inertial filtering. We further compare the $\langle w_{r}^{-}\rangle$ and RV variance $\langle w_{r}^{2}\rangle$ from experiments with DNS and theoretical predictions by Pan & Padoan (J. Fluid Mech., vol. 661, 2010, pp. 73–107). For $St\lesssim 1$, experimental $\langle w_{r}^{-}\rangle$ and $\langle w_{r}^{2}\rangle$ match these values well at $r\gtrsim 10\unicode[STIX]{x1D702}$, but they are higher than both DNS and theory at $r\lesssim 10\unicode[STIX]{x1D702}$. For $St\gtrsim 1$, $\langle w_{r}^{-}\rangle$ from all three match well, except for $r\lesssim 10\unicode[STIX]{x1D702}$, for which experimental values are higher, while $\langle w_{r}^{2}\rangle$ from experiment and DNS are much higher than theoretical predictions. We discuss potential causes of these discrepancies. What this study shows is the first experimental validation of $R_{\unicode[STIX]{x1D706}}$ and $St$ effect on inertial particle-pair $\langle w_{r}^{-}\rangle$ in homogeneous and isotropic turbulence.