Particle-pair relative velocity measurement in high-Reynolds-number homogeneous and isotropic turbulence using 4-frame particle tracking velocimetry

Abstract

The radial relative velocity (RV) between particles suspended in turbulent flow plays a critical role in droplet collision and growth. We present a simple and accurate approach to RV measurement in isotropic turbulence—planar 4-frame particle tracking velocimetry—using routine PIV hardware. It improves particle positioning and pairing accuracy over the 2-frame holographic approach by de Jong et al. (Int J Multiphas Flow 36:324–332; de Jong et al., Int J Multiphas Flow 36:324–332, 2010) without using high-speed cameras and lasers as in Saw et al. (Phys Fluids 26:111702, 2014). Homogeneous and isotropic turbulent flow (${R_\lambda }=357$) in a new, fan-driven, truncated iscosahedron chamber was laden with either low-Stokes (mean $St=0.09$, standard deviation 0.05) or high-Stokes aerosols (mean $St=3.46$, standard deviation 0.57). For comparison, DNS was conducted under similar conditions (${R_\lambda }=398$; $St=0.10$ and 3.00, respectively). Experimental RV probability density functions (PDF) and mean inward RV agree well with DNS. Mean inward RV increases with $St$ at small particle separations, $r$, and decreases with $St$ at large $r$, indicating the dominance of “path-history” and “inertial filtering” effects, respectively. However, at small $r$, the experimental mean inward RV trends higher than DNS, possibly due to the slight polydispersity of particles and finite light sheet thickness in experiments. To confirm this interpretation, we performed numerical experiments and found that particle polydispersity increases mean inward RV at small $r$, while finite laser thickness also overestimates mean inward RV at small $r$, This study demonstrates the feasibility of accurately measuring RV using routine hardware, and verifies, for the first time, the path-history and inertial filtering effects on particle-pair RV at large particle separations experimentally.

Appendix: Monte Carlo analysis to account for out-of-plane components of particle-pair RV

A certain thickness of the laser light sheet is required in the planar 4F-PTV technique to keep a substantial number of particles within the laser sheet over the multiple tracking steps to achieve tracking accuracy (Sect. 2). This finite laser thickness results in small but not-insignificant out-of-plane components of the particle separation ${\varvec{r}}$ and particle-pair RV ${{\varvec{w}}_{\varvec{r}}}$, which may contribute to the experimental error in RV (Sect. 5.4). We attempt to account for the out-of-plane components within the finite light sheet as follows.

Using Monte Carlo analysis (MCA) method, we add the out-of-plane components of particle-pair separation, ${r_z}$, and particle-pair relative velocity, ${w_{{r_z}}}$, to each particle-pair sample based on the statistical distributions of their in-plane counterparts. After this step, we then recalculate the 3D RV using the “new” experimental data samples. This process is repeated until the PDF of particle-pair RV at each $r$ is converged. The detailed procedure of MCA is described in the supplementary material (online resource, Sect. C).

To evaluate the ability of the MCA method to represent the out-of-plane components of RV, we applied the MCA correction to the top curve in Fig. 12, i.e., $\left\langle {{w_r}{{\left( r \right)}^ - }} \right\rangle$ versus $r$ from simulated planar measurement at laser thickness of ${Z_0}=8\eta$. The corrected result is plotted as the dashed red curve in Fig. 13. The corrected $\left\langle {{w_r}{{\left( r \right)}^ - }} \right\rangle$ versus $r$ matches the “true” DNS result (solid black curve) within 1% at $r=5\sim 20\eta$. When $r \gtrsim 20\eta$, all three curves (simulated planar measurement results before and after the MCA correction, and the “true” DNS result) are identical and thus not plotted. This is because when the RV of a particle pair separated by $r \gtrsim 20\eta$ is captured by planar 4-frame PTV with laser thickness of $8\eta$, the out-of-plane components of RV are negligible. This explains the overlap between the original experimental and DNS RV curves in Fig. 10 when $r \gtrsim 20\eta$.

However, at $r \lesssim 5\eta$, the simulated experimental RV after the MCA correction (dashed red curve) trends higher than “true” DNS curve, indicating that the MCA correction cannot represent the out-of-plane components of particle pair RV. This is likely due to the assumption in the MCA method that particle-pair RV in $x$-, $y$-, and $z$-directions are uncorrected (Assumption 3 in the online supplementary material, Section C). This might not be true in reality. Taking together, the MCA method is able to correct the RV overestimation for particle separation $r \gtrsim 5\eta$ caused by the finite laser thickness effect of ${Z_0}=8\eta$. In other words, with out-of-plane correction, RV measured by planar 4F-PTV is sufficiently accurate as long as $r \gtrsim 5\eta$.

We applied the MCA correction to the original experimental data $\left\langle {{w_r}{{\left( r \right)}^ - }} \right\rangle$ versus $r~$at $St=0.09$ and 3.46 (Fig. 10) and show the corrected results for $r=1\sim 20\eta$ in Fig. 14, along with the original uncorrected RV curves and DNS results. When $r=5\sim 20\eta$, the MCA corrected result is less than 5% above DNS due to the remaining polydispersity effect. When $r=1\sim 5\eta$, however, experimental RV after MCA correction is still significantly higher than DNS results, which could be due in part to the drawback of the MCA method and the particle polydispersity effect when $r \to 0$. In addition, we notice that when $r$ is around $5\sim 10\eta$, the MCA corrected approximately half of the discrepancy between experiment and DNS, and that the rest of the discrepancy is mainly due to the polydispersity effect. This implies that particle polydispersity and finite laser thickness have comparative effects on $\left\langle {{w_r}{{\left( r \right)}^ - }} \right\rangle$ at small $r$.

We have corrected all the experimental results shown in Figs. 9 and 10 by applying the MCA method to account for the out-of-plane component of RV due to the finite laser thickness. The regenerated curves are remarkably similar to the original ones, and, therefore, for conciseness, not presented in this paper.