Experimental investigation into Lagrangian statistics of droplets in homogeneous isotropic turbulence

Abstract

We experimentally investigate the Lagrangian dynamics of finite-sized, neutrally buoyant droplets in homogeneous isotropic turbulence. The droplet size follows a log-normal distribution whose average value decreases with increasing Reynolds number, reflecting enhanced turbulent breakup. While size-conditioned velocity and acceleration statistics show only weak finite-size dependence, temporal measures reveal clear size-dependent dynamics: larger droplets exhibit longer Lagrangian velocity integral times and an extended ballistic regime in their mean squared displacement. These findings indicate that though droplets exhibit mild deformation and internal circulation, they behave similarly to finite-size rigid particles in terms of Lagrangian dynamics. Our study opens the way to study droplet-laden turbulence and droplet-flow interactions.

Figures (5)

• Figure 1: Experimental setup for droplet Lagrangian tracking. (a) Photograph of the experiment conducted in the "soccer-ball-like" HIT chamber. (b) Schematic of the optical arrangement from a plan view: four high-speed cameras capture droplet motion from different viewing angles to enable accurate three-dimensional reconstruction. Three cameras are mounted on the same horizontal plane and view the flow through the large windows, while a fourth camera is tilted at $45^\circ$ to view through the smaller windows. All cameras are synchronized with a double-cavity high-speed laser that generates a $\approx 2$ cm thick pulsed illumination volume. (c) Reconstructed three-dimensional trajectories at $Re_\lambda \approx 215$, color-coded by the instantaneous velocity magnitude. The trajectories span 550 consecutive frames, corresponding to a physical duration of 1.83 s ($\approx 87\tau_\eta$), where $\tau_\eta$ is the Kolmogorov timescale. Insets illustrate the determination of the droplet effective diameter from the visual-hull-based voxel reconstruction for a small droplet ($D \approx 2.5\eta$) and a large droplet ($D \approx 8.9\eta$). (d) Variation of droplet size and number density with increasing Taylor-Reynolds number $Re_\lambda$. For clarity, droplet sizes are scaled by a factor of two.
• Figure 2: (a) PDFs of the normalized droplet diameter, $D / \langle D \rangle$, for a range of $Re_\lambda$. Solid lines denote the corresponding log-normal fits. Each PDF is constructed from approximately $\mathcal{O}(10^3)$ droplet trajectories. The inset shows the fitted standard deviation $\sigma_0$ plotted against $Re_\lambda$. (b) Mean droplet diameter $\langle D \rangle$ and the most probable diameter $D_{peak}$ as functions of $Re_\lambda$. The corresponding best-fit trends are shown by the blue dashed line and the black dash-dotted line, respectively. For comparison, the Kolmogorov-Hinze scaling $\langle D \rangle \sim Re_\lambda^{-1.2}$ is also included. The inset displays the ratio of $\langle D \rangle$ to the Hinze scale $d_H$ (Eq. \ref{['eq:Hinze']}).
• Figure 3: PDFs of (a) normalized velocity $u/\sigma_u$ and (b) normalized acceleration $a/\sigma_a$ for droplets at $Re_\lambda = 266$, conditioned on size ranges $D/\eta \in [2,4]$, $D \in [4,6]$, $D \in [6,8]$, and $D \in [8,10]$, with the results of tracer experiment at the same $Re_\lambda$ as well as the Gaussian distribution and stretched exponential fit as references. In both panels, the inset shows the ratio of velocity variance and acceleration variance to those of tracer. In the inset of panel (b), Faxén-corrected numerical results at $Re_\lambda = 75$ and $Re_\lambda = 180$calzavarini2009acceleration are included.
• Figure 4: (a) Velocity ACF $C_{\boldsymbol{vv}}(\tau)$and (b) acceleration ACF $C_{\boldsymbol{aa}}(\tau)$ at $Re_\lambda = 266$, conditioned on four droplet-size ranges: $D \in [2,4]$, $D \in [4,6]$, $D \in [6,8]$, and $D \in [8,10]$. Tracer results at the same $Re_\lambda$ are shown for reference. Insets show Lagrangian integral times: $T_v$ is defined as the characteristic time at which $C_{\boldsymbol{vv}}(\tau)$ decays to $1/e$, and $T_a = \int_0^{T_0} C_{\boldsymbol{aa}}(\tau)\,\mathrm{d}\tau$ with $T_0$ the first zero-crossing of $C_{\boldsymbol{aa}}$. Faxén-corrected numerical results at $Re_\lambda = 75$ and $Re_\lambda = 180$calzavarini2009acceleration are included for comparison.
• Figure 5: MSD of droplets with different diameters at $Re_\lambda=266$. The MSD exhibits an initial ballistic regime, $\langle |\Delta X|^2 \rangle \sim \tau^2$ (black solid line), and then progressively transitions toward a diffusive regime, $\langle |\Delta X|^2 \rangle \sim \tau$ (black dashed line). The upper-left inset presents the local slope of the MSD, from which the ballistic time scale $T_b$ is determined as the delay time at which the local slope $\xi$ first falls below 95% of its ballistic regime scaling $\xi=2$ (black dashed line in the inset). The corresponding normalized values $T_b/\tau_\eta$ are shown in the lower-right inset.