Dispersion of Air Bubbles in Isotropic Turbulence

Bubbles play an important role in the transport of chemicals and nutrients in many natural and industrial flows. Their dispersion is crucial to understanding the mixing processes in these flows. Here we report on the dispersion of millimetric air bubbles in a homogeneous and isotropic turbulent flow with a Taylor Reynolds number from 110 to 310. We find that the mean squared displacement (MSD) of the bubbles far exceeds that of fluid tracers in turbulence. The MSD shows two regimes. At short times, it grows ballistically ($\propto {\tau }^2$), while at larger times, it approaches the diffusive regime where the $\mathrm{MSD}\propto \tau$. Strikingly, for the bubbles, the ballistic-to-diffusive transition occurs one decade earlier than for the fluid. We reveal that both the enhanced dispersion and the early transition to the diffusive regime can be traced back to the unsteady wake-induced motion of the bubbles. Further, the diffusion transition for bubbles is not set by the integral timescale of the turbulence (as it is for fluid tracers and microbubbles), but instead, by a timescale of eddy crossing of the rising bubbles. The present findings provide a Lagrangian perspective towards understanding mixing in turbulent bubbly flows.

Figure 1

(a) Measurement section of the Twente Water Tunnel facility. Four high-speed cameras (Photron 1024PCI) are mounted on the right, and the bubbles are illuminated by LED spotlights through a light diffuser. (b) Sample image by one of the cameras at ${\mathrm{Re}}_{\lambda }=150$. (c) Trajectory of a bubble (colored by horizontal speed) tracked over 595 frames for a duration of 2.38 s ($\approx 26{\tau }_{\eta }$) at ${\mathrm{Re}}_{\lambda }=230$, where ${\tau }_{\eta }$ is the dissipative timescale. See also movies S1 and S2 in the Supplemental Material [44] of the 3D trajectories.

Figure 2

Mean squared displacement in the (a) horizontal and (b) vertical directions normalized by the standard deviation of fluid velocity as a function of $\tau$. The colors denote the ${\mathrm{Re}}_{\lambda }$ of the experiments. For fluid tracers, the dispersion in the ballistic and diffusive regimes are given by Eq. (1), see the gray dashed and solid lines, respectively. The colored dotted lines in (a) and (b) give rough predictions for the longtime-dispersion rates of the bubbles, obtained using hot-film data.

Figure 3

Standard deviation of the bubble velocities ($u$, $v$, $w$) vs the standard deviation of the flow velocity. The bubble velocity fluctuations exceed that of the fluid (solid line). The dashed line $\sigma (u_{\mu b})$ is for microbubbles at ${\mathrm{Re}}_{\lambda }=62$ [49], obtained from fitting their data, $\sigma (u_{\mu b})\approx 0.9\sigma (u_f)$.

Figure 4

Lagrangian time correlation of the (a) horizontal and (b) vertical components of the bubble velocity. With increasing ${\mathrm{Re}}_{\lambda }$, the oscillations dampen out. (c) Kurtosis of the horizontal velocity increment PDFs as a function of $\tau$ for the ${\mathrm{Re}}_{\lambda }=110$ case. For large $\tau$, the PDF converges to a sub-Gaussian kurtosis ($K\approx 2.9$), comparable to the kurtosis of the flow [38]. The PDFs corresponding to the colored dots marked in (c) are given in (d). The jagged arrow connects the increment PDFs, revealing the oscillatory evolution of the PDF tails with increasing $\tau$.