Dynamics of small bubbles in turbulence in non-dilute conditions

TL;DR

This study addresses whether four-way coupling—back-reaction of bubbles on the fluid and excluded-volume collisions between bubbles—significantly affects the dynamics of small bubbles in turbulence beyond conventional one-way coupling. Using high-resolution direct numerical simulations with massless microbubbles ($\rho_p/\rho_f=0$, $\mathrm{St}=1$) and a YOCO collision model across $\alpha$ from $10^{-5}$ to $10^{-1}$, the authors quantify both Eulerian fluid metrics and a range of Lagrangian bubble statistics. They find that the fluid's total kinetic energy and spectrum are largely insensitive to four-way coupling, while Lagrangian observables reveal a dilute-to-dense crossover near $\alpha_c \approx 1\%$, where preferential sampling in high-vorticity filaments weakens and clustering becomes more volumetric; the short-time roughness and the tails of first-exit-time distributions are also affected by collisions. These results clarify when four-way coupling is essential for accurate bubbly-turbulence modeling and hint at applications for probing and manipulating coherent small-scale vortex structures using bubbles, while noting the limitations of the idealized assumptions (no deformation, gravity, or non-instantaneous collisions).

Abstract

Turbulent flows laden with small bubbles are ubiquitous in many natural and industrial environments. From the point of view of numerical modeling, to be able to handle a very large number of small bubbles in direct numerical simulations, one traditionally relies on the one-way coupling paradigm. There, bubbles are passively advected and are non-interacting, implicitly assuming dilute conditions. Here, we study bubbles that are four-way coupled, where both the feedback on the fluid and excluded-volume interactions between bubbles are taken into account. We find that, while the back-reaction from the bubble phase onto the fluid phase remains energetically small under most circumstances, the excluded-volume interactions between bubbles can have a significant influence on the Lagrangian statistics of the bubble dynamics. We show that as the volume fraction of bubbles increases, the preferential concentration of bubbles in filamentary high-vorticity regions decreases as these strong vortical structures get filled up; this happens at a volume fraction of around one percent for $\textrm{Re}_λ=O(10^2)$. We furthermore study the influence on the Lagrangian velocity structure function as well as pair dispersion, and find that, while the mean dispersive behavior remains close to that obtained from one-way coupling simulations, some evident signatures of bubble collisions can be retrieved from the structure functions and the distribution of the dispersion, even at very small volume fractions. This work not only teaches us about the circumstances under which four-way coupling becomes important, but also opens up new directions towards probing and ultimately manipulating coherent vortical structures in small-scale turbulence using bubbles.

Figures (8)

• Figure 1: A snapshot of bubbles in turbulence for a simulation one-way coupled (a) and four-way coupled (b) bubbles of size $D=0.81\eta$ and volume fraction $\alpha=3.5\%$. The inset zooms show a small cross section of the full domain, where the background color indicates the local enstrophy $\Omega^2\equiv|\bm{\nabla} \times \bm{u}|^2$.
• Figure 2: Total kinetic energy (a) and kinetic energy spectra (b) for four-way coupled bubbles in turbulence at various volume fractions $\alpha$ and bubble sizes $D$. In (a) the kinetic energy $E_\textrm{tot}$ is compensated by the kinetic energy of the single-phase turbulence without bubbles $E_\textrm{tot,0}$, while (b) shows the total energy spectra with the compensated spectra in the inset. The inset in (a) shows the kinetic energy for runs with heavy particles ($\rho_p/\rho_f=2$ and $\rho_f/\rho_p=25$) for comparison. Panel (b) provides the results for bubbles with $D=0.81\eta$. The figure shows that the back-reaction of the bubbles on the fluid flow only has a very small energetic effect, even at relatively large volume fractions.
• Figure 3: The effect of four-way coupling on the preferential sampling of enstrophy as a function of the volume fraction $\alpha$. The sampled enstrophy $\Omega^2$ at the bubble position is compensated by the Eulerian domain average $\Omega_0^2$, shown for varying bubble sizes $D$ (with $\textrm{Re}_\lambda=95$) (a) and for varying Taylor-scale Reynolds numbers $\textrm{Re}_\lambda$ (with $D=0.81\eta$) (b). The inset in (b) shows the preferential sampling of enstrophy rescaled with respect to the one-way coupled sampled enstrophy as $(\Omega^2-\Omega_0^2)/(\Omega_{\textrm{one-way}}^2-\Omega_0^2)$. Panel (c) shows the same data, where the volume fraction is rescaled by a factor $\textrm{Re}_\lambda^{-1/2}$. Horizontal dashed lines indicate the results for one-way coupled bubbles. The figure shows that the vortex filaments with the highest enstrophy become filled up at around a critical volume fraction of $\alpha=\alpha_c\approx 1\%$, beyond which preferential sampling is inhibited.
• Figure 4: Fractal covering dimension $d$ for four-way coupled bubbles in turbulence at varying volume fraction $\alpha$ for different bubble sizes $D$. It shows that, as volume fraction increases, the bubble clusters transition from the filamentary line-like structures with $d\approx 1$ that are retrieved for one-way coupled bubbles to more volumetric structures with $d\to 3$. The inset shows a rescaling of the abscissa by the bubble size $D$.
• Figure 5: Lagrangian velocity structure function scaling exponent $\zeta_4(\tau)$ for four-way coupled bubbles in turbulence with $D=0.81\eta$ at various volume fractions $\alpha$. The exponents are obtained via extended self-similarity (ESS) as $\zeta_4(\tau) \equiv \mathrm{d}\log S_4 (\tau)/\mathrm{d}\log S_2 (\tau)$. Green dashed lines indicate fits through linear combination $\zeta_4^\textrm{(fit)} = \gamma \zeta_4^\textrm{(tracer)} + (1-\gamma)\zeta_4^\textrm{(one-way)}$. The horizontal dashed line indicates the Kolmogorov non-intermittent smooth solution $\zeta_4=2$. It shows that the viscous dip around $\tau = \mathcal{O}(\tau_\eta)$, that is amplified for bubbles due to preferential sampling, is reduced as the volume fraction increases.