Kolmogorov Turbulence Coexists with Pseudo-Turbulence in Buoyancy-Driven Bubbly Flows

We investigate the spectral properties of buoyancy-driven bubbly flows. Using high-resolution numerical simulations and phenomenology of homogeneous turbulence, we identify the relevant energy transfer mechanisms. We find (a) at a high enough Galilei number (ratio of the buoyancy to viscous forces) the velocity power spectrum shows the Kolmogorov scaling with a power-law exponent $-5/3$ for the range of scales between the bubble diameter and the dissipation scale ($\eta$). (b) For scales smaller than $\eta$, the physics of pseudo-turbulence is recovered.

Figure 1

The isocontour plot of the $z$ component of the vorticity ${\omega }^z=(\nabla \times \mathit{u})·\widehat{\mathit{z}}$ for (left to right) $\mathrm{Ga}=302$, 605, 1029, and 2057. We show the contour corresponding to $\pm {⟨{({\omega }^z)}^2⟩}^{1/2}$ in purple and orange, respectively. The bubbles are represented using gray contours.

Figure 2

(a) Log-log plot of the normalized velocity power spectra $E(k)$ as a function of $k\eta$ for various Ga. We observe Kolmogorov scaling $E(k)\sim k^{-5/3}$ for $k\eta \lesssim 0.3$, and the pseudo-turbulence scaling $E(k)\sim k^{-3}$ for $k\eta \gtrsim 0.3$. (b) The velocity power spectra for different Ga compensated by $k^{5/3}$. In (b) the vertical arrows show the wave number corresponding to the bubble diameter $k_{\mathrm{d}}\eta$.

Figure 3

Semi-log plot of the different terms in scale-by-scale budget for low Atwood number $\mathrm{At}=0.04$ (top row) with $\mathrm{Ga}=2057$ (a) and $\mathrm{Ga}=302$ (b); $\mathrm{At}=0.8$ (middle row) for $\mathrm{Ga}=1029$ (c), $\mathrm{Ga}=345$ (d); and $\mathrm{At}=0.98$ (bottom row), $\mathrm{Ga}=1489$ (e). The abscissa is normalized by the total energy injection rate ${\epsilon }^{\mathrm{g}}$. The dashed vertical line represents the wave number corresponding to bubble diameter, $k_{\mathrm{d}}$. The continuous vertical line shows the wave number where the advective flux and the dissipative flux crosses. The region between these two lines, where the advective contribution dominates, is shaded with light blue color.