Multifractality of fine bubbles in turbulence due to lift

We develop a theory of spatial distribution of fine bubbles in turbulence and confirm it numerically. The bubbles are a few hundred microns in size and the considered value of the turbulence's strength is typical for oceans. The fluid applies a lift force on bubbles that causes bubble clustering by acting in a direction perpendicular to the rise velocity and the horizontal vorticity. We demonstrate that the centripetal attraction of bubbles to the cores of vortices is negligible compared with this effect. The combined action of the lift and the turbulence creates a continuum flow of bubbles distinct from the flow of the fluid and identical to the Peddley-Kessler (PK) model of swimming phytoplankton cells in the limit of strong gravitaxis. This allows one to derive a whole array of results by applying the theory of this model, which is well developed in the limit. The similarity explains the patchiness and clustering of bubbles in regions of downwelling flow; these properties are understood for motile phytoplankton. The bubble clustering decreases the average rise velocity, prolonging the time until the bubble's rupture at the surface. We demonstrate that bubbles form columnar structures characterized by a multifractal distribution with the dimension deficit increasing as the eighth power of the radius. Fractal dimensions are derived from the information dimension given by the Kaplan-Yorke formula. We also provide some results for the shape-dependent PK model. We confirm the theory by direct numerical simulations of the bubbles' motion in Navier-Stokes turbulence.

Figure 1

Comparison of bubbles' velocity from (a) the full equation of motion [Eq. (1)] and (b) the weak form [Eq. (6)] and (c) the relative error between the two velocity fields defined by Eq. (7) in the same homogeneous isotropic turbulence. Here $\text{St}=0.005$ and $\text{Fr}=0.0005$. Color contours denote the vorticity component normal to the plane. Gravity acts downward and thus bubbles mostly rise. A bubble's position uniquely determines its velocity. Two velocity fields are identical even near strong vortices, confirming that the centripetal attraction of bubbles to the vortex cores is negligible. Open symbols in (c) correspond to cases with parameters outside the valid range ${10}^{-2}{\left(6r\right)}^{3/2}>\text{Fr}{\text{St}}^{-3/2}$ or $\text{Re}>100$, which will be discussed later.

Figure 2

Lyapunov exponents (a) ${\lambda }_1\tau$ (closed symbols) and $|{\lambda }_3|\tau$ (open symbols) and (b) Kaplan-Yorke codimension, obtained from the DNS (symbols) compared with theoretical estimates [lines, Eqs. (31) and (21)] for $\text{St}=0.005$, 0.01, and 0.02. The Froude number and the Lyapunov exponents ${\lambda }_1\tau$ and $|{\lambda }_3|\tau$ are rescaled by ${\text{St}}^{3/2}$ so that the range of validity of the model equation, ${10}^{-2}{\left(6r\right)}^{3/2}\lesssim \text{Fr}{\text{St}}^{-3/2}\lesssim {\left(6r\right)}^{3/2}$, is clearly shown. Here $C_{\text{KY}}$ is compensated by ${\text{St}}^4$ and then rescaled by ${\text{St}}^3$ because the theory predicts $C_{\text{KY}}\sim {\text{St}}^4/{\text{Fr}}^2$ [Eq. (21)]. Open symbols in (b) correspond to cases with parameters outside the valid range ${10}^{-2}{\left(6r\right)}^{3/2}>\text{Fr}{\text{St}}^{-3/2}$ or $\text{Re}>100$. The discrepancy for the upper rightmost point is due to ${\tau }_{\eta }/{\tau }_g\sim 5$ violating ${\tau }_{\eta }/{\tau }_g\gg 1$. The inset of (b) is provided for better reading of the value of $C_{\text{KY}}$ for the given St and Fr.

Figure 3

Snapshot of bubble distribution for $\text{St}=0.02$ and (a) $\text{Fr}=0.0005(\text{Fr}{\text{St}}^{-3/2}=0.177,C_{\text{KY}}=0.222)$; (b) $\text{Fr}=0.0002(\text{Fr}{\text{St}}^{-3/2}=0.07,C_{\text{KY}}=0.845)$. The color contour denotes the vorticity. The arrow below the colormap indicates the direction of gravity. The slightly slanted gravity direction is intentional to avoid numerical artifacts due to periodicity of flow fields. The parameters of case (a) satisfy the valid range ${10}^{-2}{\left(6r\right)}^{3/2}(\simeq 0.15)\lesssim \text{Fr}{\text{St}}^{-3/2}\lesssim {\left(6r\right)}^{3/2}\left(\simeq 15\right)$, while those of case (b) are slightly outside the valid range, for which our theoretical prediction of the fractal dimension still shows good agreement with the simulation as shown in Fig. 2. The columnar structure of the bubbles is more pronounced, and the bubble clusters are more compressible as Fr decreases. Even for larger Fr, such a cluster is present, although it is not that pronounced.