Mixing mechanism in a two-dimensional bubble column

The present contribution investigates the mixing of a passive scalar by a homogeneous bidimensional bubble swarm rising at high Reynolds number in a liquid initially at rest. Mixing experiments are performed in a Hele-Shaw cell for gas volume fractions $\alpha$ ranging from 3.0% to $14.0\%$. A weakly diffusive passive dye is injected within the swarm, and the temporal evolution of the spatial distribution of concentration is measured. The vertical distribution of concentration shows an upward propagation and a spreading due to the mixing induced by the unsteady open wakes of the bubbles. A one-dimensional large-scale model involving an intermittent and convective mechanism has been developed to describe the global evolution of the concentration distribution in the vertical direction. Based on experimental observations, it assumes that each bubble catches a given volume of fluid $V_t$ in its wake and transports it over a certain length $L$ before releasing it. A good agreement is found between the experimental concentration profiles measured in the vertical direction and the model prediction. The comparison between the model and the experiments allows the determination of the transported volume $V_t$ and the transport length $L$ as a function of the gas volume fraction. It appears that the transported volume is related to the characteristic length of the velocity deficit at the rear of the bubble. The transport length, which is related to the correlation length of the dye patches, shows two regimes. At low gas volume fraction, it is controlled by the viscous length related to the flow damping at the walls, whereas, at higher gas volume fraction, it is limited by the distance between two successive bubbles. The mixing properties are finally characterized from the first three-order moments of the dye distribution, which are determined by means of the model. The upward propagation of the dye is shown to scale as $\alpha V$, where $V$ is the bubble rise velocity. The spreading of the concentration distribution is characterized by an effective diffusivity, which presents strong similarities with the diffusion coefficient measured in a three-dimensional bubble column [Alméras et al. J. Fluid Mech. 776, 458 (2015)]. At low gas volume fraction, it increases as $\sqrt{\alpha }$, whereas it saturates at high gas volume fraction. The dye distribution also is asymmetric with a significant skewness coefficient which slowly decreases in time. Therefore, the dye transport cannot be described as a purely diffusive process over the time required for the dye to spread over the cell.

Figure 1

Experimental setup.

Figure 2

Calibration curves for different powers of the LED panel (defined by the electricity supply intensity $I)$. The symbols show the measured gray level as a function of the concentration, the lines the calibration relation given by Eq. (1).

Figure 3

Illustration of the image processing. Images taken 3 s after the dye injection beginning for $\alpha =5.3\%$. Note that lighter means greater concentration in the treated image (to the left), while it is the opposite in the raw image (to the right).

Figure 4

Spatial and temporal experimental evolutions of the concentration $\left(\alpha =5.3\%\right)$. (a) Instantaneous concentration field, (b) Vertical concentration profiles at three different instant, and (c) Temporal evolution of the concentration averaged over the region between the two horizontal (red) lines.

Figure 5

Illustration of the mixing process by capture of the dye in the bubble wakes. The time step between two images is 0.04 s.

Figure 6

Illustration of the numerical model. The initial concentration profile is indicated to the right.

Figure 7

Model validation from the global mixing time $T_m$. (The model parameters are given in Table 1.) (a) Temporal evolution of the concentration averaged over a fixed area. $\alpha =5.3\%$ and (b) Evolution of the mixing time $T_m$ with $\alpha$.

Figure 8

Influence of the parameter pair $(A_t;L)$ on the concentration profiles for $\alpha =9.7\%$. While the two pairs allow the model to predict the experimental global mixing time (equal to 5.4 s for the present gas volume fraction), only one allows it to reproduce the concentration profiles.

Figure 9

Model validation from one-dimensional concentration profiles at different instants. Continuous lines: experiments. Dashed lines: model.

Figure 10

Global displacement of the vertical distribution of the concentration. (a) Temporal evolution of the barycentre of the concentration distribution and (b) Propagation velocity normalized by the mean bubble velocity.

Figure 11

Spreading of the concentration distribution. (a) Temporal evolution of the variance of the concentration distribution, (b) Diffusion coefficient as a function of the gas volume fraction $\alpha$, and (c) Diffusion coefficient normalized by $D_{\mathrm{low}}$ as a function of the gas volume fraction $\alpha$.

Figure 12

Skewness of the concentration distribution as a function of time for different $\alpha$.