Induced mixing in stratified fluids by rising bubbles in a thin gap

This work reports a numerical study of mixing in temperature- or density-stratified fluids induced by the motion of a monodispersed swarm of bubbles in a thin gap. Simulations are run for void fractions between $3.35\%$ and $13.4\%$. The strength of stratification is varied by changing the Froude number between 4.5 and 12.74. The confinement prevents turbulence production, and mixing occurs primarily due to transport of colder liquid into the hotter layers by the bubble wake. Bubbles move in a zigzag motion attributed to the periodic vortex shedding in their wake. We report the formation of horizontal clusters and establish a direct correlation between the size of clusters and the rise velocity of the bubbles. We report an increase in the buoyancy flux across the isopycnals as the void fraction increases. The fraction of energy production due to the buoyancy flux increases with the strength of stratification, giving rise to a higher mixing efficiency. At the same time, cross isopycnal diffusion is higher at weaker stratification strengths.

Figure 1

Schematic of the computational setup. (a) View of the bubbles rising in the domain. (b) $y\text{−}z$ view of a single bubble which is confined between two solid walls.

Figure 2

Contours of nondimensional vorticity for a single bubble in computational domain at $t=1.57$ s.

Figure 3

Spatial velocity correlation for the domain in the (a) vertical direction and (b) horizontal direction for $\alpha =13.4\%.$

Figure 4

Bubble center displacements for $\alpha =$ (a) $3.35\%$, (b) $8.37\%,$ and (c) $13.4\%$ at $\text{Fr}=6.37.$

Figure 5

Bubble trajectories for $\alpha =$ (a) $3.35\%$, (b) $8.37\%,$ and (c) $13.4\%$ at $\text{Fr}=6.37.$

Figure 6

Flow field for $\alpha =8.37\%$ at $\text{Fr}=6.37.$ (a) Nondimensional $z$-vorticity contours. (b) Temperature contours normalized by its highest value at $t=0.$

Figure 7

Streamlines within the bubble. (a) $y\text{−}x$ view at $t=1.57$ s. (b) $y\text{−}x$ view $t=1.79$ s. (c) $y\text{−}z$ view at $t=1.57$ s.

Figure 8

Bubble Reynolds number for different void fraction at Fr = 6.37. (a) Instantaneous vs. time. (b) Time averaged vs. void fraction.

Figure 9

Bubble Reynolds number for different Froude numbers at $\alpha =3.35\%.$ (a) Instantaneous vs. time. (b) Time averaged vs. void fraction.

Figure 10

(a) Instantaneous bubble Reynolds number over time for different domain sizes. Cluster Formation shown in (b) $8\times 16$ domain at $t=1.1$ s and (c) $16\times 16$ domain at $t=1.54$ s.

Figure 11

Cluster size index with corresponding Reynolds number for $\alpha =$ (a) $8.37\%$ and (b) $13.4\%$ at Fr = 6.37.

Figure 12

Variance of bubble center displacements with time; (a) and (b) correspond to vertical and horizontal variance at different void fractions at Fr = 6.37, respectively.

Figure 13

Autocorrelation functions at (a) different void fractions for $\text{Fr}=6.37$ and (b) different stratification strengths at $\alpha =3.35\%.$

Figure 14

Energy spectrum.

Figure 15

Semilog plot of mixing parameters for varying $\alpha$ at $\text{Fr}=6.37.$ (a) COX number. (b) Diapycnal eddy diffusivity (nondimensionalized by $15{\text{cm}}^2/\text{s})$. (c) Mixing efficiency.

Figure 16

Semi-log plot of mixing parameters for varying Fr at $\alpha =3.35\%.$ (a) COX number. (b) Diapycnal eddy diffusivity (nondimensionalized by $15{\text{cm}}^2/\text{s})$. (c) Mixing efficiency.

Figure 17

Time averaged (a), (d) diapycnal eddy diffusivity, (b), (e) COX number, and (c), (f) mixing efficiency, plotted against $\alpha$ for two different Froude numbers and plotted against Fr for $\alpha =3.35\%,$ respectively. Red $--$ and $-·-$ correspond to lower and upper bounds, respectively, predicted by Ref. [37] in (a) and Ref. [41] in (b) and (c).

Figure 18

Total scalar diffusivity normalized by molecular diffusivity versus the normalized viscous dissipation. Fit to data taken from Ref. [41]: $--,(K_{\rho }+\kappa )/\kappa =0.2\text{Pr}\epsilon /\nu N^2$; $···,(K_{\rho }+\kappa )/\kappa =5\text{Pr}{(\epsilon /\nu N^2)}^{1/3}$; —, $(K_{\rho }+\kappa )/\kappa =2\text{Pr}{(\epsilon /\nu N^2)}^{1/2}$.

Figure 19

Variation of $K_{\rho }$ with bubble velocity fluctuations. Same colors correspond to same $\alpha$. Same shapes correspond to same Fr.