Vortex interactions between a pair of bubbles rising side by side in ordinary viscous liquids

We report on a series of numerical results about a pair of bubbles rising side by side in various silicone oils, with the terminal Reynolds number to vary between 10 and 700 and the aspect ratio of the bubble shapes to vary between 1.0 and 3.2. Through fully solving the three-dimensional Navier-Stokes equations, different interactive behaviors between the two bubbles are reproduced, such as the repelling, the attracting coalescence, the attracting bounce, and the repeated bounce, which are also observed in the experiments. First, we determine that, if there are no toroidal vortex rings attached at the rear end of the bubbles, the moderate- to high-Reynolds-number bubbles tend to attract with one another after being released, and, subsequently, whether they will coalesce or bounce after contact depends on the development of the vortex structures. In contrast, if toroidal vortex rings are generated in the bubble wake in other parameter spaces, a pair of bubbles will repel one another after being released, and a subsequent repeated bounce will be even observed. Lastly, an available force calculation model enables us to identify the relation between the forces and the vorticity produced on the bubbles when the pair of bubbles collide.

Figure 1

Sketch of the pair of bubbles rising in viscous liquid.

Figure 2

The mesh distributions (a) before and (b) after performing the topology-based AMR criterion. It is observed that only the grids inside the gap between the pair of bubbles are refined.

Figure 3

(a) Calculated trajectories of the pair of bubbles under the spatial resolution of $\mathrm{\Delta }=\frac{D}{1600}$ inside the thin film; (b) the mesh distributions close to the thin film between the two bubbles. It is observed that the bubbles will bounce off after attracting with one another at a height of $H=6$. Moreover, only the grids inside the thin film are refined adaptively to save the computational resource.

Figure 4

Bubble pair 8 rising in K2 when the bulk spatial resolutions is different, corresponding to (a) rising paths with refined mesh ($\mathrm{\Delta }=D/64$) at the interface; (b) rising paths with coarse mesh ($\mathrm{\Delta }=D/32$) at the interface; and (c) interactions on different meshes. It is observed that the fine mesh guarantees the pair of bubbles will bounce off, while they coalesce on coarse mesh.

Figure 5

Two types of unstable flow transitions around the deformable bubble. (a) Transition A from type I to type III, without forming and breaking the axisymmetric toroidal vortex rings. (b) Transition B from type II to type IV, subsequently forming and breaking the axisymmetric toroidal vortex ring.

Figure 6

Phase diagram showing the entire set of the present numerical results being positioned in the $\left(\text{Bo,Ga}\right)$ plane, while they correspond to the isolated bubble motion. Solid line: the approximate neutral curve for the unstable transition of an isolated bubble [23]. $\blacksquare$, Type I flow; $▴$, type II flow; $\square$, type III flow; and $▵$, type IV flow.

Figure 7

Four typical interactions between a pair of bubbles, corresponding to (a) bubble pair 1, repelling; (b) bubble pair 11, attracting coalescence; (c) bubble pair 12, attracting bounce; and (d) bubble pair 4, repeated bounce. The horizontal axis is the position of the bubble centroid and the vertical axis is the rising height, and both axes are nondimensionalized by the bubble diameter.

Figure 8

Repulsive interactions between bubble pair 1, corresponding to flow type I in the low-Re regime ($\text{Re}<30$). (a) Bubble trajectories. (b) The isocontours of ${\omega }_y$ around the bubbles, corresponding to $-100<{\omega }_y<100$. The result reveals that the vorticity diffusion is blocked in presence of another bubble, and hence the repulsive force is generated.

Figure 9

Interactions between bubble pair 10 and bubble pair 7 where the isolated bubble rises rectilinearly in both cases. (a) Bubble pair 10 with terminal state of $\text{Re}=133$ and $\chi =1.2$. Bubble pair 7 with terminal state of $\text{Re}=123$ and $\chi =1.4$. The right panel of each subfigure is the streamlines around the pair of bubbles and the streamwise vorticity isocontours of ${\omega }_z=\pm 2$ before the bubbles kiss each other. It is found that no torodial vortex rings or streamwise wakes are generated behind the pair of bubbles at the “kissing” moment.

Figure 10

Oscillations during the coalescence between bubble pair 7. (a) The rise velocities. (b) The bubble shape evolutions with a time interval of $\mathrm{\Delta }{t}^{*}=13$. Significant oscillations are observed during the coalescence process.

Figure 11

Rising trajectories of bubble pair 8 and the corresponding isolated bubble. (a) Front views of bubble pair 8; (b) front views of the isolated bubble; and (c) top views of the (left) bubble [note that coordinate transformations are applied to make the (left) bubble rise from $x=0$]. It is observed that although the isolated bubble rises rectilinearly, the pair of bubbles still show an attracting-bounce interaction behavior.

Figure 12

(a) Evolution process of the double-threaded wakes during the collision of bubble pair 8, while the isocontours correspond to ${\omega }_z=\pm 1$. (b) Amount of ${\omega }_z$ accumulated at the (left) bubble interface. For an isolated bubble, $\mathrm{\Omega }$ almost stays at a relative stable and low value of $\mathrm{\Omega }=5$; however, for bubble pair 8, $\mathrm{\Omega }$ increases sharply to trigger the vortex shedding as the bubbles get close, indicating that only if the streamwise vorticity accumulated on the bubble surface is more than a critical value, then the double-threaded wakes would be generated to cause bubble collision.

Figure 13

Rising trajectories of bubble pair 12 and the corresponding isolated bubble. Detailed description is provided in the caption of Fig. 11. It is observed that while the isolated bubble rises in an unstable path, the bubble pair shows an attracting-bounce interaction followed by another attracting-coalescence interaction.

Figure 14

Evolution process of the double-threaded wakes during the attracting-bounce period and the following attracting-coalescence period for bubble pair 12. [(a)–(e)] The two bubbles bounce off and move outward to the furthest position; [(f)–(j)] the two bubbles attract one another until coalescence. In all figures, isocontours correspond to ${\omega }_z=\pm 2$.

Figure 15

Interactions between the pair of bubbles rising in K2 when the initial separated distance is changed. Bubble pair 12 with $S=2.5$; bubble pair 13 with $S=2.0$; ($c$) different wake developments during bubble interactions. It is observed that with larger distance, the pair of bubbles are more likely to bounce off instead of coalescence, and this is closely related to the wake developments behind the pair of bubbles.

Figure 16

Temporal variations of the transverse velocity of the left bubble in bubble pair 12 and bubble pair 13. Point A: coalescence of bubble pair 13. Point B: bounce of bubble pair 12. Point C: coalescence of bubble pair 12. It is found greater collision velocity does not necessarily indicate bouncing interactions between the pair of bubbles.

Figure 17

Trajectories of the pair of bubbles rising in K0 when the bubble sizes are varied. (a) Bubble pair 11 with $D=1.25\text{mm}$, (b) bubble pair 12 with $D=1.5\text{mm,}$ and (c) bubble pair 14 with $D=2\text{mm}$. It is observed that by increasing the bubble size, attracting coalescence, attracting bounce, and repeated bounce appear in succession between a pair of bubbles.

Figure 18

Wake structures behind the bubble pair 14 at different time periods during a repeated-bounce period, and the isocontours correspond to ${\omega }_z=\pm 2$. It is observed that periodic vortex shedding happens while the wake is unstable at the tail due to viscous shear.

Figure 19

Interaction behaviors between bubble pair 0, corresponding to an isolated bubble rising rectilinearly within flow type II. (a) The rising trajectories; (b) the flow streamlines around the pair of bubbles. It is observed that the toroidal vortex rings become asymmetric in the presence of another bubble.

Figure 20

Different curvatures of the fluid velocity profile on the flow around the bubble. (a) Isolated bubble; (b) a pair of bubbles.

Figure 21

Rising trajectories of a pair of bubbles in K5 with varied initially separated distance. Note that these cases belong to flow type II. (a) Bubble pair 2 with $S$ = 1.3, (b) bubble pair 3 with $S$ = 1.8, (c) bubble pair 4 with $S$ = 2.5, and (d) bubble pair 5 with $S$ = 4.0. It is observed that with smaller $S$, the repeated bounce happens earlier while the bounce frequency and the bounce amplitude are almost the same.

Figure 22

(a) Temporal variation of the transverse velocity of the left bubble in bubble pair 2–5. It is observed that with a smaller separated distance, the repeated bounce happens earlier. However, once they enter the repeated-bounce motion, the frequency and amplitude of the velocity oscillations are the same, indicating that the repeated-bounce behavior is actually a particular type of the zigzag motion. (b) Isocontours ${\omega }_z=\pm 2$ at four succussive time points when the bubbles pass the average and furthermost (nearest) positions corresponding to bubble pair 4. It is observed that the double-threaded wakes are generated behind the pair of bubbles, while their signs are changed periodically during the repeated-bounce process.

Figure 23

Top views of the trajectory of the left bubble in different bubble pairs, from left to right: bubble pair 2–5 and the isolated bubble. It is observed that while an isolated bubble shows a zigzag-spiral transitional motion, in the presence of another bubble rising side by side, a more perfect zigzag motion is generated by shortening the separated distance between the pair of bubbles.

Figure 24

Amount of ${\omega }_z$ accumulated at the bubble interface by changing the initial separated distance between the pair of bubbles, with respect to bubble pair $2-5$ and the isolated bubble. It is observed that smaller $S$ accelerate the accumulation of ${\omega }_z$, and hence the critical ${\mathrm{\Omega }}_{\mathrm{cri}}$ for the onset of vortex shedding is reached earlier to trigger the repeated-bounce happening. In addition, although the isolated bubble rises in an spiral motion, the critical value of $\mathrm{\Omega }$ to trigger the vortex shedding is identical to that of the bubble pair case.

Figure 25

Rise velocities for different bubble pairs. (a) Bubble pair 4 and bubble pair 14, corresponding to repeated-bounce motion; (b) bubble pair 8 and bubble pair 12, corresponding to attracting-bounce motion. Solid line: left bubble; dashed line: right bubble; dot-dashed vertical line: time instant for the bouncing moment. As shown in the dotted ellipse in panel (b), the rise velocity drops to almost $90\%$ at collision; however, such decrease is not observed in panel (a).

Figure 26

(a) Rise velocity, (b) drag force, and (c) lift force for the left bubble in bubble pair 8 and for the isolated bubble. It is found that during the bubble collision, the rise velocity of the pair of bubbles drops as an indication of the drag enhancement. This is due to the increment of the shear rate inside the gap between the pair of bubbles that causes more vorticities to be produced at the bubble interface, as already shown in Fig. 12. For the evolution of $F_L$, it is observed that the lift force is repulsive during the bubbles collision; however, it becomes attractive gradually if the pair of bubbles are separated due to dissipation of the double-threaded vortices and the increase of the viscous force.

Figure 27

(a) Rise velocity, (b) rise height, and (c) lift force for the left bubble in bubble pair 2 and bubble pair 4 and for the isolated bubble. It is found that with smaller separated distance, the pair of bubbles experience greater drag force and the bubble rises with lower velocity. For the lift force evolvement, the isolated spiralling bubble experiences a more steady lift force due to the spiral motion while different bubble pairs display the same oscillatory trend due to zigzag motion.