# Numerical Study on the Rising Motion of Bubbles near the Wall
## 3 Numerical Simulation Results
### Analysis of the Motion of Double Bubbles near the Wall  
#### 3.2.1 Vertically Rising Bubbles  
(1) Double bubbles of equal diameter  
Figure 11 shows the upward motion of two vertically arranged bubbles in the near-wall domain. The diameter of the bubbles is 4 mm, and the two bubbles are arranged vertically with a distance of 8 mm. At 100 ms, it can be found that the two bubbles are deflected away from the wall under the influence of the wall; At 115 ms, the deflection of the tail bubble recovers and it approaches the wall again, while the head bubble continues to move away from the wall; A large bubble is formed after two bubbles come into contact at 140 ms. From the bubble motion in the figure and the previous research, we can find that two bubbles with the same diameter and the same distance from the bubble wall produce different swing amplitudes in the near-wall domain. This shows that the effect of the near wall on the double bubble is different from that of the single bubble.  
Figure 11: The shape change diagram of the rising double bubbles vertically arranged near the wall.  
Figure 10: Bubble motion path and rising speed diagram under different diameters: (**a**) Motion path, (**b**) Ascent speed.  
Figure 12 is the flow field diagram of the bubble at different moments. At 80 ms, the bubble moves away from the wall due to the influence of the wall at this time. At 110 ms, the bubble forms a wake vortex due to the deflection, and the right wake vortex of the head bubble has an impact on the tail bubble, causing the bubble to rotate counterclockwise to restore the deflection, resulting in a difference in the swing amplitude of the two bubbles.  
The upper small and the lower large double bubbles  
Figure 13 is a simulation of the rising motion of double bubbles of different sizes arranged vertically, with small head bubbles and large tail bubbles. From the figure, it can be found that the double bubbles of the same size are similar to those in the previous article. Both bubbles deflected away from the wall, and then merged.  
In Figure 14a, it can be found that the flow field of large bubbles is denser than that of small bubbles. This is the reason why the rising speed of large bubbles is faster than that of small bubbles; the big bubble is moving away from the wall, while the small bubble has recovered its deflection and is moving close to the wall. This result can be mutually confirmed with the near-wall motion of single bubbles with different diameters. Large bubbles are affected by the wall surface more obviously, with larger average swing amplitude and lower frequency.  
Figure 12: Upward flow field diagrams of vertically arranged double bubbles at different moments: (a) 80 ms, (b) 110 ms.  
Figure 13: The rising shape of double bubbles of different sizes near the wall (upper small and lower large).  
The upper large and the lower small double bubbles  
Figure 15 is a simulation of the rising motion of double bubbles of different sizes arranged vertically. The head bubble is large and the tail bubble is small. Contrary to the previous section, the bubbles still coalesce. It can be seen from the figure that it is different from the previous section. The big bubble first deflects at 70 ms, while the small bubble starts to deflect for the first time at 135 ms. The speed of the small bubble is faster than that of the big bubble. The two bubbles come into contact at 160 ms and then merge into one large bubble.  
From Figure 14b, it can be seen that when the big bubble rises, a wake vortex is left at the tail. The small bubbles rise rapidly along the jet formed by the rising of the large bubbles, thus coming into contact with the large bubbles.