# Numerical Study on the Rising Motion of Bubbles near the Wall
## 3 Numerical Simulation Results
### Analysis of Single Bubble Motion near the Wall
#### 3.1.1 The Influence of Bubble Wall Distance  
Figure 7a is the ascending path of the 4 mm bubble at \(S^{*}=\) 0.75, 1.0, 1.25, 1.5. The bubble rises linearly in a certain path, and then begins to deviate to the side away from the wall to swing up in a \"Z\" shape. Figure 7b adjusts the origin of the X axis of the four types of bubbles. The origin of the X axis is set to the X coordinate corresponding to the center of the bubble, so that the starting points of the bubble path are all fitted together. The path difference between the bubbles can be seen more intuitively. In the Table 1, \(L\) is the average swing amplitude of the bubble, and \(F\) is the average swing frequency of the bubble. When \(S^{*}\) becomes larger, the bubble gradually moves away from the wall, and the swing amplitude of the bubble gradually decreases, while the swing frequency of the bubble increases. The influence of the wall on the bubble gradually weakens as the bubble moves away. In the table, \(H_{\text{max}}\) represents the maximum rise height of the bubble during the 500 ms monitoring time. In \(H_{\text{max}}\), the rise height is the lowest when \(S^{*}=1\), and when \(S^{*}\) becomes larger, the bubble height also becomes higher. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**S\({}^{*}\)** & **L\({}_{1}\)/mm** & **L\({}_{2}\)/mm** & **L\({}_{3}\)/mm** & **L\({}_{4}\)/mm** & **L\({}_{5}\)/mm** & **L/mm** & **H\({}_{\text{max}}\)/mm** & **F\({}_{8}\)/s\({}^{-1}\)** \\ \hline
0.75 & 4.0 & 4.3 & 3.4 & 4.4 & 2.2 & 3.66 & 84.4 & 4.16 \\
1 & 5.4 & 3.5 & 2.7 & 4.3 & 1.5 & 3.48 & 80.8 & 4.42 \\
1.25 & 3.8 & 4.1 & 1.8 & 4.1 & 2.4 & 3.22 & 84 & 4.69 \\
1.5 & 3.6 & 4.0 & 2.2 & 3.8 & 1.8 & 3.08 & 88.6 & 4.85 \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical table of bubble rise swing under different S\({}^{*}\). Figure 7: Bubble motion path diagram under different S\({}^{*}\): (**a**) Different initial position, (**b**) Same initial position. Figure 8 shows the average rising speed of the bubble every 20 ms in 500 ms. From Table 2, it can be seen that when the bubble rises, the speed will gradually increase, and when it reaches a certain value, it will gradually stabilize and fluctuate up and down within a certain range. The farther the bubble is from the wall, the faster the initial rising speed increases. From the two curves of \(S^{*}=1\) and \(S^{*}=0.75\), it can be found that when the bubble rises in contact with the wall, the speed will be faster than the free rise of the bubble. It can be seen from Table 2 that the overall rising speed of the bubble within 500 ms can also prove that the bubble bounce motion will rise faster than the free motion. It can be seen from Figure 9a,b that the flow field of bubbles rising near the wall is different from the flow field of bubbles rising in unbounded still water. Due to the influence of the left wall, the flow field on the left side of the fluid flows closer to the left side of the bubble, and the flow velocity direction is more. It is the tangent direction of the left side of the bubble. The flow velocity on the left side will be faster than that of unbounded static water, resulting in the formation of the flow field at the bottom of the bubble forming a situation where the left side is less and the right side is more, which further causes the clockwise deflection of the bubble. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Bubble**} & & & & & & & \\
**Diameter** & **L\({}_{1}\)/mm** & **L\({}_{2}\)/mm** & **L\({}_{3}\)/mm** & **L\({}_{4}\)/mm** & **L\({}_{5}\)/mm** & **L\({}_{7}\)/mm** & **H\({}_{\text{max}}\)/mm** & **F/s\({}^{-1}\)** \\ \multicolumn{1}{c}{/mm} & & & & & & & & \\ \hline
4 & 3.6 & 4.0 & 2.2 & 3.8 & 1.8 & 3.08 & 88.6 & 4.85 \\
5 & 3.7 & 5 & 4.2 & 3.1 & – & 4.0 & 94.4 & 4.24 \\
6 & 3.4 & 3.7 & 7.3 & 4.6 & – & 4.75 & 99 & 3.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical table of rising swing amplitude of bubbles with different diameters.