# Numerical Study on the Rising Motion of Bubbles near the Wall
## 4 Conclusions  
This article takes the rising motion of bubbles in the near-wall domain as the research topic. Numerical simulations are used to analyze the rising motion characteristics of a single circular bubble and double circular bubbles in the domain near the wall, and explore the influence of the wall on the bubble rising, and the following conclusions are obtained:  
1. When a single bubble rises near the wall, the bubble will be affected by the wall and change its original state of motion. The bubble first rises in a straight line, and then begins to swing upward. When \(S^{*}\) becomes larger, the height of bubble rising also becomes higher.
2. Under the condition that the diameter of the bubble is fixed, when \(S^{*}<1\), the bubble will touch the wall when it rises, and then form a bouncing upward motion; and when \(S^{*}\geq 1\), the bubble will not touch the wall, and the bubble will rise straight up for a certain distance and then rise in a \"Z\" shape.
3. Numerical simulation results of the rising motion of a single bubble with a fixed diameter show that when \(S^{*}\) increases from 0.75 to 1.5, the bubble's average swing amplitude decreases from 3.66 mm to 3.08 mm, while the bubble's swing frequency decreases from 4.16 s\({}^{-1}\) increases to 4.85 s\({}^{-1}\). The results of this study show that the distance of the bubble wall has an effect on the rising movement of the bubble. It is concluded that when the diameter is fixed, when \(S^{*}\) gradually becomes larger, that is, when the bubble gradually moves away from the wall, the swing amplitude of the bubble gradually decreases, and the swing frequency of the bubble follows. As the bubble increases, the influence of the wall on the bubble gradually weakens as the bubble moves away.
4. The result of numerical simulation of the rising motion of a single bubble with \(S^{*}<1\) shows that when the diameters are 4, 5, and 6 mm, the average swing amplitude of the bubble increases from 3.08 mm to 4.75 mm, and the swing frequency of the bubble increases from 4.85 s\({}^{-1}\).The law of change reduced to 3.57 s\({}^{-1}\). The results of this study indicate that the bubble diameter has an effect on the bubble's rising motion. It is concluded that when the bubble diameter gradually increases in the area near the wall, the swing amplitude of the bubble gradually increases, while the swing frequency of the bubble decreases accordingly. It is concluded that the influence of the size of the bubble on the rising movement of the bubble gradually increases with the larger the bubble diameter.  
Figure 20: The shape change of the rising double bubbles arranged horizontally near the wall.  
5. The double bubbles in the near-wall domain will be affected by the wall and move away from the wall. The swing amplitude of the head bubble is greater than that of the tail bubble. 6. When the double bubbles are arranged horizontally, the influence of the wall surface on the bubbles is not obvious. The bubbles first move away from each other and then approach, and the flow field after the bubbles merge is symmetrically distributed. When the distance between the horizontally arranged double bubbles gradually increases, the influence of the wall surface on the left bubble will gradually become obvious. When the vertically arranged double bubbles of different sizes rise, with proper bubble spacing, coalescence can occur regardless of whether it is an arrangement of large top and small bottom or large top and small bottom.  
Conceptualization, P.L.; Formal analysis, Y.L. and K.Z.; Investigation, Q.C. and P.L.; writing-original draft preparation, K.Z.; writing-review and editing, Y.L. All authors have read and agreed to the published version of the manuscript.  
The present work is financially supported by the Key R&D Program of Zhejiang Province (Grant No. 2020C03081), the Joint Funds of the National Natural Science Foundation of China (Grant No. U2006221), the National Natural Science Foundation of China (Grant No. 51676173).  
The authors declare no conflict of interest.