# Experimental Investigations on the Cavitation Bubble Dynamics near the Boundary of a Narrow Gap

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

_{b}is the bubble centroid position. R

_{max}is the maximum bubble radius. l is the distance between the bubble centroid and the boundary. h is the width of the gap between the two plates. In the experiments, the two glass plates were large enough, and the bubble was dominantly affected by the boundary located on the Y-axis. It is worth mentioning that, as l decreased below R

_{max}, the bubble partially protruded beyond the two plates. The influence of the boundaries in other directions (such as the upper, lower, and left boundaries) can be ignored. In Figure 2b, θ is the contact angle between the water droplet and the glass plate, which was set to 30°.

^{*}represents the dimensionless distance between the bubble and the gap boundary. It can affect the degree of anisotropy of the liquid field around the bubble and, thus, affect the collapse pattern and jet of the bubble. h

^{*}represents the dimensionless width of the gap. It can reflect the influence degree of the parallel plates on the bubble collapse process and the jet characteristics. In addition, the Reynolds number is defined as [31]

_{max}= 1.78 mm; h

^{*}= 0.73, 0.79, and 0.84; l

^{*}= [0, 1.75]; μ = 10

^{−3}g/(m·s); and ρ = 1000 kg/m

^{3}. In the case of the above parameters, T is always less than 400 μs. Therefore, it can be calculated that Re > 5782.33 >> 1, and the influence of the gap boundary can be ignored.

## 3. Typical Dynamic Behaviors of the Cavitation Bubble

^{*}= 0.00, 1.00, and 1.75, respectively, including the bubble evolution during the first and subsequent oscillation periods. The dashed blue line is the boundary of the narrow gap. The left and right of the boundary represent the inside and outside of the gap, respectively. In each figure, the coordinate system in the lower right corner corresponds to the X-axis and Y-axis directions in Figure 2a. Frames 1–3 refer to the bubble growth stage; frames 4–10 refer to the collapse stage; and frames 11–15 refer to the rebound stage.

^{*}is small (Figure 3), the bubble is partially inside the gap. When the bubble reaches its maximum radius (frame 3), the left half of the bubble is restricted due to the two plates. During the collapse stage, the right side of the bubble shrinks much faster than the left side, and the bubble has completely retracted into the gap (frame 9). Subsequently, the left surface of the bubble is penetrated with a jet towards the gap (frame 10). During the rebound stage, under the influence of the collapse jet, the left side of the bubble forms a tip and continues to move to the left, which is generally called the vapor jet. Meanwhile, a counter-vapor jet appears on the right side of the bubble, and it partially extends out of the gap.

^{*}is medium (Figure 4), the bubble is completely inside the gap but still very close to the boundary. When the bubble reaches its maximum radius (frame 3), it appears as a standard cylindrical shape with its right endpoint exactly touching the boundary. During the collapse stage, a coronal structure is generated on the right side of the bubble (frame 4). As the bubble collapses and moves away from the boundary, the coronal structure gradually shrinks and disappears, and then the right side of the bubble shrinks rapidly, forming a depression (frame 8). During the rebound stage, the bubble is divided into three bubble clouds (frame 14). The bubble cloud on the left appears in an arc shape, and it keeps moving to the left. Meanwhile, the two bubble clouds behind it also move to the left, along with spin in opposite directions.

^{*}is large (Figure 5), the bubble is completely inside the gap and is far from the boundary. The bubble is only weakly affected due to the boundary. During the growth and collapse stages, the bubble maintains a regular round shape. During the rebound stage, it is in the shape of a hollow cloud, it evenly spreads, and then it dissipates rapidly.

^{*}= 0.00, which corresponds to Figure 3. The coordinate system in the lower right corner corresponds to the Y-axis and Z-axis directions in Figure 2a. During the growth stage, the curvature of the upper and lower surfaces of the bubble gradually decreases and becomes almost flat when the bubble reaches its maximum radius (frame 3). During the collapse stage, the shrinkage speed of the bubble in the horizontal direction is greater than that in the vertical direction. Meanwhile, the depressions appear on the upper and lower surfaces of the bubble. Eventually, the bubble completely shrinks into the gap.

## 4. Quantitative Analysis of the Bubble Behaviors

#### 4.1. Bubble Outlines and Time Stacks

^{*}= 0.00, 1.00, and 1.75, respectively. Among them, the sub-figures (a) and (b) correspond to the collapse and rebound stages, respectively. In each sub-figure, the gray and white areas represent the inside and outside of the gap. In terms of the time order, the outline and the centroid of the bubble correspond to green, red, blue, and purple in turn.

^{*}is small (Figure 7), during the collapse stage (Figure 7a), the right side of the bubble shrinks at the fastest speed, the left side shrinks at the slowest speed, and the upper and lower sides have a moderate shrinkage speed. Meanwhile, the bubble centroid is constantly moving into the gap. It can be considered that the inhomogeneity of the pressure around the bubble and the influence of the viscous boundary layer are the main reasons for the bubble’s anisotropic deformation and translational movement. During the rebound stage (Figure 7b), the bubble expands significantly in both the left and right directions, and the expansion on the left side is much more dramatic, causing the bubble centroid to continuously move to the left.

^{*}is medium (Figure 8), during the collapse stage (Figure 8a), the right side of the bubble shrinks irregularly, while the other sides shrink normally and are hardly affected due to the boundary. Meanwhile, the bubble centroid is still moving towards the left. During the rebound stage (Figure 8b), the bubble’s left side expands to the left, while the right side also exhibits a slight leftward movement. And the bubble centroid continues to move towards the left. When l

^{*}is large (Figure 9), there is no significant difference in the shrinkage or expansion of the bubble in all directions. And the bubble centroid just moves slightly towards the left.

^{*}= 0.79, 0.73, and 0.84, respectively. The subfigures (a–c) correspond to l

^{*}= 0.00, 1.00, and 1.75, respectively. In each sub-figure, the abscissa represents the X-axis defined in Figure 2a, and the ordinate represents the time. The left and right sides of the dashed blue line refer to the inside and outside of the narrow gap, respectively. The upper and lower sides of the dashed red line refer to the bubble’s first oscillation period and subsequent processes, respectively.

^{*}, for different l

^{*}values, there are obvious differences in the shrinkage speed of the right endpoint of the bubble during the collapse stage, as well as the moving direction of the left and right endpoints of the bubble during the rebound stage. During the collapse stage, as l

^{*}increases, the shrinkage speed of the bubble’s right endpoint gradually decreases. During the subsequent processes, as l

^{*}increases, the moving direction of the bubble’s left endpoint is always towards the left, and its moving distance gradually shortens. However, the moving direction of the bubble’s right endpoint changes. It moves towards the right at l

^{*}= 0.00, towards the left at l

^{*}= 1.00, and towards the right at l

^{*}= 1.75.

^{*}is mainly reflected in the rebound and subsequent processes. When the bubble is to l

^{*}= 0.00, as h

^{*}increases, the moving distance of the bubble’s left endpoint gradually shortens. The bubble cloud’s expansion degree is more concentrated. When the bubble is at l

^{*}= 1.75, as h

^{*}increases, the movement direction of the bubble’s right endpoint changes from the left to the right. The bubble cloud’s expansion degree is even less concentrated. In addition, the boundary has a larger spatial influence range (i.e., the range of l

^{*}) on the bubble dynamics for a smaller gap width.

#### 4.2. Bubble Feature Points

^{*}= 0.00, 1.00, and 1.75, respectively. In the figure, s represents the distance between the position of the bubble’s left/right endpoint at a certain moment and at the moment when the bubble reaches its maximum radius. When l

^{*}is small (Figure 13a), the difference in the movement of the two endpoints gradually increases over time. Among them, the moving distance of the left endpoint slowly increases, and its total moving distance during the collapse stage is quite small, about 0.22 R

_{max}. In contrast, the moving distance of the right endpoint increases significantly and becomes quite large at the end of the collapse, about 1.04 R

_{max}. When l

^{*}is medium (Figure 13b), the moving distances of the two endpoints mildly increase and are almost the same at the early collapse stage. But the difference between them suddenly increases at the end of the collapse. In this stage, the moving velocity of the right endpoint increases rapidly due to the disappearance of the coronal structure and the appearance of the depression on the bubble’s right side. During the entire collapse process, its moving distance is about 1.00 R

_{max}. When l

^{*}is large (Figure 13c), the moving distances of the left and right endpoints are similar. During the collapse process, the moving distances of the left and right endpoints of the bubble are 0.56 R

_{max}and 0.81 R

_{max}, respectively. In addition, when comparing the cases with different l

^{*}values, it can be found that, as l

^{*}increases, the total moving distance of the left endpoint during the collapse stage gradually increases and that of the right endpoint behaves conversely. With the increase in l

^{*}, the difference in the moving distance between the left and right endpoints of the bubble gradually decreases. Specifically, at l

^{*}= 1.0 and 1.75, the differences in the moving distances of the two endpoints at the end of the collapse are 68% and 10%, respectively, of that in the case at l

^{*}= 0.

^{*}. In the figure, Δs represents the difference between the moving distance of the two endpoints at the end of the collapse, which is defined in Figure 13a. As shown in Figure 14, at l

^{*}= 0, the value of Δs is the highest, about 0.84 R

_{max}. As l

^{*}increases, the difference in the moving distance between the two endpoints first decreases and then increases. At l

^{*}= 0.50, it reaches its local minimum, about 0.31 R

_{max}. At l

^{*}= 1.00, it reaches its local maximum, about 0.58 R

_{max}. Finally, it gradually decreases to zero.

#### 4.3. Bubble Centroids

_{c}represents the distance between the bubble centroid at a certain moment and at the moment when the bubble reaches its maximum radius in the horizontal direction. As shown in Figure 15, l

^{*}has a significant impact on the movement of the bubble centroid. When the bubble is at l

^{*}= 0.00, its centroid moves noticeably with a progressively increasing moving velocity, and the moving distance reaches about 0.35 R

_{max}at the end of the collapse. When the bubble is at l

^{*}= 1.00, it has no obvious moving distance at the early stage of the collapse but suddenly accelerates, and the moving distance reaches about 0.25 R

_{max}at the end of the collapse. When the bubble is at l

^{*}= 1.75, its centroid moves only slightly, and the moving distance reaches about 0.07 R

_{max}at the end of the collapse. With the increase in l

^{*}, the moving distance of the bubble centroid gradually decreases. Specifically, at l

^{*}= 1.0 and 1.75, the centroid moving distances at the end of the collapse are 73% and 20%, respectively, of that in the case at l

^{*}= 0.

^{*}on the moving distances of the bubble centroid during the collapse stage, Figure 16 shows the change in the moving distance of the bubble centroid with l

^{*}. In the figure, Δx

_{c}represents the moving distance of the bubble centroid from the moment when the bubble radius is maximum to the end of the collapse. With the increase in l

^{*}, the moving distance of the bubble centroid gradually decreases overall, and e reaches about 0.05 R

_{max}at l

^{*}= 1.75. An irregular increase occurs at l

^{*}= 1.00 with Δx

_{c}reaching 0.20 R

_{max}, which is caused due to the appearance of the crown structure at the right side of the bubble.

## 5. Conclusions

^{*}, and the dimensionless gap width h

^{*}on the jet, as well as the outline evolution and translational movement of the bubble during the bubble collapse and subsequent processes, have been revealed. The main conclusions are as follows:

- (1)
- When l
^{*}is small, the part of the bubble outside the gap shrinks quite quickly, and a violent jet is generated towards the gap during the collapse stage. Subsequently, a strong vapor jet toward the gap and a weak counter-vapor jet toward the boundary both appear during the rebound stage. - (2)
- As l
^{*}increases, both the translational distance of the bubble centroid and the difference in the moving distance of the left and right endpoints of the bubble initially decrease, then increase, and finally decrease to zero. - (3)
- Within the range of 0.73 < h
^{*}<0.84, h^{*}mainly affects the expansion degree of the bubble cloud and the moving direction of the left and right endpoints of the bubble during its rebound and subsequent processes. As h^{*}decreases, the influence range of the boundary (i.e., the range of l^{*}) on the bubble behaviors increases. In addition, the value of the Re number during the bubble collapse is much higher than 1; hence, the influence of the liquid viscosity is very weak.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Physical diagram and schematic diagram of the experimental system: (

**a**) physical diagram; (

**b**) schematic diagram.

**Figure 2.**Specific structure of the narrow gap and the physical properties of the glass plate surface: (

**a**) three-dimensional gap structure and the parameter definitions; (

**b**) the wettability of the plate surface.

**Figure 3.**High-speed photographs of the bubble at l

^{*}= 0.00. The dashed blue line is the boundary of the narrow gap. The left and right of the boundary represent the inside and outside of the gap, respectively. h

^{*}= 0.79.

**Figure 4.**High-speed photographs of the bubble at l

^{*}= 1.00. The dashed blue line is the boundary of the narrow gap. The left and right of the boundary represent the inside and outside of the gap, respectively. h

^{*}= 0.79.

**Figure 5.**High-speed photographs of the bubble at l

^{*}= 1.75. In each frame, the presented area is completely inside the gap, and the boundary is on the right side outside the frames. h

^{*}= 0.79.

**Figure 6.**High-speed photographs of the bubble at l

^{*}= 0.00 from the right-side view. The vertical dark gray strips are the two glass plates. h

^{*}= 0.79.

**Figure 7.**Bubble outlines at l

^{*}= 0.00. (

**a**) Collapse stage. (

**b**) Rebound stage. The gray area is the inside of the gap, and the white area is the outside of the gap. h

^{*}= 0.79.

**Figure 8.**Bubble outlines at l

^{*}= 1.00. (

**a**) Collapse stage. (

**b**) Rebound stage. The gray area is the inside of the gap, and the white area is the outside of the gap. h

^{*}= 0.79.

**Figure 9.**Bubble outlines at l

^{*}= 1.75. (

**a**) Collapse stage. (

**b**) Rebound stage. The gray area is the inside of the gap, and the boundary is on the right side outside the subfigure. h

^{*}= 0.79.

**Figure 10.**Composite images of high-speed photographs of the bubble over time: (

**a**) l

^{*}= 0.00; (

**b**) l

^{*}= 1.00; and (

**c**) l

^{*}= 1.75. h

^{*}= 0.79. The red dashed lines are the demarcation between the first oscillation period of the bubble and the subsequent processes. The blue dashed lines are the boundary of the narrow gap.

**Figure 11.**Composite images of high-speed photographs of the bubble over time: (

**a**) l

^{*}= 0.00; (

**b**) l

^{*}= 1.00; and (

**c**) l

^{*}= 1.75. h

^{*}= 0.73. The red dashed lines are the demarcation between the first oscillation period of the bubble and the subsequent processes. The blue dashed lines are the boundary of the narrow gap.

**Figure 12.**Composite images of high-speed photographs of the bubble over time: (

**a**) l

^{*}= 0.00; (

**b**) l

^{*}= 1.00; and (

**c**) l

^{*}= 1.75. h

^{*}= 0.84. The red dashed lines are the demarcation between the first oscillation period of the bubble and the subsequent processes. The blue dashed lines are the boundary of the narrow gap.

**Figure 13.**Moving distances of the bubble’s left and right endpoints during the collapse stage over time: (

**a**) l

^{*}= 0.00; (

**b**) l

^{*}= 1.00; and (

**c**) l

^{*}= 1.75. The black and red curves refer to the left and right endpoints, respectively. The two blue dashed lines in the sub-figure (

**a**) correspond to the moving distance of the left and right endpoints of the bubble at the end of the collapse, and the blue arrow line represents the difference of the moving distance between the two endpoints, represented by Δs.

**Figure 14.**Change in the differences of the moving distance between the left and right endpoints of the bubble with l

^{*}during the bubble collapse. h

^{*}= 0.79.

**Figure 15.**Moving distances of the bubble centroid over time during the bubble collapse stage at different l

^{*}. The black, red, and blue curves refer to l

^{*}= 0.00, 1.00, and 1.75, respectively. h

^{*}= 0.79.

**Figure 16.**Change in the moving distance of the bubble centroid with l

^{*}during the bubble collapse. h

^{*}= 0.79.

Equipment | Model | Information | Accuracy/Sensitivity |
---|---|---|---|

Laser generator | Penny-100A-SC | Maximum pulse energy: 50 mJ Wavelength: 532 nm | Power instability: 0.81% |

High-speed camera | Qianyanlang X113 | Shooting speed: 41,666 fps Picture scale: 0.078 mm/pix | Scale error: ±0.001 mm/pix |

Delay signal generator | Stanford DG535 | Distinguishability: 5 ps | Resolution: 5 ps |

Water tank | Customization | Volume: 150 × 150 × 150 mm^{3} |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Yang, Y.; Guo, Z.; Hu, Q.; Wang, X.; Zhang, Y.; Li, J.; Zhang, Y.
Experimental Investigations on the Cavitation Bubble Dynamics near the Boundary of a Narrow Gap. *Symmetry* **2024**, *16*, 541.
https://doi.org/10.3390/sym16050541

**AMA Style**

Wang Z, Yang Y, Guo Z, Hu Q, Wang X, Zhang Y, Li J, Zhang Y.
Experimental Investigations on the Cavitation Bubble Dynamics near the Boundary of a Narrow Gap. *Symmetry*. 2024; 16(5):541.
https://doi.org/10.3390/sym16050541

**Chicago/Turabian Style**

Wang, Zhifeng, Yihao Yang, Zitong Guo, Qingyi Hu, Xiaoyu Wang, Yuning Zhang, Jingtao Li, and Yuning Zhang.
2024. "Experimental Investigations on the Cavitation Bubble Dynamics near the Boundary of a Narrow Gap" *Symmetry* 16, no. 5: 541.
https://doi.org/10.3390/sym16050541