# Retardant Effects of Collapsing Dynamics of a Laser-Induced Cavitation Bubble Near a Solid Wall

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

- Check the connections between different systems, e.g., signal transmission.
- Place the solid wall in a suitable place through the 3D platform.
- Generate bubbles using focused laser. Meanwhile, trigger the camera and the flashing lights. The delays between the three elements could be adjusted for different purposes.
- Save the data and check its quality.
- Repeat the same parameter setup several times for repeatability.
- Try another distance between the bubble and wall or different bubble sizes controlled by the laser energy.

_{max}is the equivalent maximum radius of the cavitation bubble during its dynamic oscillations, also corresponding to the maximum bubble volume; L is the distance between the inception point of the bubble and the surface of the solid wall. In order to analyze the data, the Phantom Camera Control (PCC) application was employed for the accurate measurement of R

_{max}and L. In the present paper, a series of experiments with different R

_{max}(7 values) and L (13 values) were conducted with the variations of λ within [0.2, 3.0].

## 3. Qualitative Descriptions of the Cavitation Bubble Dynamic Behavior Near the Wall

#### 3.1. Typical Examples of the Collapse of a Cavitation Bubble Near a Wall

_{max}) with the left-most position at the point A and the right-most position at the point B, respectively (denoted poles A and B for short). In order to show the movement of the bubble interface more clearly, two vertical dashed lines are marked in the figure with the yellow and the blue lines passing the points A and B, respectively.

_{max}. In the present paper, the laser energy is critical for the bubble size and the phenomenon. Hence, for the same energy, R

_{max}is assumed to be the same. Specifically, for the convenience of the calculations, the value of R

_{max}without the presence of the wall is employed for the calculations of the non-dimensional standing-off distance. Afterwards, the cavitation bubble enters its first collapsing stage (subplot 9–14 of Figure 2) with a rapid shrinkage of the bubble interface. Differently from the growth stage, the bubble at the collapsing stage is highly non-spherical with great differences in different directions. As shown in Figure 2, some further phenomena (e.g., the rebound) could be also observed in our experiment. However, these phenomena are outside of our primary interest and will not be further discussed in detail in the present paper.

#### 3.2. Influences of the Bubble–Wall Distance on the Bubble Collapse

_{max}), respectively. The time interval between every two adjacent subplots is 4.17 µs. The detailed characteristics among three cases are summarized in Table 1 with the following explanations:

- Case 1:
- This case corresponds to the condition with a short distance between the cavitation bubble and the wall (e.g., λ = 1.25). It can be observed from the figure that when the cavitation bubble is very close to the wall, pole A is nearly stationary, while pole B moves to the left significantly. In addition, the overall position of the cavitation bubble also moves toward the wall. If the internal density of the cavitation bubbles is assumed to be uniform, the bubble (mass) centroid continuously moves to the left during the collapsing stage. At the end of the collapsing stage (subplot 18–20 in Figure 3), the left side of the bubble approaches the wall surface while the right side of the bubble shows a toroidal shape in the middle, indicating that the jet may be finally generated.
- Case 2:
- This case corresponds to the condition with a medium distance between the cavitation bubble and the wall (e.g., λ = 2.50). As shown in Figure 4, differently from Figure 3 with a short distance, both poles A and B move towards the inside of the cavitation bubble (Table 1). In addition, the velocity of pole B is much faster than that of pole A, leading to the obvious movement of the bubble towards the wall. Due to the relatively considerable distance, the bubble does not contact the wall during the whole process. At the end of the collapsing stage, similarly to Figure 3, there also exists a toroidal shape on the right side of the cavitation bubble (subplot 20 in Figure 4). However, the display time of the torus is obviously delayed in Figure 4 because of the weak influences of the wall.
- Case 3:
- This case corresponds to the condition with a large distance between the cavitation bubble and the wall (e.g., λ = 3.75). It can be observed from the figure that when the bubble–wall distance is large, the bubble nearly retains a spherical shape during the oscillations. In particular, poles A and B both move toward the inside of the bubble with nearly the same speed. Hence, the movement of the bubble centroid is very marginal during the collapsing stage.

## 4. The Velocities of Poles A and B of the Cavitation Bubble

**shown in Figure 6, Figure 7 and Figure 8 is defined as the value of $\tau $ corresponding to a difference in the velocities between poles A and B of 5 m/s.**

_{crit}**= 0.45, ${V}_{B}$ increases dramatically, indicating a large amount of shrinkage. However, ${V}_{A}$ is nearly zero throughout and hence it is not necessary to perform curve fitting for ${V}_{A}$. Here, because the presence of the torus at the right part of bubble leads to great difficulties for the calculations of ${V}_{B}$, only data with $\tau $ < 0.8 is shown in Figure 6. As shown in Figure 6, $\Delta V$ is quite significant for this case, especially at the late collapsing stage.**

_{crit}**= 0.60, the values of the ${V}_{A}$ and ${V}_{B}$ are both nearly zero, indicating that the collapse of the bubble is delayed. Furthermore, for $\tau $ > 0.60, ${V}_{A}$ also increases slightly but is still far less than ${V}_{B}$. For example, at the final collapsing stage ($\tau $ = 1.00), ${V}_{B}$ is about three times the value of ${V}_{A}$.**

_{crit}**= 0.64. Differently from Figure 6 and Figure 7, for nearly the whole collapsing process (e.g., $\tau $ < 0.80), ${V}_{A}$ is almost equal to ${V}_{B}$ and the bubble mainly retains a spherical collapsing shape (e.g., ${V}_{A}\approx {V}_{B}$).**

_{crit}## 5. The Motions of the Bubble Centroid

#### 5.1. The Motions of the Bubble Interface and the Position of the Centroid

#### 5.2. The Relative Movement of the Bubble Centroid

## 6. Physical Interpretation of the Phenomenon

^{3}and ${c}_{L}$ of 1500 m/s, and the wall material is steel with ${\rho}_{L}$ of 8050 kg/m

^{3}and ${c}_{L}$ of 5790 m/s. The assumptions employed for Equation (7) include the adiabatic status. In the present paper, the cavitation bubbles show fast movement during the collapse and this assumption could be safely satisfied. As shown in Section 4, a typical collapse velocity (${V}_{C}$) measured in our experiment is of the order of 60 m/s, yielding a rough estimation of ${P}_{WH}$ of approximately 87.19 MPa. Hence, one can find a vivid example of the tremendous damage potential of the cavitation bubble collapse.

_{max}). The speed of the wave could be considered to be 1500 m/s, which is the speed of sound in the water. Then, during the bubble collapse, the traveling distance of the wave can be estimated as 0.15 m, which is far beyond the typical size of the generated bubbles (with the maximum bubble radius being about 1 mm). Hence, the pressure wave could easily reach the surrounding fluids near the bubble during its dynamic oscillations. Clearly there are still some limitations of the aforementioned mechanisms (e.g., wave reflections by the wall and the bubble). These complex scenarios will be further considered in future work, together with non-spherical bubble oscillations [41]. Furthermore, the cavitation-induced vortex generations are of great interest for many industrial flows, which could also be revealed with the aid of the newly developed vortex identification method [42,43].

## 7. Conclusions

- (1)
- The presence of the wall could significantly alter the collapse contour, leading to a great difference between the movements of poles A and B.
- (2)
- The retardant effect can be observed due to the presence of the wall. With the increase of λ, the retardant effects of the wall will be dismissed and the bubble finally recovers the spherical oscillations.
- (3)
- The cavitation bubble could induce a significant water hammer pressure (up to 87 MPa in our experiment) during the bubble collapse.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Zhang, Y.; Liu, K.; Xian, H.; Du, X. A review of methods for vortex identification in hydroturbines. Renew. Sustain. Energy Rev.
**2018**, 81, 1269–1285. [Google Scholar] [CrossRef] - Arndt, R.E.; Voigt, R.L., Jr.; Sinclair, J.P.; Rodrique, P.; Ferreira, A. Cavitation erosion in hydroturbines. J. Hydraul. Eng.
**1989**, 115, 1297–1315. [Google Scholar] [CrossRef] - Brennen, C.E. Hydrodynamics of Pumps; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Reuter, F.; Mettin, R. Mechanisms of single bubble cleaning. Ultrason. Sonochem.
**2016**, 29, 550–562. [Google Scholar] [CrossRef] [PubMed] - Zeng, Q.; Gonzalez-Avila, S.R.; Dijkink, R.; Koukouvinis, P.; Gavaises, M.; Ohl, C.D. Wall shear stress from jetting cavitation bubbles. J. Fluid Mech.
**2018**, 846, 341–355. [Google Scholar] [CrossRef] [Green Version] - Philipp, A.; Lauterborn, W. Cavitation erosion by single laser-produced bubbles. J. Fluid Mech.
**1998**, 361, 75–116. [Google Scholar] [CrossRef] - Benjamin, T.B.; Ellis, A.T. The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Philos. Trans. R. Soc. Lond. Ser. A
**1966**, 260, 221–240. [Google Scholar] [CrossRef] - Zhang, S.; Duncan, J.H.; Chahine, G.L. The final stage of the collapse of a cavitation bubble near a rigid wall. J. Fluid Mech.
**1993**, 257, 147–181. [Google Scholar] [CrossRef] - Brujan, E.A.; Keen, G.S.; Vogel, A.; Blake, J.R. The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys. Fluids
**2002**, 14, 85–92. [Google Scholar] [CrossRef] [Green Version] - Popinet, S.; Zaleski, S. Bubble collapse near a solid boundary: A numerical study of the influence of viscosity. J. Fluid Mech.
**2002**, 464, 137–163. [Google Scholar] [CrossRef] - Zhang, Y.L.; Yeo, K.S.; Khoo, B.C.; Wang, C. 3D jet impact and toroidal bubbles. J. Comput. Chem. Phys.
**2001**, 166, 336–360. [Google Scholar] [CrossRef] - Tomita, Y.; Robinson, P.B.; Tong, S.P.; Blake, J.R. Growth and collapse of cavitation bubbles near a curved rigid boundary. J. Fluid Mech.
**2002**, 466, 259–283. [Google Scholar] [CrossRef] - Klaseboer, E.; Hung, K.C.; Wang, C.; Wang, C.W.; Khoo, B.C.; Boyce, P.; Debono, S.; Charlier, H. Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. J. Fluid Mech.
**2005**, 537, 387–413. [Google Scholar] [CrossRef] - Yusof, N.S.M.; Babgi, B.; Alghamdi, Y.; Aksu, M.; Madhavan, J.; Ashokkumar, M. Physical and chemical effects of acoustic cavitation in selected ultrasonic cleaning applications. Ultrason. Sonochem.
**2016**, 29, 568–576. [Google Scholar] [CrossRef] [PubMed] - Prentice, P.; Cuschieri, A.; Dholakia, K.; Prausnitz, M.; Campbell, P. Membrane disruption by optically controlled microbubble cavitation. Nat. Phys.
**2005**, 1, 107–110. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Xie, X.; Zhang, Y.; Du, X. Experimental study of influences of a particle on the collapsing dynamics of a laser-induced cavitation bubble near a solid wall. Exp. Fluid Sci.
**2019**, 105, 289–306. [Google Scholar] [CrossRef] - Zhang, Y.; Chen, F.; Zhang, Y.; Zhang, Y.; Du, X. Experimental investigations of interactions between a laser-induced cavitation bubble and a spherical particle. Exp. Fluid Sci.
**2018**, 98, 645–661. [Google Scholar] [CrossRef] - Zhang, Y.; Xie, X.; Zhang, Y.; Zhang, Y.X. High-speed experimental photography of collapsing cavitation bubble between a spherical particle and a rigid wall. J. Hydrodyn.
**2018**, 30, 1012–1021. [Google Scholar] [CrossRef] - Gonzalez-Avila, S.R.; Klaseboer, E.; Khoo, B.C.; Ohl, C.D. Cavitation bubble dynamics in a liquid gap of variable height. J. Fluid Mech.
**2011**, 682, 241–260. [Google Scholar] [CrossRef] - Brujan, E.A.; Takahira, H.; Ogasawara, T. Planar jets in collapsing cavitation bubbles. Exp. Fluid Sci.
**2019**, 101, 48–61. [Google Scholar] [CrossRef] - Li, T.; Zhang, A.M.; Wang, S.P.; Li, S.; Liu, W.T. Bubble interactions and bursting behaviors near a free surface. Phys. Fluids
**2019**, 31, 042104. [Google Scholar] [CrossRef] - Pearson, A.; Cox, E.; Blake, J.R.; Otto, S.R. Bubble interactions near a free surface. Eng. Anal. Bound. Elem.
**2004**, 28, 295–313. [Google Scholar] [CrossRef] - Brujan, E. Cavitation in Non-Newtonian Fluids: With Biomedical and Bioengineering Applications; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Brujan, E.A.; Nahen, K.; Schmidt, P.; Vogel, A. Dynamics of laser-induced cavitation bubbles near an elastic boundary. J. Fluid Mech.
**2001**, 433, 251–281. [Google Scholar] [CrossRef] - Hopfes, T.; Wang, Z.; Giglmaier, M.; Adams, N.A. Collapse dynamics of bubble pairs in gelatinous fluids. Exp. Fluid Sci.
**2019**, 108, 104–114. [Google Scholar] [CrossRef] - Wu, J.H.; Wang, Y.; Ma, F.; Gou, W.J. Cavitation erosion in bloods. J. Hydrodyn.
**2017**, 29, 724–727. [Google Scholar] [CrossRef] - Wan, C.R.; Liu, H. Shedding frequency of sheet cavitation around axisymmetric body at small angles of attack. J. Hydrodyn.
**2017**, 29, 520–523. [Google Scholar] [CrossRef] - Blake, J.R.; Gibson, D.C. Cavitation bubbles near boundaries. Annu. Rev. Fluid Mech.
**1987**, 19, 99–123. [Google Scholar] [CrossRef] - Lauterborn, W.; Kurz, T. Physics of bubble oscillations. Rep. Prog. Phys.
**2010**, 73, 106501. [Google Scholar] [CrossRef] - Brennen, C.E. Cavitation and Bubble Dynamics; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Wang, S.P.; Zhang, A.M.; Liu, Y.L.; Zhang, S.; Cui, P. Bubble dynamics and its applications. J. Hydrodyn.
**2018**, 30, 975–991. [Google Scholar] [CrossRef] - Best, J.P. The formation of toroidal bubbles upon the collapse of transient cavities. J. Fluid Mech.
**1993**, 251, 79–107. [Google Scholar] [CrossRef] - Lee, M.; Klaseboer, E.; Khoo, B.C. On the boundary integral method for the rebounding bubble. J. Fluid Mech.
**2007**, 570, 407–429. [Google Scholar] [CrossRef] - Wang, Q. Multi-oscillations of a bubble in a compressible liquid near a rigid boundary. J. Fluid Mech.
**2014**, 745, 509–536. [Google Scholar] [CrossRef] - Wang, Q.; Liu, W.; Zhang, A.M.; Sui, Y. Bubble dynamics in a compressible liquid in contact with a rigid boundary. Interface Focus
**2015**, 5, 20150048. [Google Scholar] [CrossRef] [PubMed] - Han, R.; Zhang, A.M.; Li, S.; Zong, Z. Experimental and numerical study of the effects of a wall on the coalescence and collapse of bubble pairs. Phys. Fluids
**2018**, 30, 042107. [Google Scholar] [CrossRef] - Koch, M.; Lechner, C.; Reuter, F.; Köhler, K.; Mettin, R.; Lauterborn, W. Numerical modeling of laser generated cavitation bubbles with the finite volume and volume of fluid method, using OpenFOAM. Comput. Fluids
**2016**, 126, 71–90. [Google Scholar] [CrossRef] - Lechner, C.; Lauterborn, W.; Koch, M.; Mettin, R. Fast, thin jets from bubbles expanding and collapsing in extreme vicinity to a solid boundary: A numerical study. Phys. Rev. Fluids
**2019**, 4, 021601. [Google Scholar] [CrossRef] - Thoroddsen, S.T.; Etoh, T.G.; Takehara, K. High-speed imaging of drops and bubbles. Annu. Rev. Fluid Mech.
**2008**, 40, 257–285. [Google Scholar] [CrossRef] - Wang, Q.; Liu, W.; Leppinen, D.M.; Walmsley, A.D. Microbubble dynamics in a viscous compressible liquid near a rigid boundary. IMA J. Appl. Math.
**2019**, 84, 696–711. [Google Scholar] [CrossRef] - Klapcsik, K.; Hegedűs, F. Study of non-spherical bubble oscillations under acoustic irradiation in viscous liquid. Ultrason. Sonochem.
**2019**, 54, 256–273. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Qiu, X.; Chen, F.; Liu, K.; Zhang, Y.; Liu, C. A selected review of vortex identification methods with applications. J. Hydrodyn.
**2018**, 30, 767–779. [Google Scholar] [CrossRef] - Dong, X.; Wang, Y.; Chen, X.; Dong, Y.; Zhang, Y.; Liu, C. Determination of epsilon for Omega vortex identification method. J. Hydrodyn.
**2018**, 30, 541–548. [Google Scholar] [CrossRef]

**Figure 1.**The definitions of the paramount control parameters of the bubble–wall interaction system. In the figure, L is the distance between the inception point of the cavitation bubble and the upper surface of the wall; R

_{max}is the equivalent maximum radius of the bubble during its whole oscillations. The red arrow represents the direction of gravity.

**Figure 2.**Representative high-speed photos of the dynamic process of the cavitation bubble near the wall from the bubble inception to the rebound. fps = 240,000, λ = 1.00.

**Figure 3.**The obtained experimental high-speed photos of the dynamic process of the cavitation bubble with a short distance to the wall. The time interval between two adjacent subplots is 4.17 µs. The two vertical dashed yellow and blue lines correspond to the bubble positions during the maximum volume, respectively. fps = 240,000, λ = 1.25.

**Figure 4.**The obtained experimental high-speed photos of the dynamic process of the cavitation bubble with a medium distance to the wall. The time interval between two adjacent subplots is 4.17 µs. The two vertical dashed yellow and blue lines correspond to the bubble positions during the maximum volume, respectively. fps = 240,000, λ = 2.50.

**Figure 5.**The obtained experimental high-speed photos of the dynamic process of the cavitation bubble with a large distance to the wall. The time interval between two adjacent subplots is 4.17 µs. The two vertical dashed yellow and blue lines correspond to the bubble positions during the maximum volume respectively. fps = 240,000, λ = 3.75.

**Figure 6.**The velocities of the poles A and B of the cavitation bubble interface during the collapsing stage with a short bubble–wall distance (λ = 0.80). The solid black line represents the velocity of pole A (${V}_{A}$) and the dotted-dashed red line represents the velocity of pole B (${V}_{B}$).

**Figure 7.**The velocities of the poles A and B of the cavitation bubble interface during the collapsing stage with a medium bubble–wall distance (λ = 1.59). The solid black line represents the velocity of pole A (${V}_{A}$) and the dotted-dashed red line represents the velocity of pole B (${V}_{B}$).

**Figure 8.**The velocities of the poles A and B of the cavitation bubble interface during the collapsing stage with a large bubble–wall distance (λ = 2.39). The solid black line represents the velocity of pole A (${V}_{A}$) and the dotted-dashed red line represents the velocity of pole B (${V}_{B}$).

**Figure 9.**The motions of the bubble interface and the movement of the bubble centroid during the collapsing stage with λ = 0.80. The four dashed lines with different colors represent the four representative interfaces of the cavitation bubble during its collapsing stage (τ = 0.00–1.00). The four symbols with different colors and shapes represent the calculated centroids of the cavitation bubbles for the given non-dimensional time. The black line represents the position of the wall.

**Figure 10.**The motions of the bubble interface and the movement of the bubble centroid during the collapsing stage with λ = 1.59. The four dashed lines with different colors represent the four representative interfaces of the cavitation bubble during its collapsing stage (τ = 0.00–1.00). The four symbols with different colors and shapes represent the calculated centroids of the cavitation bubbles for the given non-dimensional time. The black line represents the position of the wall.

**Figure 11.**The motions of the bubble interface and the movement of the bubble centroid during the collapsing stage with λ = 2.39. The four dashed lines with different colors represent the four representative interfaces of the cavitation bubble during its collapsing stage (τ = 0.00–1.00). The four symbols with different colors and shapes represent the calculated centroids of the cavitation bubbles for the given non-dimensional time. The black line represents the position of the wall.

**Figure 12.**The variations of the moving distance ($\Delta D$) of the bubble centroids with non-dimensional time (τ) during the collapsing stage.

**Table 1.**Comparisons of the directions and the magnitude of the velocities of the bubble interface poles (${V}_{A}$ and ${V}_{B}$) during the bubble collapsing stage in the present experiments.

λ | $\mathbf{Direction}\text{}\mathbf{of}\text{}{\mathit{V}}_{\mathit{A}}$ | $\mathbf{Direction}\text{}\mathbf{of}\text{}{\mathit{V}}_{\mathit{B}}$ | $\Delta \mathit{V}$ |
---|---|---|---|

Small | No Move | ← | Large |

Medium | → | ← | Medium |

Large | → | ← | Small |

Parameters | Case 1 | Case 2 | Case 3 | |||
---|---|---|---|---|---|---|

Point A | Point B | Point A | Point B | Point A | Point B | |

a | 0.00 | 108.98 ± 18.63 | −23.63 ± 3.36 | 71.81 ± 5.14 | −32.40 ± 2.96 | 92.72 ± 7.60 |

b | 0.00 | 3.81 ± 0.50 | 6.82 ± 1.29 | 5.53 ± 0.54 | 5.03 ± 0.65 | 8.88 ± 0.91 |

Adjusted ${R}^{2}$ | --- | 0.90 | 0.76 | 0.92 | 0.88 | 0.92 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, X.; Duan, Y.; Zhang, Y.; Tang, N.; Zhang, Y.
Retardant Effects of Collapsing Dynamics of a Laser-Induced Cavitation Bubble Near a Solid Wall. *Symmetry* **2019**, *11*, 1051.
https://doi.org/10.3390/sym11081051

**AMA Style**

Li X, Duan Y, Zhang Y, Tang N, Zhang Y.
Retardant Effects of Collapsing Dynamics of a Laser-Induced Cavitation Bubble Near a Solid Wall. *Symmetry*. 2019; 11(8):1051.
https://doi.org/10.3390/sym11081051

**Chicago/Turabian Style**

Li, Xiaofei, Yaxin Duan, Yuning Zhang, Ningning Tang, and Yuning Zhang.
2019. "Retardant Effects of Collapsing Dynamics of a Laser-Induced Cavitation Bubble Near a Solid Wall" *Symmetry* 11, no. 8: 1051.
https://doi.org/10.3390/sym11081051