# Numerical Study of the Interaction between a Collapsing Bubble and a Movable Particle in a Free Field

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Modeling

## 3. Validation

#### 3.1. Vapor–Liquid Two-Phase Flow

#### 3.1.1. The Collapse of a Vapor Bubble in a Free Flow Field

#### 3.1.2. The Collapse of a Vapor Bubble near a Rigid Wall

#### 3.2. Assessment of the Overset Grid Method

## 4. Results and Discussion

## 5. Conclusions

- The collapsing forms of cavitation bubbles in cases with stand-off parameter $\chi $ > 1 experience spherical-shaped collapse under the influence of the approaching particle, which is attracted by the collapsing bubble. These bubbles no longer collapse towards particles, which happens in cases with stationary particles.
- As for the dynamics of particles induced by cavitation bubbles, an analytical model based on force balance shows that the velocity of particle inversely depends on the size and density of the particle, and approximately on the second power of the initial distance from the bubble. It is an extension of Poulain et al.’s theory in low Reynolds number range.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic showing the configuration of a collapsing bubble and a spherical particle; (

**b**) 2D illustration of an overset mesh system: the background mesh (blue) and overset mesh (red).

**Figure 2.**(

**a**) Illustration of computational domain; (

**b**) 2D illustration of the local domain containing a bubble with grid refinement; (

**c**) comparisons of the temporal evolution of bubble radius between analytical solution and numerical results with different cells in the bubble; (

**d**) evolution of L2 error of bubble radius versus time.

**Figure 3.**(

**a**) Schematic showing the configuration of a bubble near a wall; (

**b**) collapse time for the cases in Table 1.

**Figure 4.**Variations of the profile of a collapsing bubble for the case $\gamma =1.10$: (

**a**) results of the experiment [35]; (

**b**) results of the simulation.

**Figure 5.**The comparisons of profiles of collapsing bubbles under the influence of static particles from experimental (left) and numerical (right) results.

**Figure 6.**The comparisons of profiles of collapsing bubbles under the influence of static particles from experimental (left) and numerical (right) results.

**Figure 7.**The comparisons of profiles of collapsing bubbles under the influence of static particles from experimental (left) and numerical (right) results.

**Figure 8.**The patterns of streamline for case 8 in Table 2 with static particles (

**a**–

**e**) and movable particles (

**f**–

**j**) respectively.

**Figure 9.**Displacement of particles due to collapsing bubbles for cases with different initial distance between their centers ${d}_{0}$ (

**a**), for cases with different particle radius ${R}_{p}$ (

**b**) and for cases with different initial bubble radius ${R}_{0}$ (

**c**) in Table 2; the constant velocity is given for each case.

Case No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

$\gamma $ | 1.5 | 1.65 | 1.85 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 12.5 |

L (mm) | 3.0 | 3.3 | 3.7 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 25.0 |

**Table 2.**Information of parameters in the initial flow field shown in Figure 1 under each case.

Case No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{0}$ (mm) | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.5 | 3.0 | 4.0 | 5.0 |

${\mathit{R}}_{p}$ (mm) | 3.0 | 3.0 | 3.0 | 3.0 | 1.0 | 1.5 | 3.0 | 5.0 | 3.0 | 3.0 | 3.0 |

${\mathit{d}}_{0}$ (mm) | 13.6 | 9.3 | 7.2 | 3.8 | 4.0 | 5.0 | 7.0 | 9.0 | 8.0 | 8.0 | 8.0 |

$\chi $ | 3.03 | 1.80 | 1.20 | 0.23 | 0.86 | 1.00 | 1.14 | 1.14 | 1.70 | 1.25 | 1.00 |

**Table 3.**The profiles of collapsing bubbles under the influence of movable particles in simulations of cases in Table 2.

Case No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

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**MDPI and ACS Style**

Zheng, Y.; Chen, L.; Liang, X.; Duan, H. Numerical Study of the Interaction between a Collapsing Bubble and a Movable Particle in a Free Field. *Water* **2020**, *12*, 3331.
https://doi.org/10.3390/w12123331

**AMA Style**

Zheng Y, Chen L, Liang X, Duan H. Numerical Study of the Interaction between a Collapsing Bubble and a Movable Particle in a Free Field. *Water*. 2020; 12(12):3331.
https://doi.org/10.3390/w12123331

**Chicago/Turabian Style**

Zheng, Yuxin, Linya Chen, Xiaoyu Liang, and Hangbo Duan. 2020. "Numerical Study of the Interaction between a Collapsing Bubble and a Movable Particle in a Free Field" *Water* 12, no. 12: 3331.
https://doi.org/10.3390/w12123331