# Study on the Dynamic Characteristics of Single Cavitation Bubble Motion near the Wall Based on the Keller–Miksis Model

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## Abstract

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## 1. Introduction

## 2. Theoretical Background and Numerical Methods

#### 2.1. Cavitation Bubble Dynamics Model

_{G}

_{0}represents the partial pressure of the non-condensable gas, with gas properties exhibiting exponential variation. The effects of thermal effects are neglected, and there is no mass transfer between the gas and liquid phases. ${\upsilon}_{L}$ denotes the dynamic viscosity coefficient of the liquid, $S$ stands for surface tension, and $\kappa $ represents the polytropic index of the gas. When c approaches infinity, the weakly compressible Keller–Miksis model (1) can be simplified to the incompressible Rayleigh–Plesset model.

#### 2.2. Translational Motion Equations of Near-Wall Cavitation Bubble

_{i}represents the center of the primary bubble, while O

_{j}represents the center of the mirrored bubble. In the subsequent formula derivations, i and j, respectively, denote the primary cavitation bubble and the mirrored cavitation bubble, with the outward normal direction of the wall defined as the positive direction.

_{i}represents the added mass matrix, and ${\overrightarrow{v}}_{i}$ denotes the translational velocity of the cavitation bubble. The added mass of the bubble near the wall is calculated using the following formula [15]:

_{i}= 0.5, the additional mass is equal to half of the displaced fluid mass. Now, substituting Equations (5) and (6) into Equation (4), we obtain:

#### 2.3. Numerical Solution Methods

_{1}, k

_{2}, k

_{3}, k

_{4}are the function values at different points. The Runge–Kutta method is a commonly used and highly effective technique. Its advantages include its simplicity in programming, requiring only the initial values to progressively extrapolate with a fixed or variable step size, and its exhibiting of good numerical stability [20].

## 3. Model Reliability Validation

## 4. Results

_{0}is the distance from the bubble center to the wall, and R

_{0}is the initial radius of the bubble. To exclusively study the effect of the wall on the bubbles, it is assumed that all bubbles begin to collapse from a maximum initial radius of 1 mm. Six sets of data are selected for analysis, with γ values of 1.3, 1.5, 1.8, 2.5, 3.0, and 5.0, respectively.

#### 4.1. Analysis of the Velocity Field of Bubbles in the Near-Wall

_{i}represents the effective radius of the bubble, ${\dot{R}}_{i}$ is the velocity on the bubble wall, r denotes the position of each point in the flow field relative to the center of the bubble, and ${\overrightarrow{v}}_{i}$ and ${\overrightarrow{u}}_{i}$ represents the velocity components at each point. Figure 6 depicts the velocity field distribution near the bubble in the near-wall region with γ = 1.5, obtained based on Equation (15).

#### 4.2. Analysis of Bubble Dynamics Characteristics in the Near-Wall

#### 4.3. Analysis of Bubble Acoustic Pressure Radiated in Near a Wall

_{0}is the distance from the bubble center to the measurement point. The radiated sound pressure of the bubble is directly proportional to the second derivative of the varying volume and inversely proportional to the distance. Expanding the second derivative of the varying volume, the value of ${p}_{a}$ is directly related to the radial oscillation of the bubble, which, in turn, is related to the distance from the wall. Therefore, there exists a corresponding functional relationship between the distance from the bubble to the wall.

## 5. Conclusions

- (1)
- Under near-wall conditions, as the dimensionless distance γ between the bubble and the wall decreases, the time for the bubble to collapse increases, and the minimum radius gradually decreases in a linear fashion.
- (2)
- The movement of the bubble’s center of mass varies significantly with different values of γ. During the initial stages of collapse, the bubble’s center of mass remains nearly unchanged. However, towards the end of collapse, the bubble’s center of mass rapidly moves towards the wall, with a larger displacement observed for smaller values of γ.
- (3)
- The peak velocity of bubble translation decreases with increasing γ values, and the peak velocity aligns with the time when the bubble collapses to its minimum radius. For conditions γ = 1.2 and γ = 1.5, the bubble exhibits a phenomenon of moving away from the wall during the rebound phase, with maximum velocities of 22.27 m/s and 7.79 m/s, respectively.
- (4)
- As γ decreases, the secondary Bjerknes force during the initial stage of bubble collapse gradually increases in the form of attraction. Subsequently, during the later stages of collapse and rebound, the secondary Bjerknes force alternates between attraction and repulsion. However, averaging over the entire computational period reveals that bubbles in the near-wall region are primarily subject to attraction, causing them to move towards the wall.
- (5)
- In analyzing the radiated sound pressure at the wall, it was observed that the radiated sound pressure of the bubble decreases in an inverse function trend with the increase in the dimensionless parameter γ. In the case where γ equals 1.3, the radiated sound pressure at the wall is 33.73 times the ambient pressure. Moreover, this paper introduces a radiated sound pressure coefficient to characterize the radial vibration behavior of the bubble. It was found that the distance to the wall has a minor influence on the radiated sound pressure coefficient, providing a basis for future research on bubbles of different scales.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Experimental Setup. (1—High-speed camera (Revealer) with frame rate up to 1 million fps, 2—Water tank (250 mm × 150 mm × 250 mm) and electric discharge device, 3—LED light source, 4—Adjustable DC power supply, 5—Charging and discharging circuit, 6—Computer).

Parameter | Numerical Value |
---|---|

Liquid Density ρ/(kg/m³) | 1000 |

Surface Tension Coefficient S/(N/m) | 0.072 |

Dynamic Viscosity Coefficient ${\upsilon}_{L}$/Pa·s | 1.02 × 10^{−3} |

Gas Polytropic Index $\kappa $ | 5/3 |

Speed of Sound c/(m/s) | 1450 |

Ambient Pressure ${p}_{\infty}$/Pa | 1 × 10^{5} |

Initial Bubble Radius R_{0}/mm | 1 |

Gas Partial Pressure ${p}_{G0}$/Pa | 3000 |

Initial Bubble Velocity $U$/(m/s) | 0 |

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**MDPI and ACS Style**

Han, W.; Gu, Z.; Li, R.; Mi, J.; Bai, L.; Deng, W.
Study on the Dynamic Characteristics of Single Cavitation Bubble Motion near the Wall Based on the Keller–Miksis Model. *Processes* **2024**, *12*, 826.
https://doi.org/10.3390/pr12040826

**AMA Style**

Han W, Gu Z, Li R, Mi J, Bai L, Deng W.
Study on the Dynamic Characteristics of Single Cavitation Bubble Motion near the Wall Based on the Keller–Miksis Model. *Processes*. 2024; 12(4):826.
https://doi.org/10.3390/pr12040826

**Chicago/Turabian Style**

Han, Wei, Zhenye Gu, Rennian Li, Jiandong Mi, Lu Bai, and Wanquan Deng.
2024. "Study on the Dynamic Characteristics of Single Cavitation Bubble Motion near the Wall Based on the Keller–Miksis Model" *Processes* 12, no. 4: 826.
https://doi.org/10.3390/pr12040826