# Study of Cavitation Bubble Collapse near a Wall by the Modified Lattice Boltzmann Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Principle of the LBM

## 3. Physical Model

_{0}is the radius of the cavitation bubble, b is the distance from the centre of the cavitation bubble, P

_{v}is the pressure in the cavitation bubble, and P

_{∞}is the pressure outside the cavitation bubble. The left and right boundaries are infinite areas, the upper boundary is the pressure inlet, and the lower boundary is the rigid wall. (b) is the physical model for the study of the evolution of the double cavitation bubble in the near-wall region. b

_{1}is the distance from the centre of the left cavitation bubble to the rigid wall, b

_{2}is the distance from the centre of the right cavitation bubble to the rigid wall, and b

_{3}is the horizontal distance of the centre of the two cavitation bubbles. The other settings are the same as those of the physical model of a single cavitation bubble in the study of the collapse of the near-wall region.

## 4. Simulation Content and Parameter Initialization Settings

_{c}= 0.689 is used to simulate and the equilibrium pressure p = 0.0028 $\mathrm{mu}\xb7{\mathrm{lu}}^{-1}{\mathrm{ts}}^{-2}$ is gotten by the equal area rule is. In the C-S state of equation, a = 1, b = 4 and R = 1 [34] are adopted. The initial temperature is set to a specific temperature, the velocity is zero, and the density field is initialized as follows:

_{1}and y

_{1}is the location of the middle of the bubble at the initial moment, the hyperbolic tangent function $tanh=\left({e}^{x}-{e}^{-x}\right)/\left({e}^{x}+{e}^{-x}\right)$ and the phase interface width is W = 4. Since the cavitation bubble radius is approximately ten times the width of the phase interface, it can obtain better numerical stability of the model, but it is essential to ensure that the bubble collapse is not affected by other boundaries and that the calculation cost is reduced. Therefore, our simulation calculation area is 401 × 401.

_{l}are the physical quantities that characterize the equilibrium gas pressure and the equilibrium liquid pressure, respectively. In addition, the coexistence densities ${\rho}_{v}$ and ${\rho}_{l}$ of gas and liquid, respectively, can also be determined by phase separation simulation with a slight random disturbance of the initial density. In the calculation process, ${\rho}_{l}$ need to be slightly adjusted to ensure ${\rho}_{l}$ has the same density as the pressure boundary so that an additional pressure difference between the inside and the outside of the bubble is obtained after the fluid balance in the entire calculation domain.

## 5. Study of the Evolution of a Single Cavitation Bubble

_{0}was introduced, which is the amount that characterizes the distance from the centre of the bubble to the wall. The figures below show cases with λ = 1.6, ${R}_{0}$ = 80 lu and λ = 2.5, ${R}_{0}$ = 80 lu. The cavitation bubble is initially a circle. The bubble size and the thickness of the gas‒liquid boundary layer are controlled by Equation (11). Since the pressure difference exists inside and outside the bubble, the bubble is deformed by extrusion. Due to the influence of the bottom rigid sidewall, the longitudinal flow is blocked, and a negative pressure forms under the bubble to induce longitudinal expansion of the bubble. Due to the shrinkage and deformation of the bubbles, the volume is decreasing, and the surrounding liquid fills the space created by the bubbles, resulting in a decrease in the density and pressure around them. Then, the pressure of the upper pressure boundary is first transmitted to the upper surface of the bubble, and a high-pressure zone is formed in the upper part of the bubble that acts together with the low-pressure zone of the cavitation bubble to form a depression at the upper portion (t = 470). Due to the rebound effect of the liquid and the relatively high speed of movement of the upper portion of the bubble, a relatively large conical high-pressure region is formed in the upper portion, which is crucial in the subsequent deformation. Over time, the depression continues to expand (t = 530), and with the influence of the surrounding high pressure, the bubble gradually shrinks, assuming a crescent shape (t = 570). When the sag causes the upper surface of the bubble to touch the lower surface, a large pressure difference directly breaks down the empty bubble, forming a micro shock wave, which has a destructive effect on the wall surface (t = 588). At the same time, a complex sound field is generated, which causes additional damage to the rigid wall. By comparison, the morphology of cavitation bubble collapse differs for different dimensionless parameters λ and ΔP is confirmed. In our simulation, a crescent-shaped bubble is formed at λ = 1.6 and is then broken down to form two bubbles, which generates a micro-shock; however, at λ = 2.5, the bubble is squashed directly. A crescent-shaped bubble is formed, but the bubble does not break in the middle and finally collapses in the form of a small bubble. The calculation results are consistent with the results of Philipp [35]. When the bubble is too far from the wall surface, the sidewall has a small retarding effect on the bubble, and a strong negative pressure is unlikely to form under the bubble, so when the bubble collapses, the bubbles are gradually crushed and collapsed, and the impact of the formed pressure on the wall surface is also alleviated.

## 6. Study of the Evolution of a Double Cavitation Bubble

_{6}direction. Similar to the previous simulation, the dimensionless quantities λ

_{1}, λ

_{2}, and λ

_{3}is introduced, where λ

_{1}= b

_{1}/R

_{0}, which is the amount that characterizes the distance of the LB centre from the wall; λ

_{2}= b

_{2}/R

_{0}, which is the amount that characterizes the distance of the LB centre from the wall surface; and λ

_{3}= b

_{3}/R

_{0}, which is the amount that represents the horizontal distance between the LB and RB centres.

#### 6.1. Case 1: Numerical Simulation of Tilt Distribution Cavitation with Two Cavitation Bubbles

_{1}= 1.2, λ

_{2}= 2.7, and λ

_{3}= 1.5, the line connecting the centres of the two cavitation bubbles is at an angle of 45° to the wall surface. The density field and the pressure field coupling with the velocity field of the cavitation bubble collapse process are shown in Figure 7 and Figure 8, respectively.

_{5}direction. The e

_{6}direction of the LB and the e

_{8}direction of the RB are attracted to each other, and the opposite direction is deformed. If the gas density and the liquid density are in equilibrium at this time, the two cavitation bubbles would attract each other and eventually merge into a single bubble. However, since the liquid has been pressurized during the simulated initialization, the additional generated pressure prevents the emergence of the two bubbles. As the simulation progresses, the deformation is further aggravated, and the bubbles decrease under the action of additional pressure. Between them, the change of LB is the most easily detected. When the time reaches t = 350 ts, the LB is elongated and deformed due to the low pressure resulting from the blockage of the wall surface; the e

_{6}direction of RB appears the same as the change of a single cavitation bubble in the near-wall region, and a depression occurs. The change becomes more apparent at 400 ts. Around the shrinking bubble, the liquid fills the newly available space, and a slightly lower pressure appears around the cavitation bubble. The transition can be observed in the velocity field. Around the bubble, the velocity direction tends to support cavitation. Note that in the figure of the pressure field and the velocity field, for the sake of convenience, only the velocity of the liquid is shown and we do not plot the velocity inside the bubble. The LB exhibits an elliptical shape that exhibits an inclination in the e

_{6}–e

_{8}direction under the action of the sidewall, the attraction of RB and the external extreme pressure. The RB is also elliptical due to the attraction of the two bubbles, but the deformation is not as strong as with the bubble on the left. At this time, relative with the collapse rule of a single cavitation bubble in the near-wall region, LB is equivalent to rigidity avoidance for RB, so a low-pressure zone is generated between the two bubbles, and high pressure is generated by the upper pressure boundary. A pressure difference is generated in the e

_{6}–e

_{8}direction of the RB caused the deformation to continue to produce. Inspection of the velocity field reveals that the maximum velocity occurs in the same direction of the RB. At t = 450 ts, the cavitation bubble is crescent-shaped. The bubble is strongly compressed from the upper part of the depression until the entire bubble is completely collapsed from the upper part of the bubble, which occurs because the flow field is not completely symmetrical on both sides of the e

_{6}–e

_{8}direction. The upper part is first crushed by the pressure transferred from the pressure boundary. The other reason is that the most attractive part of LB is located at the nearest position of the two bubbles. The velocity field indicates that the maximum velocity direction has been deflected, and RB begins to collapse from top to bottom under the pressure difference, producing a huge jet and complex sound field. The huge pressure generated by the collapse acts in conjunction with other factors to promote the continued collapse of LB. At t = 550 ts, the pressure field and velocity field images indicate that the flow field changes due to the collapse pressure of RB, and the micro-jet generated by RB in the e

_{6}direction collides with the downward flowing liquid to generate a high voltage and noise, and energy begins to be consumed; furthermore, the micro-jet in the e

_{7}direction overlaps with the original flow field, and the flow state becomes complex. As the simulation proceeds, the flow field exhibits vortices, and LB begins to collapse. First, the cavitation bubble e

_{6}direction begins to shrink under the influence of the pressure of RB, which causes a protrusion in the upper portion of the cavitation bubble. At this time, the energy generated by the RB collapse is insufficient to continue to compress LB. Under the joint action of the upper pressure boundary pressure and the high pressure generated by the RB collapse, a high-pressure region (t = 550 ts) is formed at the upper convex portion, and collapse from the upper portion of the cavitation bubble is induced. As RB is destroyed, the e

_{9}direction finally collapses, and a jet from the e

_{7}to the e

_{9}direction is generated in the flow field, so that the entire flow field exhibits vortices. Finally, LB collapses under the combined action of various factors, and the generated pressure impacts the wall surface to avoid impact. The pressure field diagram of t = 700 ts indicates that a high pressure is generated on the wall surface. Note that after RB collapse, LB collapse occurs under the influence of various factors, and the mechanism of action is relatively complex. The LB high-pressure zone rotates around the cavitation bubble. This visualization is a powerful way to illustrate the complex vortices and other phenomena in the flow field, involving other physical quantities. The parameters of this study cannot describe the more detailed characteristics. The specific mechanism needs further study.

#### 6.2. Case 2: Numerical Simulation of Two Parallel Cavitation Bubbles

_{1}= 1.2, λ

_{2}= 1.2, and λ

_{3}= 2.4, that is, the line connecting the centres of the two cavitation bubbles is parallel to the wall surface, and the density field of the cavitation bubble collapse process is shown in Figure 9 as well as the pressure field and the velocity field shown in Figure 10.

## 7. Maximum Wall Pressure

_{3}= 2.4 is used and, likewise, the value of λ (from 1.05 to 2.0) with a series of gradients was used to simulate the maximum wall pressure resulting from the collapse of two cavitation bubbles under different initial conditions, where λ = λ

_{1}= λ

_{2}. Since the cavitation bubble collapse process is complicated, the maximum wall pressure is not generated at the same position for the collapse process under different initial conditions. The wall pressure below the cavitation bubble (${\mathrm{e}}_{5}$ direction) and the pressure change zone is recorded to find the maximum value The comparison of the maximum wall pressure under different initial conditions is shown in Figure 11.

- The maximum wall pressure generated by the collapse of a single cavitation bubble in the near-wall region decreases with increasing distance from the wall surface, and it first drops sharply, then slowly decreases and finally becomes relatively stable. This is because the closer to the wall, the smaller the thickness of the fluid that the micro shock wave generated by cavitation passes to the wall and the smaller the blockage effect of the flow field, the greater the wall pressure generated.
- It is complex for the case where a double parallel cavitation bubble collapses in the near-wall region because the maximum wall pressure produces a different position under different initial conditions. When λ = 1.05–1.10, the wall below the cavitation bubble is subjected to the maximum wall pressure, because the cavitation bubble is closer to the wall surface, and the generated micro-shock is transmitted to the wall surface almost unimpeded and the cavitation bubble of the pressure change region is not fully developed, which is not enough to generate create a huge pressure with the lifting force to bubble. When λ = 1.15–1.25, the maximum wall pressure occurs in the pressure change zone where new cavitation bubbles are generated and collapse rapidly, resulting in a large wall pressure exceeding the maximum wall pressure at other locations. When λ = 1.40–2.0, the area where the maximum wall pressure is generated is the pressure change zone, but the pressure generation at this time is not when the new cavitation bubble collapses but the pressure after the collapse of the two cavitation bubbles overlaps each other on the wall. The change in wall pressure depends mainly on the size of the new cavitation bubble induced and the pressure generated by the collapse and the lifting force of the bubbles.
- When λ < 1.7, the wall pressure generated by the collapse of a single cavitation bubble is relatively large. This is because, for the case where the double cavitation bubble collapses in the near-wall region, the pressure generated by the collapse of the new cavitation bubble induced in the pressure change region has an effect of lifting force on the cavitation bubble. Thereby, the bottom pressure of the cavitation bubble collapse is increased, and the blockage effect of the wall surface is weakened, which is equivalent to an increase of λ, and the resulting wall pressure is relatively small. When λ > 1.7, the wall pressure of the pressure change zone is formed by the superposition of pressure generated by the collapse of two cavitation bubbles, so the generated wall pressure is greater than the pressure generated by the collapse of a single cavitation bubble.

## 8. Conclusions

- For the case where a single cavitation bubble collapses in the near-wall region, when λ = 1.6, a crescent-shaped bubble is formed that is broken down to form two bubbles, and a micro-shock is generated; when λ = 2.5, the bubbles are crushed to form crescent-shaped bubbles, but the bubbles do not break in the middle but rather ultimately collapse in the form of a small bubble. The shape of the cavitation bubble is related to the distance of the cavitation bubble from the rigid wall.
- For the numerical simulation of tilted distribution cavitation with two cavitation bubbles, in the early stage of simulation, the collapse behaviour is similar to that of the single-bubble case. Subsequently, the e
_{6}direction of RB has a concave deformation, which is very important in the collapse of the two bubbles. The velocity field indicates that the maximum velocity appears in the depression of RB. The tremendous pressure generated by this velocity directly penetrates the bubble; thereafter, the bubble is crescent-shaped until collapsing completely. Due to the asymmetry of the collapse, the flow field becomes complex after RB collapses and particularly so after LB collapses, and vortices appear in the flow field. - For the numerical simulation of parallel distribution cavitation of two bubbles, the bubble collapses under the blocking effect of the rigid wall and the attraction of the two bubbles, and the two bubbles collapse simultaneously. Therefore, the mutual influence during the collapse process is smaller than that of Case 1; in addition, the erosion effect of the bubble collapse on the wall surface is the result of superimposing the pressure fields formed by the collapse of the two bubbles. However, for each bubble, since the collapse does not occur instantaneously but is collapsed from the upper part of the closest position of the two cavitation bubbles, a complex vortex phenomenon occurs in the flow field.
- By comparing the maximum wall pressure generated by cavitation under different initial conditions, the factors affecting the maximum wall pressure are obtained. For a single cavitation bubble, the distance from the wall is the most important factor. For two cavitation bubbles, the lifting effect of the new induced cavitation bubble collapse is the most important factor.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

${\mathit{f}}_{\mathit{i}}$ | Single Particle Density Distribution Function |
---|---|

${f}_{i}^{eq}$ | equilibrium particle distribution function |

$\tau $ | relaxation time |

$\nu $ | kinematic viscosity |

$c$ | grid velocity |

$\Delta x$ | grid step |

$\Delta t$ | time step |

${c}_{s}$ | lattice sound velocity |

${\omega}_{i}$ | weighting factor |

u | fluid velocity |

${F}_{i}$ | interaction force |

$\psi $ | interaction potential |

${T}_{c}$ | critical temperature |

${P}_{c}$ | critical pressure |

F | total interaction force |

P_{v} | pressure of the cavitation bubble |

P_{∞} | the pressure outside the cavitation bubble |

R_{0} | radius of the cavitation bubble |

b (${b}_{1},{b}_{2}$,${b}_{3}$) | distance between corresponding points |

T | lattice temperature |

a (in Equation (11)) | parameter of the C-S state of equation |

b (in Equation (11)) | parameter of the C-S state of equation |

R (in Equation (11)) | parameter of the C-S state of equation |

${V}_{m,g}$ | the equilibrium gas pressure |

${V}_{m,l}$ | the equilibrium liquid pressure |

λ(${\lambda}_{1},{\lambda}_{2},{\lambda}_{3}$) | dimensionless value that characterizes the distance |

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**Figure 2.**Physical model. (

**a**) (single bubble model); (

**b**) (double bubbles model) (R

_{0}—bubble initial radius; b

_{i}—distance; P

_{v}—vapour pressure in bubble; P

_{∞}—ambient pressure).

**Figure 10.**Pressure field and velocity field of cavitation bubble collapse (800 ts is the pressure field and streamline diagram).

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**MDPI and ACS Style**

Mao, Y.; Peng, Y.; Zhang, J.
Study of Cavitation Bubble Collapse near a Wall by the Modified Lattice Boltzmann Method. *Water* **2018**, *10*, 1439.
https://doi.org/10.3390/w10101439

**AMA Style**

Mao Y, Peng Y, Zhang J.
Study of Cavitation Bubble Collapse near a Wall by the Modified Lattice Boltzmann Method. *Water*. 2018; 10(10):1439.
https://doi.org/10.3390/w10101439

**Chicago/Turabian Style**

Mao, Yunfei, Yong Peng, and Jianmin Zhang.
2018. "Study of Cavitation Bubble Collapse near a Wall by the Modified Lattice Boltzmann Method" *Water* 10, no. 10: 1439.
https://doi.org/10.3390/w10101439