# Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking

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## Abstract

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## 1. Introduction

## 2. LE Two-Way Coupling Model

#### 2.1. Governing Equations

#### 2.2. Gas Volume Fraction

#### 2.3. Fluid-Mixture Pressure

## 3. Numerical Methods

#### 3.1. Bubble Dynamics

#### 3.2. Spatial Discretization

#### 3.3. Time Integration

- if $\mid \Delta {R}_{n}\mid /{R}_{n}<0.02$, $\Delta {tb}_{n+1}=\Delta {tb}_{n}$;
- if $\mid \Delta {R}_{n}\mid /{R}_{n}>0.02$ and $\Delta {R}_{n}<0$, $\Delta {tb}_{n+1}=\Delta {tb}_{n}/1.3$;
- if $\mid \Delta {R}_{n}\mid /{R}_{n}>0.02$ and $\Delta {R}_{n}>0$, $\Delta {tb}_{n+1}=1.3\Delta {tb}_{n}$.

#### 3.4. Computing Procedure

- Lagrangian computation. Update the gas bubbles positions ${\mathbf{x}}_{b}$ and radii ${R}_{b}$. Here, $\frac{d{\mathbf{x}}_{b}}{dt}$ is directly derived from the velocity field $\mathbf{u}$ of the mixture. The radius ${R}_{b}$ is updated by Equation (4). The ${p}_{\infty}$ and ${\rho}_{\infty}$ in Equation (4) are derived by Equation (13).
- Update the volume fraction in each computation cell by Equation (8).

## 4. Validation

#### 4.1. Isolated Bubble

#### 4.2. 1D Bubble Advection

#### 4.3. Single Bubble Oscillating

## 5. Application of a Bubble Cloud Interacting with Pressure Wave

## 6. Rayleigh Collapse of a Bubble Cloud

#### 6.1. Rayleigh Collapse

#### 6.2. Results and Discussion

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of three interface types with respect to the computational grid: the fully-resolved, under-resolved, and sub-grid dispersed interfaces.

**Figure 2.**Isolated bubble: The time history of the bubble size ${R}^{*}={R}_{b}/{R}_{b0}$ in 1D simulation. Results of grid resolutions of 100 ($\mathrm{d}x=4{R}_{b0}$) and 200 ($\mathrm{d}x=2{R}_{b0}$) are compared with the analytical solution.

**Figure 3.**Isolated bubble: The time history of the bubble size ${R}^{*}={R}_{b}/{R}_{b0}$ in 3D simulation.

**Figure 4.**1D bubble advection: The solid line represents the initial fluid condition, and ∘ represents the results when the Lagrangian bubble reaches $x=2.86$ at $t=3.0$.

**Figure 5.**Single bubble oscillating: Volume fraction ${\alpha}_{l}$ along the bubble radius for different resolutions.

**Figure 6.**Single bubble oscillating: Pressure wave propagation induced from an oscillating gas bubble at ${t}^{*}=tf=0.0,1.0,3.0,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}5.0$.

**Figure 7.**Single bubble oscillating: Pressure wave propagation induced from a continuous oscillating gas bubble at ${t}^{*}=tf=0.0,1.0,3.0,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}5.0$.

**Figure 8.**Initial setup of the simulation of a bubble cloud interacting with pressure wave. The radius of the bubble cloud is ${A}_{0}=2\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$, and it has 200 bubbles with random size distribution between $1\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$ and $5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$. The computation domain is $2\phantom{\rule{0.166667em}{0ex}}L\times L$ (L = 10.24 mm). The distribution of the gas bubbles is visualized by both the volume fraction ${\alpha}_{g}$ and the radius ${R}_{b}$ of Lagrangian spherical particles. Four bubbles (bubble A, B, C, and D) located at the four boxes’ centers are marked for future reference.

**Figure 9.**The pressure field induced by the bubble cluster when the bubble cluster interacts with a sinusoidal pulse. The initial state and the pressure field at $1.0\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $2.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $3.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $4.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, and $5.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$ are shown.

**Figure 10.**The time history of pressure at the bubble cloud center without (the dashed line) and with bubble cloud for two different resolutions.

**Figure 11.**The gas volume fraction ${\alpha}_{g}$ when the bubble cluster interacts with a sinusoidal pulse at $1.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $2.0\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $2.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $3.0\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, $3.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$, and $4.5\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{s}$.

**Figure 12.**The time history of bubble radius ${R}_{b}$ of Bubble A (on the

**left**side), Bubble B (on the

**bottom**), Bubble C (at bubble cloud

**center**), and Bubble D (on the

**right**side).

**Figure 13.**Schematic of the set-up of the bubble cloud Rayleigh Collapse. A spherical bubble cloud with ${N}_{0}$ vapor bubbles is placed inside the bulk water. $A\left(t\right)$ is the radius of the bubble cloud.

**Figure 14.**The snapshots of the collapsing process of a bubble cloud with ${N}_{0}$ = 200 vapor bubbles (at t = 0 $\mathsf{\mu}$s, 6 $\mathsf{\mu}$s, 9 $\mathsf{\mu}$s, 12 $\mathsf{\mu}$s, 14 $\mathsf{\mu}$s, and 16 $\mathsf{\mu}$s). The distribution of the vapor bubbles is visualized both by the iso-surfaces of the volume fraction ${\alpha}_{g}=0.002$ (${\alpha}_{l}=0.998$) and the Lagrangian spherical particles of the radius ${R}_{b}$. Slices of the high pressure area (higher than 10 atm) at the center of the bubble cloud are also plotted, which indicate the cloud collapse induced violent pressures.

**Figure 15.**The time history of the non-dimensional active bubble number ${N}_{b}/{N}_{0}$ (${N}_{0}$ = 200) and the averaged gas fraction $\beta /{\beta}_{0}$, here being $\beta ={\Sigma}_{{V}_{c}}{\alpha}_{g}/{V}_{c}$.

**Figure 16.**The snapshots of the vapor bubble cloud collapse process with ${N}_{0}$ = 300 vapor bubbles (t = 0 $\mathsf{\mu}$s, 6 $\mathsf{\mu}$s, 9 $\mathsf{\mu}$s, 12 $\mathsf{\mu}$s, 13 $\mathsf{\mu}$s, and 14 $\mathsf{\mu}$s). Distribution of the vapor bubbles is visualized both by the iso-surfaces of the volume fraction ${\alpha}_{g}=0.002$ (${\alpha}_{l}=0.998$) and the radius ${R}_{b}$ of Lagrangian spherical particles. Slices of the extreme pressure area (higher than 10 atm) at the center of the bubble cloud are also plotted, which indicate the cloud collapse induced violent pressures.

**Figure 17.**The snapshots of the vapor bubble cloud collapse process initially with ${N}_{0}$ = 400 vapor bubbles (t = 0 $\mathsf{\mu}$s, 6 $\mathsf{\mu}$s, 9 $\mathsf{\mu}$s, 12 $\mathsf{\mu}$s, 13 $\mathsf{\mu}$s, and 14 $\mathsf{\mu}$s). Distribution of the vapor bubbles is visualized both by the iso-surface of the volume fraction ${\alpha}_{g}=0.002$ (${\alpha}_{l}=0.998$) and the radius ${R}_{b}$ of Lagrangian spherical particles. Slices of the high pressure area (higher than 10 atm) at the center of the bubble cloud are also plotted, which indicate the cloud collapse induced violent pressures.

**Figure 18.**The time history of the non-dimensional active bubble number ${N}_{b}/{N}_{0}$ (${N}_{0}$ = 400) and the averaged gas fraction $\beta /{\beta}_{0}$ ($\beta ={\Sigma}_{{V}_{c}}{\alpha}_{g}/{V}_{c}$).

**Figure 19.**The bubble distributions (radii density bar graphs) of the bubbles’ radii at t = 0 $\mathsf{\mu}$s, 6 $\mathsf{\mu}$s, 9 $\mathsf{\mu}$s, 12 $\mathsf{\mu}$s, 13 $\mathsf{\mu}$s, and 14 $\mathsf{\mu}$s. The first bar from the left side indicates the density of the collapsed vapor bubbles.

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**MDPI and ACS Style**

Lyu, X.; Zhu, Y.; Zhang, C.; Hu, X.; Adams, N.A.
Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking. *Water* **2021**, *13*, 2684.
https://doi.org/10.3390/w13192684

**AMA Style**

Lyu X, Zhu Y, Zhang C, Hu X, Adams NA.
Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking. *Water*. 2021; 13(19):2684.
https://doi.org/10.3390/w13192684

**Chicago/Turabian Style**

Lyu, Xiuxiu, Yujie Zhu, Chi Zhang, Xiangyu Hu, and Nikolaus A. Adams.
2021. "Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking" *Water* 13, no. 19: 2684.
https://doi.org/10.3390/w13192684