# Modeling Acoustic Cavitation Using a Pressure-Based Algorithm for Polytropic Fluids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Polytropic Closure

## 4. Numerical Framework

#### 4.1. Finite-Volume Discretization

#### 4.2. Advecting Velocity

#### 4.3. Discretized Governing Equations

#### 4.4. Solution Procedure

## 5. Interface Treatment

#### 5.1. Interface Advection

#### 5.2. Fluid Properties

## 6. Results

#### 6.1. Acoustic Waves

#### 6.2. Rayleigh Collapse

#### 6.3. Wall-Bounded Cavitation

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of mesh cell P and its neighbor cell Q, with their shared face f and its unit normal vector ${\mathit{n}}_{f}$ (pointing out of cell P).

**Figure 2.**Pressure profiles of acoustic waves in different fluids obtained using the proposed algorithm. The pressure amplitudes based on linear acoustic theory, $\pm \Delta {p}_{0}$, are shown as a reference.

**Figure 3.**Profiles of pressure pulses in different air–water systems obtained using the proposed algorithm. The amplitudes of the pressure pulses reflected and transmitted at the fluid interface based on linear acoustic theory are shown as a reference. (

**a**) Air–water (Water NASG) system at $t=1.5\times {10}^{-3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The pressure pulse is initialized with $\Delta {p}_{0}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}10\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}$, ${x}_{0}=0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ and $\sigma =0.03$. The air-water interface is located at ${x}_{\Sigma}=0.5$. (

**b**) Water-air (Water Tait) system at $t=1.0\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The reflected pressure pulse in water and the transmitted pressure pulse in air are shown in separate graphs, due to their very different amplitudes. The pressure pulse is initialized with $\Delta {p}_{0}=1000\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}$, ${x}_{0}=0.6\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ and $\sigma =0.1$. The air-water interface is located at ${x}_{\Sigma}=1.5$.

**Figure 4.**Rayleigh collapse of a spherical bubble. Dimensionless radius $R/{R}_{0}$ as a function of dimensionless time $t/{t}_{\mathrm{c}}$, where ${R}_{0}$ is the initial radius and ${t}_{\mathrm{c}}=0.915\phantom{\rule{0.166667em}{0ex}}{R}_{0}\sqrt{{\rho}_{0,\mathrm{water}}/{p}_{\infty}}$ is the Rayleigh collapse time, computed using (

**a**) different time-steps $\Delta t$ on a mesh with $\Delta x={R}_{0}/400$ and (

**b**) on meshes with different mesh spacings $\Delta x$ and a time-step of $\Delta t={10}^{-4}\phantom{\rule{0.166667em}{0ex}}{t}_{\mathrm{c}}$, compared against the solution of the Gilmore model [9].

**Figure 5.**Wall-bounded cavitation. (

**a**) Simulation setup. (

**b**–

**d**) Instantaneous bubble shape and velocity contours of a bubble with $\gamma =0.59$ and ${R}_{\mathrm{max}}=388\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ at times $t=39.0\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{s}$, $t=70.5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{s}$ and $t=85.0\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{s}$, respectively. The wall is explicitly illustrated in all figures.

**Figure 6.**Wall-bounded cavitation. (

**a**) Results of the minimum liquid film thickness ${\ell}_{\mathrm{min}}$ as a function of the dimensionless stand-off distance $\gamma ={\ell}_{0}/{R}_{\mathrm{max}}$ obtained with the proposed algorithm, compared against the correlation ${\ell}_{\mathrm{min}}=29.2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\gamma}^{4.86}+4.74\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ obtained by experimental measurements in the range $\gamma =0.47-1.07$ [51]. (

**b**,

**c**) Evolution of the maximum wall shear stress ${\tau}_{\mathrm{w}}$ and maximum wall pressure ${p}_{\mathrm{w}}$, respectively, for selected stand-off distances $\gamma $.

Fluid | $\mathit{\kappa}$ | b [m${}^{3}$ kg${}^{-1}$] | $\mathsf{\Pi}\phantom{\rule{0.166667em}{0ex}}$ [Pa] |
---|---|---|---|

Air | $1.400$ | 0 | 0 |

JA2 propellant gas [48] | $1.225$ | $1.00\times {10}^{-3}$ | 0 |

Water Tait [23] | $7.150$ | 0 | $3.046\times {10}^{8}$ |

Water NASG [32] | $1.187$ | $6.61\times {10}^{-4}$ | $7.028\times {10}^{8}$ |

**Table 2.**The excitation frequency f of the acoustic waves, the reference density ${\rho}_{0}$ and the associated speed of sound ${a}_{0}$ at the reference pressure of ${p}_{0}={10}^{5}\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}$, the wavelength ${\lambda}_{0}$ and pressure amplitude $\Delta {p}_{0}$ of the acoustic waves given by linear acoustic theory, and the wavelength $\lambda $ and the pressure amplitude $\Delta p$ of the acoustic waves predicted by the proposed algorithm.

Fluid | f [s${}^{-1}$] | ${\mathit{\rho}}_{0}$ [kg m${}^{-3}$] | ${\mathit{a}}_{0}$ [m s${}^{-1}$] | ${\mathit{\lambda}}_{0}$ [m] | $\mathbf{\Delta}{\mathit{p}}_{0}$ [Pa] | $\mathit{\lambda}$ [m] | $\mathbf{\Delta}\mathit{p}$ [Pa] |
---|---|---|---|---|---|---|---|

Air | 1750 | $1.157$ | $347.85$ | $0.199$ | $4.025$ | $0.199$ | $4.019$ |

JA2 propellant gas [48] | 1750 | $0.997$ | $350.70$ | $0.200$ | $3.496$ | $0.201$ | $3.492$ |

Water Tait [23] | 7500 | 1000 | $1476.0$ | $0.197$ | $14,760$ | $0.198$ | $14,741$ |

Water NASG [32] | 7500 | 1000 | $1568.8$ | $0.209$ | $15,688$ | $0.210$ | $15,673$ |

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**MDPI and ACS Style**

Denner, F.; Evrard, F.; van Wachem, B.
Modeling Acoustic Cavitation Using a Pressure-Based Algorithm for Polytropic Fluids. *Fluids* **2020**, *5*, 69.
https://doi.org/10.3390/fluids5020069

**AMA Style**

Denner F, Evrard F, van Wachem B.
Modeling Acoustic Cavitation Using a Pressure-Based Algorithm for Polytropic Fluids. *Fluids*. 2020; 5(2):69.
https://doi.org/10.3390/fluids5020069

**Chicago/Turabian Style**

Denner, Fabian, Fabien Evrard, and Berend van Wachem.
2020. "Modeling Acoustic Cavitation Using a Pressure-Based Algorithm for Polytropic Fluids" *Fluids* 5, no. 2: 69.
https://doi.org/10.3390/fluids5020069