# Numerical Simulation of Breathing Mode Oscillation on Bubble Detachment

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

#### 2.1. Experimental Apparatus

#### 2.2. Overview of Experimental Results

## 3. Vibrational Motion in the Breathing Mode

## 4. Computational Fluid Dynamics Simulation

#### 4.1. Governing Equation and Scheme

#### 4.2. Computational Parameters and Mesh Validation

_{a}in water became 250 if Δt = 25 μs, and the minimum mesh size was 150 μm. After carried out, supplemental studies in the same mesh system in Figure 5b during one cycle oscillation of breathing mode for Δt = 12.5, 5, 2.5, 1.25, and 0.1 μs, i.e., C

_{a}= 125, 50, 25, 12.5, and 1, we confirmed that the deviation of pressure variation and frequency was less than 3% in the breathing mode oscillation.

#### 4.3. Overview of Density and Pressure Change

#### 4.4. Deformation and Pressure Diagram for Long-Term

#### 4.5. Pressure Distribution for One Cycle

#### 4.6. Radial Variation in the Breathing Mode

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$Eo$ | Eötvös number |

L | length of the nozzle |

Q | air flow rate |

${R}^{*}$ | equilibrium radius of the bubble |

φ | inner diameter of the nozzle |

$Re$ | Reynolds number |

$d$ | distance between the bubble center and hydrophone |

$f$ | frequency |

$g$ | gravitational acceleration |

${p}_{\infty}$ | ambient pressure |

$\Delta {p}_{p}$ | pressure drop near the bubble |

$\Delta {p}_{p}^{\prime}$ | pressure drop measured by the hydrophone |

${\rho}_{a}$ | air density |

${\rho}_{w}$ | water density |

$v$ | rising velocity of the bubble |

γ | polytropic exponent |

${\mu}_{w}$ | viscosity coefficient of water. |

σ | surface tension of water |

κ | specific heat ratio of air |

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**Figure 2.**Typical deformation of bubble detachment and the corresponding pressure waveform in the experiment. The nozzle inner diameter, φ = 6 mm and air flow rate, Q = 0.186 mL/s. (

**a**) Flow visualization of bubble detachment. (

**b**) Experimental pressure signal by hydrophone.

**Figure 6.**Density and pressure variation in computational fluid dynamics (CFD), φ = 6 mm. (

**a**) Density of water and air bubble. (

**b**) Pressure. (

**c**) Data sampling timing on pressure diagram at measuring point.

**Figure 7.**Bubble deformation process. (

**a**) Photo images in experiment. (

**b**) Bubble deformation in CFD. (

**c**) Pressure profile of CFD.

**Figure 9.**Comparison of pressure profile at the position of the hydrophone. (

**a**) φ = 6 mm. (

**b**) φ = 10 mm.

**Figure 11.**Pressure diagram for the nozzle lengths, L = 10, 20, and 40 mm. (

**a**) L = 10 mm. (

**b**) L = 20 mm. (

**c**) L = 40 mm.

**Figure 13.**Radius variation of bubble and pressure profile at the measuring point. (

**a**) φ = 6 mm. (

**b**) φ = 10 mm.

φ mm | ${R}^{*}$ mm | $v$ m/s | Re | Eo |
---|---|---|---|---|

4 | 3.2 | 0.24 | 7.7 × 10^{2} | 1.39 |

6 | 3.5 | 0.19 | 6.6 × 10^{2} | 1.66 |

8 | 4.8 | 0.17 | 8.1 × 10^{2} | 3.13 |

10 | 5.0 | 0.17 | 8.5 × 10^{2} | 3.39 |

φ mm | R* mm | Images | d mm | Δp′_{p} Pa | Δp_{p} Pa | ΔR by Equation (4) μm |
---|---|---|---|---|---|---|

4 | 3.2 | 18 | −86 | −480 | ±3.2 | |

6 | 3.5 | 17 | −152 | −740 | ±5.7 | |

8 | 4.8 | 16 | −270 | −900 | ±9.9 | |

10 | 5.0 | 16 | −329 | −1050 | ±11.9 |

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**MDPI and ACS Style**

Oku, T.; Hirahara, H.; Akimoto, T.; Tsuchida, D. Numerical Simulation of Breathing Mode Oscillation on Bubble Detachment. *Fluids* **2020**, *5*, 96.
https://doi.org/10.3390/fluids5020096

**AMA Style**

Oku T, Hirahara H, Akimoto T, Tsuchida D. Numerical Simulation of Breathing Mode Oscillation on Bubble Detachment. *Fluids*. 2020; 5(2):96.
https://doi.org/10.3390/fluids5020096

**Chicago/Turabian Style**

Oku, Takao, Hiroyuki Hirahara, Tomohiro Akimoto, and Daiki Tsuchida. 2020. "Numerical Simulation of Breathing Mode Oscillation on Bubble Detachment" *Fluids* 5, no. 2: 96.
https://doi.org/10.3390/fluids5020096