# Numerical Analysis of the Free-Falling Process of a Water Droplet at Different Temperatures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Force Description of a Falling Water Droplet

_{1}, the buoyancy F

_{2}and the drag F

_{3}[26]. These forces acting on a water droplet can be expressed by the following formulas:

^{2}), ${V}_{wd}$ is the volume of a water droplet (m

^{3}), ${\rho}_{wd}$ is water density (kg/m

^{3}), ${\rho}_{a}$ is air density (kg/m

^{3}), $g$ is gravity acceleration and ${\lambda}_{d}$ is drag coefficient. The drag coefficient ${\lambda}_{d}$ is related to the Reynolds number ${R}_{e}$ [27], and its expression is

## 3. Two-Phase Flow Model of Water Droplet and Air

#### 3.1. Numerical Model and Governing Equations

#### 3.2. Grid Study and Validation

## 4. Numerical Results of the Free-Falling Process

## 5. A Test of Free Falling

## 6. Discussion of Characteristic Parameters

#### 6.1. Velocity of Free-Falling Water Droplet

_{w}is determined by the size of the major axis and minor axis of the water droplet [3].

#### 6.2. Pressure Distribution of Water Droplet

#### 6.3. Temperature Field

#### 6.4. Water Droplet Impacting on Bottom

#### 6.5. Water Droplet Freezing on Bottom

## 7. Conclusions

- (1)
- In the process of free falling, the water droplet deforms due to its surrounding forces. Before impinging on the bottom, the water droplet is the shape of a flat ellipse at 293.15 K, a drum at 283.15 K, a melon seed at 263.15 K and a human face at 253.15 K and 243.15 K when the falling time is 0.10 s.
- (2)
- The falling velocity of the water droplet is affected by the temperature. In the process of falling in air at a sub-zero temperature, the water droplet experiences a continuous increase in its falling velocity. When the water droplet falls in air at a mild temperature, the falling velocity is slightly larger than that in air at a sub-zero temperature, and there is a velocity drop at 0.10 s due to the impinging on the bottom surface.
- (3)
- There are great differences in the distribution of inner pressure and temperature. When the water droplet falls in air at a sub-zero temperature, there is a temperature increase behind the water droplet due to the latent heat released by ice freezing.
- (4)
- In the air at a sub-zero temperature, the phase change of the water droplet first occurs at the bottom, and then propagates inwards from the top of the water droplet, and at last the phase change finishes at the inner of the water droplet with a line.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Grid distribution of water droplet and its shape falling in air at 283.15 K, 0.05 s: (

**a**) case 1; (

**b**) case 2; (

**c**) case 3; (

**d**) case 4.

**Figure 7.**Pressure distribution cloud of the free-falling water droplet at t = 0.02 s. The white circle is the interface of water droplet and air.

**Figure 8.**Pressure distribution cloud of the free-falling water droplet at t = 0.06 s. The white circle is the interface of water droplet and air.

**Figure 9.**Pressure distribution cloud before water droplet impinging on bottom at t = 0.10 s. The white circle is the interface of water droplet and air.

Case | Mean Quality of Grid | The Lowest Quality of Grid |
---|---|---|

Case 1 | 0.8540 | 0.7316 |

Case 2 | 0.8368 | 0.6929 |

Case 3 | 0.8740 | 0.7633 |

Case 4 | 0.8768 | 0.6960 |

Tg (K) | t = 0.02 s | t = 0.04 s | t = 0.06 s | t = 0.08 s | t = 0.09 s | t = 0.10 s |
---|---|---|---|---|---|---|

293.15 | 7.84 | 34.86 | 73.16 | 115.52 | 135.87 | 120.46 |

283.15 | 8.27 | 33.97 | 71.87 | 118.80 | 146.67 | 120.46 |

263.15 | 7.30 | 30.53 | 60.78 | 93.87 | 110.69 | 125.49 |

253.15 | 7.59 | 30.53 | 60.78 | 93.87 | 110.69 | 127.19 |

243.15 | 7.30 | 29.70 | 60.78 | 93.87 | 110.69 | 125.49 |

Time | Tg (K) | V (m/s) | D_{w} (mm) | ${\mathit{R}}_{\mathit{e}}$ | ${\mathit{F}}_{\mathit{r}}$ | W_{e}(10^{−2}) | Oh(10^{−4}) |
---|---|---|---|---|---|---|---|

0.04 s | 293.15 | 0.78 | 3.11 | 161.56 | 4.47 | 3.17 | 11.02 |

283.15 | 0.77 | 3.03 | 165.04 | 4.47 | 3.11 | 10.68 | |

263.15 | 0.73 | 3.03 | 177.60 | 4.24 | 3.01 | 9.76 | |

253.15 | 0.73 | 3.10 | 194.97 | 4.19 | 3.20 | 9.18 | |

243.15 | 0.72 | 3.03 | 202.13 | 4.16 | 3.17 | 8.81 | |

0.09 s | 293.15 | 1.54 | 3.27 | 335.21 | 8.60 | 12.98 | 10.75 |

283.15 | 1.60 | 3.00 | 340.68 | 9.32 | 13.32 | 10.71 | |

263.15 | 1.39 | 3.16 | 352.69 | 7.90 | 11.37 | 9.56 | |

253.15 | 1.39 | 2.96 | 354.36 | 8.16 | 11.08 | 9.39 | |

243.15 | 1.39 | 2.96 | 380.85 | 8.16 | 11.54 | 8.92 | |

0.10 s | 293.15 | 1.45 | 4.27 | 412.15 | 7.09 | 15.02 | 9.40 |

283.15 | 1.45 | 4.00 | 411.48 | 7.32 | 14.58 | 9.28 | |

263.15 | 1.48 | 4.16 | 494.46 | 7.33 | 16.97 | 8.33 | |

253.15 | 1.49 | 3.96 | 508.16 | 7.56 | 17.04 | 8.12 | |

243.15 | 1.48 | 3.96 | 542.48 | 7.51 | 17.51 | 7.71 |

Grids | Normal | Refined 1 | Refined 2 |
---|---|---|---|

Numbers | 1548 | 2948 | 6292 |

Mean mesh quality | 0.8183 | 0.8527 | 0.8337 |

Time(s) | 7.8 | 5.4 | 5.8 |

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**MDPI and ACS Style**

Song, Y.; Zhang, Y.; Gao, H.
Numerical Analysis of the Free-Falling Process of a Water Droplet at Different Temperatures. *Processes* **2023**, *11*, 258.
https://doi.org/10.3390/pr11010258

**AMA Style**

Song Y, Zhang Y, Gao H.
Numerical Analysis of the Free-Falling Process of a Water Droplet at Different Temperatures. *Processes*. 2023; 11(1):258.
https://doi.org/10.3390/pr11010258

**Chicago/Turabian Style**

Song, Yuchao, Yafei Zhang, and Hongtao Gao.
2023. "Numerical Analysis of the Free-Falling Process of a Water Droplet at Different Temperatures" *Processes* 11, no. 1: 258.
https://doi.org/10.3390/pr11010258