# A Numerical Study on the Water Entry of Cylindrical Trans-Media Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Problem Definition

## 3. Numerical Methods

#### 3.1. Governing Equations

#### 3.2. VOF Method

#### 3.3. 6DOF Motion Solver

#### 3.4. Grid, Boundary and Initial Conditions

#### 3.5. Grid Convergence Study

#### 3.6. Validation of Numerical Methods

## 4. Results and Discussion

#### 4.1. The Effect of Body Mass

#### 4.2. The Effect of Diameter-to-Length Ratio

#### 4.3. The Effect of Water-Entry Angle

#### 4.4. The Effect of Head Shape

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A concept of the next-generation unmanned trans-media vehicle developed by the first and second authors [26].

**Figure 9.**Grid convergence study. (

**a**) Time history of vertical velocity; (

**b**) Time history of vertical acceleration; (

**c**) Time history of vertical displacement; (

**d**) Time history of vertical force.

**Figure 10.**The setup of the validation case. (

**a**) Geometry of the 3D wedge; (

**b**) Computational region.

**Figure 11.**Instantaneous phase contours of the 3D wedge. Blue color represents air and red color represents water.

**Figure 12.**Comparisons between the numerical results and the experimental data. (

**a**) Time history of vertical velocity; (

**b**) Time history of vertical displacement.

**Figure 13.**Instantaneous pressure contours of the cylinders with varying masses. (

**a**) M = 15.708 kg; (

**b**) M = 31.416 kg; (

**c**) M = 47.124 kg.

**Figure 14.**Instantaneous phase contours of the cylinders with varying masses. (

**a**) M = 15.708 kg; (

**b**) M = 31.416 kg; (

**c**) M = 47.124 kg.

**Figure 15.**Comparisons between the time histories of the cylinders with varying masses. (

**a**) Vertical velocity; (

**b**) Vertical displacement; (

**c**) Vertical acceleration; (

**d**) Vertical force.

**Figure 17.**Instantaneous pressure contours of the cylinders with varying diameter-to-length ratios. (

**a**) Model 1; (

**b**) Model 2; (

**c**) Model 3.

**Figure 18.**Comparisons between the time histories of the cylinders with varying diameter-to-length ratios. (

**a**) Vertical velocity; (

**b**) Vertical displacement; (

**c**) Vertical acceleration; (

**d**) Vertical force.

**Figure 19.**Instantaneous phase contours of the cylinders with varying diameter-to-length ratios. (

**a**) Model 1; (

**b**) Model 2; (

**c**) Model 3.

**Figure 20.**Instantaneous pressure contours of the cylinders with varying water-entry angles. (

**a**) α = 30°; (

**b**) α = 45°; (

**c**) α = 60°; (

**d**) α = 75°; (

**e**) α = 90°.

**Figure 21.**Comparisons between the time histories of the cylinders with varying water-entry angles. (

**a**) Vertical velocity; (

**b**) Vertical displacement; (

**c**) Vertical acceleration; (

**d**) Vertical force.

**Figure 22.**Instantaneous phase contours of the cylinders with varying water-entry angles. (

**a**) α = 30°; (

**b**) α = 45°; (

**c**) α = 60°; (

**d**) α = 75°; (

**e**) α = 90°.

**Figure 23.**Instantaneous pressure contours of the cylindrical bodies with varying head shapes. (

**a**) Cylinder; (

**b**) Sphere; (

**c**) Ellipsoid; (

**d**) Cone.

**Figure 24.**Instantaneous phase contours of the cylindrical bodies with varying head shapes. (

**a**) Cylinder; (

**b**) Sphere; (

**c**) Ellipsoid; (

**d**) Cone.

**Figure 25.**Comparisons between the time histories of the cylindrical bodies with varying head shapes. (

**a**) Vertical velocity; (

**b**) Vertical displacement; (

**c**) Vertical acceleration; (

**d**) Vertical force.

Year | Authors | Research Objects | Numerical Methods |
---|---|---|---|

2004 | Takagi [12] | 3D elliptic paraboloids | Potential method |

2006 | Aquelet et al. [13] | 2D wedges | Explicit FEM |

2006 | Stenius et al. [14] | 2D wedges | Explicit FEM |

2007 | Stenius et al. [15] | 2D elastomers | Explicit FEM |

2007 | Wick et al. [16] | A UAV | VOF method |

2012 | Yang and Qiu [17] | 2D and 3D wedges | CIP method |

2014 | Abraham et al. [18] | 3D balls | VOF method |

2014 | Van Nuffel et al. [11] | Horizontal cylinders | VOF method |

2015 | Qu et al. [19] | An airplane | VOF method |

2016 | Facci et al. [20] | Multi-curvature structures | VOF method |

2017 | Xiao et al. [21] | A helicopter | SPH method |

2019 | Shi et al. [23] | AUVs | Explicit FEM |

2020 | Xiang et al. [22] | Horizontal cylinders | VOF method |

2021 | Wang et al. [24] | AUVs | VOF method |

2022 | Wu et al. [25] | An air-launched underwater glider | VOF method |

Grid Name | Minimum Size of Moving Grid Region (mm) | Minimum Size of Overall Grid Region (mm) | $\mathbf{Number}\mathbf{of}\mathbf{Grid}\mathbf{Cells}(\times {10}^{6})$ |
---|---|---|---|

Grid 1 | 1.210 | 37.372 | 2.754 |

Grid 2 | 0.903 | 37.372 | 3.878 |

Grid 3 | 0.903 | 29.066 | 4.565 |

Mass (kg) | Ixx (kg·m^{2}) | Iyy (kg·m^{2}) | Izz (kg·m^{2}) | Ixy (kg·m^{2}) | Ixz (kg·m^{2}) | Iyz (kg·m^{2}) |
---|---|---|---|---|---|---|

15.708 | 1.388 | 1.388 | 0.157 | 0.000 | 0.000 | 0.000 |

31.416 | 2.775 | 2.775 | 0.314 | 0.000 | 0.000 | 0.000 |

47.124 | 4.163 | 4.163 | 0.417 | 0.000 | 0.000 | 0.000 |

Model | Ixx (kg·m^{2}) | Iyy (kg·m^{2}) | Izz (kg·m^{2}) | Ixy (kg·m^{2}) | Ixz (kg·m^{2}) | Iyz (kg·m^{2}) |
---|---|---|---|---|---|---|

Model 1 | 0.812 | 0.812 | 0.314 | 0.000 | 0.000 | 0.000 |

Model 2 | 2.775 | 2.775 | 0.314 | 0.000 | 0.000 | 0.000 |

Model 3 | 3.246 | 3.246 | 1.257 | 0.000 | 0.000 | 0.000 |

Water-Entry Angle | Ixx (kg·m^{2}) | Iyy (kg·m^{2}) | Izz (kg·m^{2}) | Ixy (kg·m^{2}) | Ixz (kg·m^{2}) | Iyz (kg·m^{2}) |
---|---|---|---|---|---|---|

30° | 2.775 | 0.929 | 2.160 | 0.000 | 0.000 | −1.066 |

45° | 2.775 | 1.545 | 1.545 | 0.000 | 0.000 | −1.230 |

60° | 2.775 | 2.160 | 0.929 | 0.000 | 0.000 | −1.066 |

75° | 2.775 | 2.610 | 0.479 | 0.000 | 0.000 | −0.615 |

90° | 2.775 | 2.775 | 0.314 | 0.000 | 0.000 | 0.000 |

Head Shape | Ixx (kg·m^{2}) | Iyy (kg·m^{2}) | Izz (kg·m^{2}) | Ixy (kg·m^{2}) | Ixz (kg·m^{2}) | Iyz (kg·m^{2}) |
---|---|---|---|---|---|---|

Cylinder | 2.775 | 2.775 | 0.314 | 0.000 | 0.000 | 0.000 |

Sphere | 2.278 | 2.278 | 0.314 | 0.000 | 0.000 | 0.000 |

Ellipsoid | 1.833 | 1.833 | 0.314 | 0.000 | 0.000 | 0.000 |

Cone | 2.278 | 2.278 | 0.314 | 0.000 | 0.000 | 0.000 |

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**MDPI and ACS Style**

Deng, F.; Sun, X.; Chi, F.; Ji, R.
A Numerical Study on the Water Entry of Cylindrical Trans-Media Vehicles. *Aerospace* **2022**, *9*, 805.
https://doi.org/10.3390/aerospace9120805

**AMA Style**

Deng F, Sun X, Chi F, Ji R.
A Numerical Study on the Water Entry of Cylindrical Trans-Media Vehicles. *Aerospace*. 2022; 9(12):805.
https://doi.org/10.3390/aerospace9120805

**Chicago/Turabian Style**

Deng, Feng, Xiaoyuan Sun, Fenghua Chi, and Ruixue Ji.
2022. "A Numerical Study on the Water Entry of Cylindrical Trans-Media Vehicles" *Aerospace* 9, no. 12: 805.
https://doi.org/10.3390/aerospace9120805