# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD

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## Abstract

**:**

## 1. Introduction

## 2. Theory

**y**, is used to express the instantaneous velocities and positions of the submarine.

**y**is determined by solving the solid-body equations of motion (EOM) and the fluid EOM.

#### 2.1. Solid Body EOM

#### 2.2. Fluid EOM

#### 2.3. Stokes Waves

## 3. CFD Implementation

#### 3.1. Submarine Model

#### 3.2. Meshing

^{+}values is approximately 15; it becomes approximately 120 when the submarine is moving at maximum speed (i.e., ∼1 m/s). The mesh in the boundary layer was sufficiently refined to ensure simulation accuracy, given that the average of wall y

^{+}should be 20∼100 [24,25].

#### 3.3. Boundary and Initial Conditions

#### 3.4. Solution of the Fluid EOM

## 4. Results and Discussion

#### 4.1. Effect of Beam Sea

#### 4.2. Effect of Head Wave and Following Wave

#### 4.3. Effect of Oblique Waves

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

B | buoyancy |

$BG$ | metacentric height |

$CB,CG$ | center of buoyancy, center of gravity |

D | maximum diameter of submarine model |

${F}_{B}$ | body force |

${I}_{xx},{I}_{yy},{I}_{zz},{I}_{xz}$ | inertial moments |

${I}_{\nabla F}$ | inertial moments of form displacement |

${I}_{blown}$ | inertial moments of blown loads |

${I}_{casing}$ | inertial moments of casing |

L | length of submarine model |

${m}_{\nabla F}$ | weight of form displacement |

${m}_{blown}$ | weight of blown loads |

${m}_{casing}$ | weight of water in all casings |

$\mu $ | viscosity of water |

${\mu}_{w},{\mu}_{a}$ | viscosities of water and air |

${O}_{b}-xyz$ | body-fixed coordinate system |

${O}_{E}-\xi \eta \zeta $ | inertia-fixed coordinate system |

$p,q,r$ | angular velocities in body axis |

P | pressure |

$\phi ,\theta ,\psi $ | roll, pitch, and yaw angles in inertial axis |

$\rho $ | density of water |

$\rho \overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ | Reynolds stress |

${\rho}_{w},{\rho}_{a}$ | densities of water and air |

${R}_{e}=\frac{UL}{\nu}$ | Reynolds number |

${R}_{0}$ | position vector in inertial coordinates |

${r}_{1}$ | vector representing a fluid particle in body axes |

${\alpha}_{w}$ | volume fraction of water in control volume |

${\alpha}_{a}$ | volume fraction of air in control volume |

$u,v,w$ | velocity in body axis |

U | fluid velocity |

${U}_{i},{u}_{i}$ | mean and fluctuating velocity of fluid |

W | weight |

${x}_{B},{y}_{B},{z}_{B}$ | coordinates of $CB$ in body axes |

${x}_{G},{y}_{G},{z}_{G}$ | coordinates of $CG$ in body axes |

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**Figure 3.**Mesh of the overset region in the simulation: (

**a**) Mesh of transverse plane in the vicinity of the submarine; (

**b**) Mesh of symmetry plane in the vicinity of the submarine; (

**c**) Mesh near the stern of the submarine; (

**d**) Mesh near the sail of the submarine.

**Figure 7.**Effect of beam sea on the submarine posture on water surface: a wave spreads from right to left . Red water surface indicates wave peak, whereas blue water surface indicates wave valley.

**Figure 10.**Position and posture of the submarine when experiencing the peak and valley of the head wave: the wave spreads from right to left. Red water surface indicates wave peak, whereas blue water surface indicates wave valley.

**Figure 11.**Falling procedure of the submarine in the following wave: the wave spreads from left to right. Red water surface indicates wave peak, whereas blue water surface indicates wave valley.

**Figure 14.**Posture of the submarine rising in bow wave: the wave spreads from right to left. Red water surface indicates wave peak, whereas blue water surface indicates wave valley.

**Figure 15.**Posture of the submarine rising in a quartering wave: the wave spreads from left to right. Red water surface indicates wave peak, whereas blue water surface indicates wave valley.

Submarine Property | Value |
---|---|

Length | 5045.2 mm |

Width | 812.8 mm |

Height | 958.6 mm |

Maximum Diameter | 812.4 mm |

${m}_{\nabla F}$ | 1147.2 Kg |

${x}_{\nabla F}$ | 223.8 mm |

${y}_{\nabla F}$ | 0 mm |

${z}_{\nabla F}$ | −9.4 mm |

$\overline{BG}$ | 13.65 mm |

${I}_{\nabla F\left(x\right)}$ | 57.2 Kg · m${}^{2}$ |

${I}_{\nabla F\left(y\right)}$ | 2813.2 Kg · m${}^{2}$ |

${I}_{\nabla F\left(z\right)}$ | Kg · m${}^{2}$ |

${I}_{\nabla F\left(xz\right)}$ | 0.77 Kg · m${}^{2}$ |

${m}_{casing}$ | 244.4 Kg |

${x}_{casing}$ | 155.4 mm |

${y}_{casing}$ | 0 mm |

${z}_{casing}$ | −127.3mm |

${I}_{casing\left(x\right)}$ | 17.5 Kg · m${}^{2}$ |

${I}_{casing\left(y\right)}$ | 665.2 Kg · m${}^{2}$ |

${I}_{casing\left(z\right)}$ | 650.8 Kg · m${}^{2}$ |

${m}_{blown}$ | 44.89 Kg |

${x}_{blown}$ | 400 mm |

${y}_{blown}$ | 0 mm |

${z}_{blown}$ | −13.3 mm |

${I}_{blown\left(x\right)}$ | 2.63 Kg · m${}^{2}$ |

${I}_{blown\left(y\right)}$ | 245.2 Kg · m${}^{2}$ |

${I}_{blown\left(z\right)}$ | 248.6 Kg · m${}^{2}$ |

Mesh | Number of Cells | ${\mathit{\phi}}_{\mathbf{max}}(\circ )$ | ${\mathit{\theta}}_{\mathbf{max}}(\circ )$ |
---|---|---|---|

Coarse mesh | 1.25 million | 5.86 | 20.25 |

Medium mesh | 3.53 million | 6.61 | 18.73 |

Fine mesh | 10.37 million | 6.59 | 18.70 |

No. | Submarine Station | Wave Station | Wave Angle |
---|---|---|---|

1 | baseline depth: 4 m metacentric height: 13.65 mm ${m}_{blown}$: 44.89 Kg ${x}_{blown}$: 400 mm | static water | |

2 | wave height: 0.4 m wave length: 15 m | ${0}^{\circ}$ | |

3 | ${30}^{\circ}$ | ||

4 | ${60}^{\circ}$ | ||

5 | ${90}^{\circ}$ | ||

6 | ${120}^{\circ}$ | ||

7 | ${150}^{\circ}$ | ||

8 | ${180}^{\circ}$ |

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**MDPI and ACS Style**

Chen, Q.; Li, H.; Zhang, S.; Wang, J.; Pang, Y.; Wang, Q.
Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD. *Appl. Sci.* **2020**, *10*, 8403.
https://doi.org/10.3390/app10238403

**AMA Style**

Chen Q, Li H, Zhang S, Wang J, Pang Y, Wang Q.
Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD. *Applied Sciences*. 2020; 10(23):8403.
https://doi.org/10.3390/app10238403

**Chicago/Turabian Style**

Chen, Qinglong, Hongwei Li, Shudi Zhang, Jian Wang, Yongjie Pang, and Qingyun Wang.
2020. "Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD" *Applied Sciences* 10, no. 23: 8403.
https://doi.org/10.3390/app10238403