# The Effect of the Fillets on Submarine Wake Field and Propeller Unsteady Bearing Force

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## Abstract

**:**

## 1. Introduction

## 2. Simulation Details

#### 2.1. Governing Equations

_{i}and u

_{j}(i, j = 1, 2, 3) represent the time-averaged velocity components, p is the time-averaged pressure, and S

_{i}is the general source term of the momentum equation. The Reynolds stress tensor τ

_{ij}is defined as

_{t}can be expressed in terms of an intermediate variable $\tilde{v}$ as ${v}_{t}=\tilde{v}{f}_{v1}\left(\chi \right)$. Here, $\chi =\frac{\tilde{v}}{v}$, and ${f}_{v1}=\frac{{\chi}^{3}}{{\chi}^{3}+{c}_{v1}^{3}}$ is a dumping function. The intermediate variable $\tilde{v}$ can be calculated using the following dimensionless transport equation based on the Spalart–Allmaras turbulent model [32]:

_{DES}. Different expressions of Δ have been proposed in the literature. In this work, the most primitive expression is adopted.

#### 2.2. Numerical Models

_{x}, Δ

_{y}, Δ

_{z}). According to the minimal spacing in the mesh, the time step used in the DES simulation must satisfy the requirement that CFL < 1.0.

#### 2.3. Test Case

## 3. Mesh Verification

#### 3.1. Meshing

#### 3.2. Verification of Resistance

#### 3.3. Verification of Wake Flow

_{0}is the incoming flow speed, u indicates the axial velocity of the flow field, u/U

_{0}is the dimensionless axial wake velocity, θ = 0° is the position at the top of the midline of the longitudinal section, and θ = 180° is the position at the bottom of the section, as shown in Figure 5.

## 4. Results Analysis and Discussion

#### 4.1. Distribution of Velocity

#### 4.2. Uniformity of Wake Field

#### 4.3. Analysis of Unsteady Bearing Force

^{3}, and D is the diameter of the propeller, unit m.

## 5. Conclusions

- The velocity distribution on the propeller disk shows that the fillets improve the wake field of the submarine by reducing the horseshoe vortex of the tail rudder, and they mainly affect the flow field characteristics in the inner radius area.
- The analysis of the harmonic components shows that the fillets can effectively reduce the axial velocity pulsation from 0.15 to 0.5, and the wake field of the improved model is more uniform.
- The frequency–domain curve shows that the fillets can reduce the bearing force pulsation value from 0.07 to 0.03, indicating that the fillets are conducive to controlling the vibration and noise performance of the submarine propeller.
- The cavitation is not considered in this paper. In future research, the influence of the fillets on the unsteady excitation force of the submarine will be studied under the condition of considering cavitation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Submarine length with full appendages L | 4.356 m |

Maximum hull diameter D_{max} | 0.508 m |

Sail length | 0.368 m |

Tail section of airfoil | NACA0020 |

Scale ratio | 1:24 |

geometric parameters | DTMB4383 | |

Diameter (D/m) | 0.305 | |

Number of blades, Z | 5 | |

Hub diameter ratio, d_{h}/D | 0.2 | |

Area ratio, A_{E}/A_{0} | 0.725 |

Original Model | Improved Model | ||
---|---|---|---|

Resistance Calculation | Self-Propulsion Calculation | Resistance Calculation | Self-Propulsion Calculation |

Without propeller | With propeller | Without propeller | With propeller |

Mesh | Grid Number | CFD | EFD | Error |
---|---|---|---|---|

Fine | 5.12 M | 106.5 | 102.3 | 4.11% |

Medium | 3.85 M | 107.3 | 102.3 | 4.89% |

Corse | 2.25 M | 112.6 | 102.3 | 10.1% |

Force | — | BPF | 2BPF | 3BPF | Time-Average |
---|---|---|---|---|---|

K_{Tx} × 10^{4} | Improved | 312 | 8.19 | 6.36 | 1932 |

Original | 722 | 7.41 | 5.61 | 1926 | |

K_{Ty} × 10^{4} | Improved | 78 | 6.21 | 16.80 | 963 |

Original | 96 | 5.15 | 0.00 | 825 | |

K_{Tz} × 10^{4} | Improved | 67 | 6.24 | 5.33 | 716 |

Original | 86 | 5.13 | 1.54 | 521 |

Moment | — | BPF | 2BPF | 3BPF | Time-Average |
---|---|---|---|---|---|

10K_{Qx} × 10^{4} | Improved | 512 | 13.44 | 10.05 | 4086 |

Original | 1018 | 12.36 | 11.33 | 4063 | |

10K_{Qy} × 10^{4} | Improved | 632 | 24.15 | 13.31 | 762 |

Original | 628 | 23.22 | 15.27 | 636 | |

10K_{Qz} × 10^{4} | Improved | 1148 | 15.21 | 11.69 | 714 |

Original | 1434 | 19.83 | 17.65 | 447 |

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**MDPI and ACS Style**

Li, X.; Cai, W.; Ren, N.; Sun, S. The Effect of the Fillets on Submarine Wake Field and Propeller Unsteady Bearing Force. *J. Mar. Sci. Eng.* **2023**, *11*, 727.
https://doi.org/10.3390/jmse11040727

**AMA Style**

Li X, Cai W, Ren N, Sun S. The Effect of the Fillets on Submarine Wake Field and Propeller Unsteady Bearing Force. *Journal of Marine Science and Engineering*. 2023; 11(4):727.
https://doi.org/10.3390/jmse11040727

**Chicago/Turabian Style**

Li, Xiang, Wenyu Cai, Nianxin Ren, and Shuai Sun. 2023. "The Effect of the Fillets on Submarine Wake Field and Propeller Unsteady Bearing Force" *Journal of Marine Science and Engineering* 11, no. 4: 727.
https://doi.org/10.3390/jmse11040727