# Sound Field Properties of Non-Cavitating Marine Propellers

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## Abstract

**:**

## 1. Introduction

#### 1.1. State of the Art

#### 1.2. Contributions of Current Work

## 2. Methodology

#### 2.1. Boundary Element Method

#### 2.2. Ffowcs Williams-Hawkings Equation

#### 2.3. Formulation of FWH Equation on the Potential Wake Sheet

#### 2.3.1. Thickness Terms

#### 2.3.2. Loading Terms

#### 2.4. Coupling of Permeable FWH Approach with BEM

## 3. Numerical Results

#### 3.1. Open Water Case with the Direct FWH Approach

#### 3.2. Behind-Hull Case with the Direct FWH Approach

#### 3.3. Behind-Hull Case with the BEM/P-FWH Hybrid Method

#### 3.4. Verification of the FWH Formulation on the Wake Sheet

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | to distinguish it from the P-FWH approach, the approach in which the FWH terms are integrated directly on the blade and wake sheet surfaces are denoted as the direct FWH approach in current work, although no direct volume integration is carried out. |

**Figure 1.**The domain boundaries and discretization in the boundary element method. Courtesy of Reference [24].

**Figure 2.**Different perspectives to consider the evolution of wake sheet. For the selected black panel in the current time step, the same panel in the next time step would be the blue one in the earth-fixed perspective and be the red one in the propeller-fixed perspective.

**Figure 3.**The FWH (Ffowcs William-Hawkings) integration surfaces enveloping the wake sheet. The gap between the upper and lower surface $\u03f5$ is set to be infinitely close to zero.

**Figure 4.**SPL (Sound Pressure Level) obtained with different approaches for the open water case. The x-axis is the distance of observers to the propeller centre.

**Figure 5.**Comparison of the pressure variation obtained with the direct FWH approach and the BEM (Boundary Element Method) solver. The observer has a distance of 8 m to the propeller centre. The pressure obtained with BEM solver has been shifted by its time-averaged value.

**Figure 8.**SPLs in the behind-hull case calculated with different wake sheet lengths. The right plot is an enlarged view for the region 20 m ~100 m. In the legend blade means only sound produced by the blade, wake means only sound produced by the wake sheet, total means the combination of both, and 2 rev ~8 rev denote the length of wake sheet in revolutions.

**Figure 9.**Contributions of different FWH terms in the blade induced sound pressure for the behind-hull case.

**Figure 10.**Contributions of different FWH terms in the wake sheet induced sound pressure for the behind-hull case. The thickness terms are always zero and are therefore not shown.

**Figure 11.**Different permeable surfaces and the size parameters. The size parameters are depicted on the basic permeable surface. Two wake revolutions are shown here.

**Figure 12.**The total potential source strength on propeller blades and the volume flux on the basic permeable surface.

**Figure 13.**The SPL variation obtained using the basic permeable surface with and without virtual source correction.

**Figure 14.**Influence of permeable surface’s panel size on SPL. The results are for the observer being 550 m from the propeller centre. The permeable surface being investigated is the basic one in Table 2.

**Figure 15.**The influence of whether including the ship wake velocity on the permeable surfaces. Left: the SPL calculated using the basic permeable surface, with and without considering the ship wake velocity on the permeable surface. Right: contributions of different FWH terms on the permeable surfaces as a function of distance to the propeller centre.

**Figure 16.**Comparison of the pressure variation obtained with the BEM solver, direct FWH approach, and P-FWH approach. The observer has a distance of 8m to the propeller centre.

**Figure 17.**SPL obtained with the direct FWH approach and P-FWH approach using different permeable surface dimensions.

**Figure 18.**The SPL generated by only the wake sheets, calculated with the direct FWH approach and P-FWH approach.

Parameter | Symbol | Value |
---|---|---|

diameter | D | 7.9 m |

number of blades | Z | 5 |

designed advance ratio | ${J}_{0}$ | 0.7 |

rotation rate | n | 1.25 s${}^{-1}$ |

angular speed | $\omega $ | 7.85 s${}^{-1}$ |

BPF (Blade Pass Frequency) | ${f}_{1}$ | 6.25 Hz |

acoustic wave length for BPF | ${\lambda}_{1}$ | 240 m |

**Table 2.**Size and discretization parameters for the permeable surfaces with names basic, long, and large. The meaning of size parameters are explained in Figure 11.

Parameters | Basic | Long | Large |
---|---|---|---|

front length | 1.5 m | 50 m | 13 m |

rear length | 21.5 m | 21.5 m | 33 m |

radius | 0.6D | 0.6D | 2D |

panel size | 0.5 m | 0.5 m | 2 m |

**Table 3.**The SPL (re 1 $\mathsf{\mu}$Pa) obtained with different permeable surface’s panel size. The basic permeable surface is used.

1st BPF | 3rd BPF | |||
---|---|---|---|---|

Panel Size | 100 m | 550 m | 100 m | 550 m |

0.2 m | 105.62 | 90.06 | 102.33 | 87.00 |

0.5 m | 105.70 | 90.13 | 102.33 | 87.01 |

0.8 m | 105.86 | 90.27 | 102.36 | 87.04 |

1.0 m | 105.41 | 89.89 | 102.40 | 87.08 |

1.5 m | 108.15 | 91.43 | 105.33 | 89.34 |

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**MDPI and ACS Style**

Wang, Y.; Göttsche, U.; Abdel-Maksoud, M.
Sound Field Properties of Non-Cavitating Marine Propellers. *J. Mar. Sci. Eng.* **2020**, *8*, 885.
https://doi.org/10.3390/jmse8110885

**AMA Style**

Wang Y, Göttsche U, Abdel-Maksoud M.
Sound Field Properties of Non-Cavitating Marine Propellers. *Journal of Marine Science and Engineering*. 2020; 8(11):885.
https://doi.org/10.3390/jmse8110885

**Chicago/Turabian Style**

Wang, Youjiang, Ulf Göttsche, and Moustafa Abdel-Maksoud.
2020. "Sound Field Properties of Non-Cavitating Marine Propellers" *Journal of Marine Science and Engineering* 8, no. 11: 885.
https://doi.org/10.3390/jmse8110885