# Application of Large Eddy Simulation to Predict Underwater Noise of Marine Propulsors. Part 1: Cavitation Dynamics

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## Abstract

**:**

## 1. Introduction

#### 1.1. State of the Art

- Turbulence;
- Two-phase flow;
- Noise source description and propagation.

#### 1.2. Contributions of Current Work

## 2. Methods

#### 2.1. Methodology

#### 2.1.1. Resolving Turbulence

#### 2.1.2. Two-Phase Flow

- Viscous effects;
- Nonspherical bubble shape;
- Interface surface tension and sliding interfaces;
- Bubble interactions;
- Compressibility.

#### 2.2. Numerical Setup

- To ensure that non-axisymmetric turbulence and cavity structures at the symmetry axis are resolved;
- In subsequent behind-ship condition setups, the non-axisymmetric inflow caused by the ship wakefield or inclined shaft lines and possible structural obstacles in the propeller slipstream, such as rudders or azimuthing propulsor housings, have to be considered.

## 3. Results

#### 3.1. PPTC’11

- The source of the formation of the cavity is the reduced pressure due to the accelerated flow (a–c), in accordance with Bernoulli’s principle;
- The assumed elliptical shape of the vortex is the result of the orientation of the detaching sheet along the blade surface;
- The conserved angular momentum along the running length maintains the cavity diameter, in accordance with Kelvin’s circulation theorem;
- The deformation of the ellipse to an energetically more favorable circular cross-sectional shape of the cavity takes place in the regions (d,e) vs. (f,g), as the surface tension is not considered in the Schnerr-Sauer model.

#### 3.2. Newcastle Propeller Test Case

## 4. Discussion

#### 4.1. Forces Prediction

#### 4.2. Noise Generation Mechanisms

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Different modes of a vibrating core vortex: (

**a**) breathing mode $n=0$; (

**b**) bending mode $n=1$; (c) double helix mode $n=2$, where $n$ is the azimuthal wave number based on the formulation in a cylindrical coordinate system, reproduced from [9] with permission from Verlag Schriftenreihe Schiffbau, year 2021.

**Figure 3.**PPTC’11. Distance refinement in tip vortex region: (

**a**) first mesh refinement step based on $Q$-criterion $Q=5\cdot {10}^{4}{\mathrm{s}}^{-2}$; (

**b**) second mesh refinement step based on $\alpha =0.5$ with additional hub vortex refinement.

**Figure 4.**PPTC’11. (

**a**) Experiment, reproduced from [26], with permission from SVA Potsdam, 2021; (

**b**) initial mesh; (

**c**) mesh refinement 1; (

**d**) mesh refinement 2.

**Figure 5.**PPTC’11. Tip vortex cavitation isosurface with investigated stations and normalized arc length approximation along helix.

**Figure 6.**PPTC’11. (

**a**–

**h**) Relative velocity plots with pressure contours in helix-normal plane at stations ${t}_{0}$–${t}_{7}$, respectively, with phase interface in magenta.

**Figure 7.**PPTC’11. Mesh comparison: (

**a**) helix-normal plane station ${t}_{3}$; (

**b**) helix-normal plane station ${t}_{4}$; (

**c**) shaft-normal plane at $x=0.4\cdot {D}_{P}$.

**Figure 8.**PPTC’11. Comparison of relative in-plane velocity with inviscid 2D analytical vortices angular velocity: (

**a**) station ${t}_{3}$; (

**b**) station ${t}_{4}$.

**Figure 10.**PPTC’11. Pressure distribution in the slipstream: (

**a**) pressure contours in midplane with investigated lines; (

**b**) pressure plots along investigated lines.

**Figure 11.**Newcastle. Comparison with experiment, reproduced from [15], with permission from Elsevier, year 2021: (

**a**) C1; (

**b**) C2; (

**c**) C3; (

**d**) C6.

**Figure 12.**Newcastle. Tip vortex cavitation isosurface with stations and normalized arc length approximation along helix: (

**a**) C1; (

**b**) C2; (

**c**) C3; (

**d**) C6.

**Figure 13.**Newcastle. Tip vortex isosurfaces for cavitation ${\alpha}_{0.5}$ (blue) and Q-criterion $Q=5\cdot {10}^{4}{\mathrm{s}}^{-2}$ (green): (

**a**) overview; (

**b**) single vortex view.

**Figure 14.**Newcastle. C2 tip vortex cavitation isosurface with two stations and two different refinement levels: (

**a**) Base a priory; (

**b**) 8× higher cell count in the tip vortex region.

**Figure 15.**Newcastle. Tip vortex longitudinal evolution analysis: (

**a**) C2: shape; (

**b**) C2: flow rate; (

**c**) C6: shape; (

**d**) C6: flow rate.

**Figure 16.**Possible mechanisms of tip vortex cavity shape formation: (

**a**) unwrapped lateral view; (

**b**) cross-section detail.

Parameter | ${\mathit{n}}_{\mathbf{0}}$ | ${\mathit{d}}_{\mathit{Nuc}}$ |
---|---|---|

Unit | $1/{m}^{3}$ | $m$ |

Value | $1\cdot {10}^{12}$ | $1\cdot {10}^{-4}$ |

Condition | C1 | C2 | C3 | C6 |
---|---|---|---|---|

$J\left[-\right]$ | $0.4$ | $0.4$ | $0.4$ | $0.5$ |

${\sigma}_{n}\left[-\right]$ | $2.22$ | $1.3$ | $0.72$ | $1.13$ |

$n$ [Hz] | 35 | 35 | 35 | 35 |

**Table 3.**PPTC’11. Effect of time step size on the propeller slipstream vorticity for an identical mesh without mesh refinement step at $J=1.019$, partly based on [25].

Type | Step [°] | Q-Isovalue of $\mathit{Q}=5\cdot {10}^{4}{\mathbf{s}}^{-2}$ with Axial Velocity | $\mathit{CFL}$ [-] | Effort (2 rot/100 CPUs) [d] |
---|---|---|---|---|

RANS | 10 | 1.38–550.49 | 0.1 | |

RANS | 1 | 0.13–72.38 | 0.32 | |

RANS | 0.1 | 0.01–7.33 | 3.4 | |

ILES | 1 | 0.13–66.20 | 0.4 | |

ILES | 0.1 | 0.01–6.95 | 3.0 | |

ILES | 0.1 | 0.001–0.66 | 28.0 |

**Table 4.**Newcastle. Effect of different time steps on the propeller slipstream vorticity with ILES for Newcastle propeller.

Step [°] | Q-Isovalue of $\mathit{Q}=5\cdot {10}^{4}{\mathbf{s}}^{-2}$ with Axial Velocity | $\mathit{CFE}$ [-] | Effort (2 rot/100 CPUs) [d] |
---|---|---|---|

$0.33$ | 0.03–11.51 | 0.72 | |

$0.1$ | 0.00–3.44 | 1.87 |

Cell Count [${10}^{6}$ Cells] | ||
---|---|---|

Mesh | PPTC’11 | Newcastle |

Initial mesh | $11.4$ | $13.0$ |

Refinement step 1 | $65.2$ | C1: $32.3$ C2: $24.9$ C3: $27.5$ C6: $40.1$ |

Refinement step 2 | $35.5$ | - |

Investigation | ${\mathit{k}}_{\mathit{T}}$ [-] | Δ [%] | $10{\mathit{k}}_{\mathit{Q}}$ | Δ [%] |
---|---|---|---|---|

Experiment (Cav Off) | 0.381 | - | 0.976 | - |

Experiment (Cav On) | 0.374 | - | 0.970 | - |

RANS k-ω-SSTawt: tdiCoupled (Cav Off) | 0.381 | 0.1 | 1.008 | 3.3 |

ILES: pimpleDyMFoam (Cav Off) | 0.396 | 4.1 | 0.995 | 1.9 |

LES Sm..: interPhaseChangeDyMFoam (Cav Off) | 0.383 | 0.4 | 1.027 | 5.2 |

ILES: interPhaseChangeDyMFoam (Cav On) | 0.429 | 14.9 | 0.995 | 2.6 |

Investigation | ${\mathrm{k}}_{\mathrm{T}}$ [-] | Δ [%] | $10{k}_{Q}$ | Δ [%] |
---|---|---|---|---|

$J=0.4$-Condition: C1, C2, C3 | ||||

Experiment | 0.242 | - | 0.341 | - |

RANS k-ω-SST: sdiCoupled (steady, Cav Off) | 0.243 | −0.6 | 0.340 | 0.3 |

ILES: interPhaseChangeDyMFoam (Cav On) | 0.231 | 4.6 | 0.358 | 0.9 |

$J=0.5$-Condition: C6 | ||||

Experiment | 0.190 | - | 0.284 | - |

RANS k-kl-ω: interPhaseChangeDyMFoam (Cav Off) | 0.182 | −4.1 | 0.271 | −4.5 |

ILES: interPhaseChangeDyMFoam (Cav On) | 0.204 | 7.2 | 0.318 | 11.9 |

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**MDPI and ACS Style**

Kimmerl, J.; Mertes, P.; Abdel-Maksoud, M.
Application of Large Eddy Simulation to Predict Underwater Noise of Marine Propulsors. Part 1: Cavitation Dynamics. *J. Mar. Sci. Eng.* **2021**, *9*, 792.
https://doi.org/10.3390/jmse9080792

**AMA Style**

Kimmerl J, Mertes P, Abdel-Maksoud M.
Application of Large Eddy Simulation to Predict Underwater Noise of Marine Propulsors. Part 1: Cavitation Dynamics. *Journal of Marine Science and Engineering*. 2021; 9(8):792.
https://doi.org/10.3390/jmse9080792

**Chicago/Turabian Style**

Kimmerl, Julian, Paul Mertes, and Moustafa Abdel-Maksoud.
2021. "Application of Large Eddy Simulation to Predict Underwater Noise of Marine Propulsors. Part 1: Cavitation Dynamics" *Journal of Marine Science and Engineering* 9, no. 8: 792.
https://doi.org/10.3390/jmse9080792