# Prediction of the Open-Water Performance of Ducted Propellers with a Panel Method

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Problem Definition

#### 2.2. Panel Code PROPAN

#### 2.3. RANS Code ReFRESCO

## 3. Results

#### 3.1. Grids and Numerical Set-Up

#### 3.2. Influence of the Wake Model

#### 3.3. Comparison Between PROPAN and ReFRESCO

#### 3.4. Prediction of the Open-Water Performance

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

IST | Instituto Superior Técnico |

MARIN | Maritime Research Institute Netherlands |

QUICK | Quadratic upwind interpolation for convective kinematics |

RANS | Reynolds-averaged Navier-Stokes |

RWM | Rigid wake model |

SIMPLE | Semi-implicit method for pressure linked equations |

SST | Shear stress transport |

WAM | Wake alignment model |

## References

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**Figure 3.**Pressure Kutta condition for the duct trailing edge with a thick round geometry and L denoting the duct length.

**Figure 4.**Overview of the surface grids used for the inviscid calculations with PROPAN (

**a**) and RANS calculations with ReFRESCO (

**b**).

**Figure 5.**Panel arrangement for propeller blades, duct, hub and blade wake at $J=0.2$. Rigid wake model (

**a**); Wake alignment model (

**b**). Only one wake surface is shown.

**Figure 6.**Wake alignment model with reduced gap pitch of $P/D=0.9$. Detail of the gap strip and blade wake sheet (

**a**); Overview of one blade wake geometry for $J=0.2$ (

**b**).

**Figure 7.**Vortex pitch ${\beta}_{v}$ distribution at $x/R=0.2$ (

**a**) and $x/R=0.4$ (

**b**) for $J=0.2$. Influence of the wake model. The filled symbols refer to the tip vortex.

**Figure 8.**Wake geometry at $x/R=0.2$ (

**a**) and $x/R=0.4$ (

**b**) for $J=0.1$. The contours represent the ReFRESCO total vorticity field $|\overrightarrow{\omega}|/\mathrm{\Omega}$. The symbols represent the PROPAN wake geometry: rigid wake model (squares), wake alignment model (diamonds) and wake alignment model with reduced gap pitch (circles). The filled symbols refer to the tip vortex.

**Figure 9.**Blade chordwise pressure distribution at $r/R=0.95$ (

**a**); Duct chordwise pressure distribution at $\theta =0$ degrees (

**b**). $J=0.1$.

**Figure 10.**Wake geometry at $x/R=0.2$ (

**a**) and $x/R=0.4$ (

**b**) for $J=0.2$. The contours represent the ReFRESCO total vorticity field $|\overrightarrow{\omega}|/\mathrm{\Omega}$. The symbols represent the PROPAN wake geometry: rigid wake model (squares), wake alignment model (diamonds) and wake alignment model with reduced gap pitch (circles). The filled symbols refer to the tip vortex.

**Figure 11.**Blade chordwise pressure distribution at $r/R=0.95$ (

**a**); Duct chordwise pressure distribution at $\theta =0$ degrees (

**b**). $J=0.2$.

**Figure 12.**Wake geometry at $x/R=0.2$ (

**a**) and $x/R=0.4$ (

**b**) for $J=0.5$. The contours represent the ReFRESCO total vorticity field $|\overrightarrow{\omega}|/\mathrm{\Omega}$. The symbols represent the PROPAN wake geometry: rigid wake model (squares), wake alignment model (diamonds) and wake alignment model with reduced gap pitch (circles). The filled symbols refer to the tip vortex.

**Figure 13.**Blade chordwise pressure distribution at $r/R=0.95$ (

**a**); Duct chordwise pressure distribution at $\theta =0$ degrees (

**b**). $J=0.5$.

**Figure 14.**Wake geometry at $z=0$ for $J=0.2$. The contours represent the ReFRESCO total vorticity field $|\overrightarrow{\omega}|/\mathrm{\Omega}$. The symbols represent the PROPAN wake geometry: rigid wake model (squares), wake alignment model (diamonds) and wake alignment model with reduced gap pitch (circles). The filled symbols refer to the tip vortex.

**Figure 15.**Comparison between numerical and experimental data from open-water tests. Propeller and duct thrust (

**a**); Propeller torque and open-water efficiency (

**b**).

**Table 1.**Variation of open-water characteristics with different grid sizes compared to the finest grid. Panel method computations at $J=0.5$.

Grid Size (Blade + Duct + Hub) ^{1} | $\mathbf{\Delta}{\mathit{K}}_{{\mathit{T}}_{\mathit{P}}}$ | $\mathbf{\Delta}{\mathit{K}}_{{\mathit{T}}_{\mathit{D}}}$ | $\mathbf{\Delta}{\mathit{K}}_{\mathit{Q}}$ |
---|---|---|---|

20 × 11 + 100 × 40 + 39 × 32 | −4.76% | −7.42% | −5.97% |

30 × 16 + 130 × 80 + 51 × 48 | −0.60% | −3.34% | −1.42% |

40 × 21 + 160 × 120 + 63 × 64 | −0.79% | −1.30% | −1.23% |

50 × 26 + 190 × 160 + 75 × 80 | −0.49% | −0.37% | −0.74% |

60 × 31 + 220 × 200 + 87 × 96 | −0.15% | −0.19% | −0.25% |

70 × 36 + 250 × 240 + 98 × 112 | – | – | – |

^{1}The grid size refers to the chordwise and spanwise radial directions for each propeller blade, and to the streamwise and circumferential directions for the duct and hub.

**Table 2.**Variation of open-water characteristics with different grid sizes compared to the finest grid with M denoting million. RANS computations at $J=0.5$.

Grid Size | $\mathbf{\Delta}{\mathit{K}}_{{\mathit{T}}_{\mathit{P}}}$ | $\mathbf{\Delta}{\mathit{K}}_{{\mathit{T}}_{\mathit{D}}}$ | $\mathbf{\Delta}{\mathit{K}}_{\mathit{Q}}$ |
---|---|---|---|

1.1 M | 3.25% | 2.39% | 3.55% |

2.6 M | 2.78% | 2.78% | 2.75% |

7.7 M | 0.67% | 1.99% | 0.69% |

12.9 M | 0.21% | 0.99% | 0.28% |

26.8 M | – | – | – |

**Table 3.**Inviscid thrust and torque coefficients for $J=0.1$, $0.2$ and $0.5$ and comparison with experimental data [23]. Influence of the wake model.

Model | ${\mathit{K}}_{{\mathit{T}}_{\mathit{P}}}$ | ${\mathit{K}}_{{\mathit{T}}_{\mathit{D}}}$ | $10{\mathit{K}}_{\mathit{Q}}$ |
---|---|---|---|

$\mathit{J}=\mathbf{0.1}$ | |||

RWM | 0.412 | 0.206 | 0.5882 |

WAM | 0.313 | 0.231 | 0.4664 |

WAM with Reduced Gap Pitch | 0.284 | 0.226 | 0.4228 |

Experiments | 0.254 | 0.214 | 0.4387 |

$\mathit{J}=\mathbf{0.2}$ | |||

RWM | 0.383 | 0.160 | 0.5538 |

WAM | 0.297 | 0.176 | 0.4456 |

WAM with Reduced Gap Pitch | 0.273 | 0.171 | 0.4083 |

Experiments | 0.248 | 0.166 | 0.4279 |

$\mathit{J}=\mathbf{0.5}$ | |||

RWM | 0.266 | 0.054 | 0.4041 |

WAM | 0.208 | 0.057 | 0.3246 |

WAM with Reduced Gap Pitch | 0.193 | 0.055 | 0.3010 |

Experiments | 0.196 | 0.053 | 0.3506 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Baltazar, J.M.; Rijpkema, D.; Falcão de Campos, J.; Bosschers, J.
Prediction of the Open-Water Performance of Ducted Propellers with a Panel Method. *J. Mar. Sci. Eng.* **2018**, *6*, 27.
https://doi.org/10.3390/jmse6010027

**AMA Style**

Baltazar JM, Rijpkema D, Falcão de Campos J, Bosschers J.
Prediction of the Open-Water Performance of Ducted Propellers with a Panel Method. *Journal of Marine Science and Engineering*. 2018; 6(1):27.
https://doi.org/10.3390/jmse6010027

**Chicago/Turabian Style**

Baltazar, João Manuel, Douwe Rijpkema, José Falcão de Campos, and Johan Bosschers.
2018. "Prediction of the Open-Water Performance of Ducted Propellers with a Panel Method" *Journal of Marine Science and Engineering* 6, no. 1: 27.
https://doi.org/10.3390/jmse6010027