# Efficiency Enhancement of Marine Propellers via Reformation of Blade Tip-Rake Distribution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model and CFD Code

#### 2.1. The Vortex-Lattice Model

#### 2.2. CFD Code

**ω**. Let

**r**= (x, y, z) be the position vector, re-writing Equation (9) with respect to the relative velocity vector ${\mathbf{u}}_{r}=\mathbf{u}-\mathsf{\omega}\times \mathbf{r}$

**,**and after some algebraic manipulations the following expressions are obtained:

**u**), vector ${\mathbf{S}}_{q}$ contains the various source terms of the equations, such as the Coriolis forces, ${\mathbf{F}}_{c}$ and ${\mathbf{F}}_{v}$ are the vectors of convective and diffusive fluxes normal to a face, respectively. The two flux vectors are given by Equation (11). By $\mathsf{\Delta}V$, we express the velocity difference between the contravariant velocity ${V}_{n}=\mathbf{u}\xb7\mathbf{n}$ and the grid face velocity due to the mesh motion ${V}_{g}=\left(\mathsf{\omega}\times \mathbf{r}\right)\xb7\mathbf{n}$, where $\mathbf{n}=({n}_{x},{n}_{y},{n}_{z})$. The convective and viscous fluxes are

## 3. Parameterization and Optimization Methodology

- Select a reference propeller geometry and produce the open water curves using both computational tools. During this process, if available, the VLM coefficients (leading-edge suction force, friction drag) can be calibrated using the available experimental data.
- Determine the upper/lower bounds of the selected design variables (rake, pitch, maximum camber, etc.) and perform an optimization study using VLM. Gradient-based methods can be sensitive to initial design vector selection and may be prone to locate the local optima. A remedy to this, which is considered common practice, is to solve the same optimization problem starting from different initial design vectors and keeping the best candidate solution among the results. This approach to optimization is implemented here.
- Then, perform viscous-CFD simulations using MaPFlow to predict the open water performance of the modified propeller near the design point and quantify the performance gain due to geometry modification.
- Finally, calibrate VLM using available data from the viscous-CFD simulations and predict the open water curves at J = {0.4–1}. Essentially, calibration occurs in the sense of best fit between the CFD data and the open water curves.

#### 3.1. Parametric Model for the Tip-Rake Reformed Geometry

#### 3.2. Optimization Problem

## 4. Results

#### 4.1. Verification

#### 4.1.1. VLM Sensitivity Analysis

_{T}), the torque coefficient (10 K

_{Q}), and the efficiency (η).

#### 4.1.2. CFD Sensitivity Analysis

#### 4.1.3. Open Water Curves for Reference Propeller Models 4381 and 4382

#### 4.2. Propeller Performance Improvement by Blade Tip Geometry Reformation

_{1}), for both propellers the optimal value is close to 70% of tip radius R.

_{2}) in both cases corresponds to a suction-side rake, as shown in Figure 9 and Figure 10 for the modified 4381 and 4382 propellers, respectively. Finally, the optimal pitch and maximum camber proportional coefficients are smaller than the corresponding value of the reference geometry. The geometric parameters for the reference and modified propellers are provided in Appendix A. Regarding the CFD simulations for both the initial and modified geometries, we present below the selected plots. In Figure 9, contour plots concerning calculated velocity and pressure fields are presented on the vertical xy plane for both the initial 4381 propeller geometry and the modified one.

_{T}and K

_{Q}for 4382 obtained by using VLM, an increase in both coefficients is observed using CFD. This observation relies on the fact that VLM is calibrated using the experimental data of the original geometry.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Propeller Geometries

r/R | c/D | P/D | t_{max}/c | f_{max}/c | θs | X_{R}/R |
---|---|---|---|---|---|---|

0.2000 | 0.1740 | 1.3320 | 0.2494 | 0.0351 | 0 | 0 |

0.2500 | 0.2020 | 1.3380 | 0.1960 | 0.0369 | 0 | 0 |

0.3000 | 0.2290 | 1.3450 | 0.1562 | 0.0368 | 0 | 0 |

0.4000 | 0.2750 | 1.3580 | 0.1068 | 0.0348 | 0 | 0 |

0.5000 | 0.3120 | 1.3360 | 0.0768 | 0.0307 | 0 | 0 |

0.6000 | 0.3370 | 1.2800 | 0.0566 | 0.0245 | 0 | 0 |

0.7000 | 0.3470 | 1.2100 | 0.0421 | 0.0191 | 0 | 0 |

0.8000 | 0.3340 | 1.1370 | 0.0314 | 0.0148 | 0 | 0 |

0.9000 | 0.2800 | 1.0660 | 0.0239 | 0.0123 | 0 | 0 |

0.9500 | 0.2100 | 1.0310 | 0.0229 | 0.0128 | 0 | 0 |

1.0000 | 0.0100 | 0.9950 | 0.0160 | 0.0123 | 0 | 0 |

r/R | c/D | (P/D) | t_{max}/c | (f_{max}/c) | θs | (X_{R}/R) |
---|---|---|---|---|---|---|

0.2000 | 0.1740 | 1.3120 | 0.2494 | 0.0342 | 0 | 0 |

0.2500 | 0.2020 | 1.3173 | 0.1960 | 0.0358 | 0 | 0 |

0.3000 | 0.2290 | 1.3244 | 0.1562 | 0.0356 | 0 | 0 |

0.4000 | 0.2750 | 1.3369 | 0.1068 | 0.0336 | 0 | 0 |

0.5000 | 0.3120 | 1.3135 | 0.0768 | 0.0295 | 0 | 0 |

0.6000 | 0.3370 | 1.2568 | 0.0566 | 0.0234 | 0 | 0 |

0.7000 | 0.3470 | 1.1869 | 0.0421 | 0.0182 | 0 | 0 |

0.8000 | 0.3340 | 1.1139 | 0.0314 | 0.0141 | 0 | 0.0213 |

0.9000 | 0.2800 | 1.0454 | 0.0239 | 0.0119 | 0 | 0.1010 |

0.9500 | 0.2100 | 1.0122 | 0.0229 | 0.0123 | 0 | 0.1628 |

1.0000 | 0.0100 | 0.9795 | 0.0160 | 0.0119 | 0 | 0.2392 |

r/R | c/D | P/D | t_{max}/c | f_{max}/c | θs | X_{R}/R |
---|---|---|---|---|---|---|

0.2000 | 0.1740 | 1.4550 | 0.2494 | 0.0430 | 0 | 0 |

0.2500 | 0.2020 | 1.4440 | 0.1960 | 0.0395 | 2.3280 | 0 |

0.3000 | 0.2290 | 1.4330 | 0.1562 | 0.0370 | 4.6550 | 0 |

0.4000 | 0.2750 | 1.4120 | 0.1068 | 0.0344 | 9.3630 | 0 |

0.5000 | 0.3120 | 1.3610 | 0.0768 | 0.0305 | 13.9480 | 0 |

0.6000 | 0.3370 | 1.2850 | 0.0566 | 0.0247 | 18.3780 | 0 |

0.7000 | 0.3470 | 1.2000 | 0.0421 | 0.0199 | 22.7470 | 0 |

0.8000 | 0.3340 | 1.1120 | 0.0314 | 0.0161 | 27.1450 | 0 |

0.9000 | 0.2800 | 1.0270 | 0.0239 | 0.0134 | 31.5750 | 0 |

0.9500 | 0.2100 | 0.9850 | 0.0229 | 0.0140 | 33.7880 | 0 |

1.0000 | 0.0100 | 0.9420 | 0.0160 | 0.0134 | 36.000 | 0 |

r/R | c/D | P/D | t_{max}/c | f_{max}/c | θs | X_{R}/R |
---|---|---|---|---|---|---|

0.2000 | 0.1740 | 1.4539 | 0.2494 | 0.0426 | 0 | 0 |

0.2500 | 0.2020 | 1.4439 | 0.1960 | 0.0395 | 2.3280 | 0 |

0.3000 | 0.2290 | 1.4293 | 0.1562 | 0.0373 | 4.6550 | 0 |

0.4000 | 0.2750 | 1.3945 | 0.1068 | 0.0361 | 9.3630 | 0 |

0.5000 | 0.3120 | 1.3395 | 0.0768 | 0.0322 | 13.9480 | 0 |

0.6000 | 0.3370 | 1.2631 | 0.0566 | 0.0260 | 18.3780 | 0 |

0.7000 | 0.3470 | 1.1783 | 0.0421 | 0.0209 | 22.7470 | 0.0027 |

0.8000 | 0.3340 | 1.0902 | 0.0314 | 0.0168 | 27.1450 | 0.0432 |

0.9000 | 0.2800 | 1.0080 | 0.0239 | 0.0142 | 31.5750 | 0.1216 |

0.9500 | 0.2100 | 0.9680 | 0.0229 | 0.0149 | 33.7880 | 0.1751 |

1.0000 | 0.0100 | 0.9290 | 0.0160 | 0.0143 | 36.000 | 0.2379 |

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**Figure 1.**(

**a**) Vortex element mesh on propeller blades with positive tip−rake (towards suction side) and corresponding trailing vortex wake mesh. The trailing edge of the blades is shown by using black lines. (

**b**) Schematic representation of the vortex−element mesh and control points on the mean camber surface.

**Figure 2.**The basic computational setup used for the CFD simulation. Typically, one blade is modelled with a periodic boundary condition.

**Figure 4.**(

**a**) Pitch and (

**b**) maximum camber distributions, using 1-dof parametrization and modification based on a B-Spline interpolation. Active control points (for r/R > 0.5) are highlighted with red squares.

**Figure 5.**Vortex−ring element mesh 15 × 8 with cosine−spacing spanwise for propeller 4381. (

**a**) 3D view, (

**b**) side view, and (

**c**) plan view.

**Figure 6.**Open water curves for 4381. Comparison between experimental data [29], vortex-lattice results with ${C}_{Drag}$ = 0.0045, ${C}_{LE}$ = 0.93, and MaPFlow (dashed line). Symbol characterization: thrust coefficient (triangles), moment coefficient (squares), and efficiency (circles).

**Figure 7.**Open water curves for 4382. Comparison between experimental data [29], vortex-lattice results, with ${C}_{Drag}$ = 0.0050, ${C}_{LE}$ = 0.90, and MaPFlow (dashed line). Symbol characterization: thrust coefficient (triangles), moment coefficient (squares), and efficiency (circles).

**Figure 8.**Post−processing vortex-lattice results for (

**a**) 4381 and (

**b**) 4382. Pressure difference on the mean camber blade surface with calculated mean velocity for J = 0.889.

**Figure 9.**xy−plane view of the flow domain for 4381 by means of CFD simulations at J = 0.889: (

**a**,

**b**) pressure contour (in kPa), (

**c**,

**d**) x−component of the velocity (in m/s), (

**a**,

**c**) original geometry (zero rake), and (

**b**,

**d**) modified geometry (positive rake close to the tip).

**Figure 10.**xy−plane view of the flow domain for propeller 4382 by means of CFD simulations at J = 0.889: (

**a**,

**b**) pressure contour (in kPa), (

**c**,

**d**) x−component of the velocity (in m/s), (

**a**,

**c**) original geometry (zero rake), and (

**b**,

**d**) modified geometry (positive rake close to the tip).

**Figure 11.**Pressure contour (in kPa) for 4381 from CFD simulations at J = 0.889 on a cylindrical section at 95% of the propeller tip radius: (

**a**) original geometry (zero rake), and (

**b**) modified geometry (positive rake close to the tip).

**Figure 12.**Pressure contour (in kPa) for propeller 4382 by means of CFD simulations at J = 0.889 on a cylindrical section at 95% of the propeller tip radius: (

**a**) original geometry (zero rake), and (

**b**) modified geometry (positive rake close to the tip).

**Figure 13.**Pressure contour plot (in kPa) on the suction side and stream lines based on CFD results for propeller model 4381, operating at J = 0.889: (

**a**) reference and (

**b**) tip-rake-modified blade geometry.

**Figure 14.**Pressure contour plot (in kPa) on the suction side and streamlines based on CFD results for the skewed propeller model 4382, operating at J = 0.889: (

**a**) reference and (

**b**) tip-rake-modified blade geometry.

**Figure 15.**Calculated pressure coefficient (−Cp) at four blade sections at r/R = 0.3, 0.5, 0.7, 0.9, between initial (dash–dot lines) and tip-rake-modified (solid lines) for (

**left**) 4381 and (

**right**) 4382 propeller models. In the horizontal axis, 0.0 indicates the position of the leading edge.

**Figure 16.**Comparison between the pressure coefficient (−Cp) distributions obtained using viscous-CFD and VLM for the 4381 reference and modified propellers for the radial position r/R = 0.7.

**Figure 17.**Comparison between the reference and modified propeller open water curves obtained using VLM and CFD for 4381 [

**left**] and 4382 [

**right**] propeller models.

ID | Description | Lower Bound | Upper Bound |
---|---|---|---|

x_{1} | Rake transition point | 0.30 | 0.90 |

x_{2} | Maximum rake (Xr/R) | −0.12D/R | 0.12D/R |

x_{3} | Pitch proportional coeff. | 0.95 | 1.05 |

x_{4} | Max. camber proportional coeff. | 0.85 | 1.25 |

Grid Mesh Sizes. Diff% Compared to Finer Grid Results | |||||||
---|---|---|---|---|---|---|---|

Exp.data | (11 × 6) | (13 × 7) | (15 × 8) | (20 × 10) | (30 × 15) | (40 × 20) | |

K_{T} | 0.208 | 4.39 | 3.90 | 1.95 | 0.970 | 0.00 | - |

10 K_{Q} | 0.445 | 4.35 | 3.89 | 2.28 | 0.91 | −0.22 | - |

η | 0.661 | −0.301 | −0.301 | −0.301 | −0.151 | −0.151 | - |

Cells (Million) | Err K_{T} (%) | Err K_{Q} (%) | |
---|---|---|---|

Coarse | 2.3 | 4.7 | 3.42 |

Mid | 7.5 | 0.72 | 1.127 |

Dense | 12.7 | - | - |

Cells (Million) | Err K_{T} (%) | Err K_{Q} (%) | |
---|---|---|---|

Coarse | 2.3 | 7.01 | 10.08 |

Mid | 7.5 | 5.00 | 7.19 |

Dense | 12.7 | 3.90 | 6.40 |

ID | Description | 4381 | 4382 |
---|---|---|---|

x_{1} | Rake transition point | 0.7136 | 0.6430 |

x_{2} | Maximum rake (Xr/R) | 0.2397 | 0.2414 |

x_{3} | Pitch proportional coeff. | 0.9845 | 0.9862 |

x_{4} | Max. camber proportional coeff. | 0.9689 | 1.0635 |

Active control points | all | r/R > 0.5 |

4381 | 4382 | |||
---|---|---|---|---|

Reference | Modified | Reference | Modified | |

${C}_{Drag}$ | 0.0055 | 0.0037 | 0.0054 | 0.0041 |

${C}_{LE}$ | 0.90 | 0.97 | 0.89 | 0.97 |

4381 | 4382 | |||
---|---|---|---|---|

VLM | MaPFlow | VLM | MaPFlow | |

dK_{T} (%) | −3.228 | −4.265 | −3.478 | 2.787 |

dK_{Q} (%) | −3.989 | −5.326 | −4.637 | 2.308 |

dη (%) | 1.857 | 1.122 | 1.216 | 0.468 |

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## Share and Cite

**MDPI and ACS Style**

Anevlavi, D.; Zafeiris, S.; Papadakis, G.; Belibassakis, K.
Efficiency Enhancement of Marine Propellers via Reformation of Blade Tip-Rake Distribution. *J. Mar. Sci. Eng.* **2023**, *11*, 2179.
https://doi.org/10.3390/jmse11112179

**AMA Style**

Anevlavi D, Zafeiris S, Papadakis G, Belibassakis K.
Efficiency Enhancement of Marine Propellers via Reformation of Blade Tip-Rake Distribution. *Journal of Marine Science and Engineering*. 2023; 11(11):2179.
https://doi.org/10.3390/jmse11112179

**Chicago/Turabian Style**

Anevlavi, Dimitra, Spiros Zafeiris, George Papadakis, and Kostas Belibassakis.
2023. "Efficiency Enhancement of Marine Propellers via Reformation of Blade Tip-Rake Distribution" *Journal of Marine Science and Engineering* 11, no. 11: 2179.
https://doi.org/10.3390/jmse11112179