# Numerical Analysis on Hydrodynamic Characteristics of Surface Piercing Propellers in Oblique Flow

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Geometry Model

_{t}= h/D. h represents the distance from blade tip to free surface and D is the diameter of the propeller. The vertical adjustable angle of SPP-1 is ±7°. The hydrodynamic characteristics of SPP-1 are predicted in horizontal flow, φ = 0°, and oblique flow, φ = 7°. φ represents the inclination angle as shown in Figure 3.

_{X}, and vertical resultant force, F

_{Z}, are obtained from the flowing equations:

_{x}and F

_{z}represent the axial thrust and vertical force in local coordinate, respectively.

#### 2.2. Governing Equations

_{f}is the film thickness. The volume V and the surface A are functions of the film thickness and its spatial distribution. ρ

_{f}is the film density,

**v**

_{f}is the film velocity,

**v**

_{g}is the grid velocity, and the subscript f denotes the fluid film values. The quantity s

_{m}is the mass source/sink per unit area. p

_{f}is the pressure,

**f**

_{b}is the body force,

**T**

_{f}is the viscous stress tensor within the film.

**s**

_{m}is the momentum source corresponding to the mass source s

_{m}.

#### 2.3. Grid Generation

#### 2.4. Boundary and Initial Conditions

^{−5}s, with the surface piercing propeller rotating 0.5 per time step. The time step is chosen to make courant number no more than 1, which is suggested by Blocken and Gualtieri [23].

## 3. Verification of Numerical Method

#### 3.1. Grid Independence

#### 3.2. Hydrodynamic Performance

_{A}. Figure 7 shows the comparison between the predicted results and experimental data of PSP-841B. As we can see, the calculated values are in good agreement with the experimental ones. With the growth of the advance ratios J, the error of thrust and torque coefficients varies from positive to negative, and the discrepancy near the design point is less than 5%. We can also investigate the transition zone of PSP-841B where the thrust and torque coefficients are unstable when 0.90 < J < 1.0. The predicted results demonstrate that the present numerical schemes are effective and accurate for the prediction of hydrodynamic characteristic of surface piercing propellers. So, the present numerical method is adopted in the simulations of SPP-1.

## 4. Numerical Results and Analysis

_{t}= 0.75. The inclination angles of SPP-1 in vertical direction are adjustable from −7° to 7°. Both conditions are selected to analyze the hydrodynamic characteristics of SPP-1 in horizontal and oblique flow, in which the inclination angles are 0° and 7°, respectively.

#### 4.1. Free Surface

#### 4.2. Hydrodynamic Performance

_{t}and 10 Kq are consistent with the experiments in the literature [7].

_{y}and KF

_{z}represent the forces in lateral and vertical directions. KM

_{y}and KM

_{z}represent the moments in lateral and vertical directions. At high load conditions, a larger amount of air attached to the blade makes the cascade effect more pronounced, and the air makes the frictional resistance of the blades less. As the advance ratio increases, the ventilation mode of blades changes from the full ventilation to partially vented. In horizontal flow, the vertical force KF

_{z}and lateral moment KM

_{y}at J = 1.0 increase more dramatically than that at J = 0.8 because the vented mode of SPP-1 changes from fully ventilation to partially vented mode. While SPP-1 is fully vented at J = 1.0 in oblique flow, the divergence of forces/moments are very little. At J = 1.2, the lateral force KF

_{y}and moment KM

_{y}increase greatly in oblique flow than that at J = 1.0. This indicates that the transition zone is between at J = 1.0 and J = 1.2. So, the transition zone is postponed in oblique flow, compared with the horizontal flow case. At the highest advance ratio, J = 1.4, only the blades’ tips are vented in both cases and there is very little effect of air cavities on the hydrodynamic performance.

#### 4.3. Pressure on the Blades

_{t}= 0.75. θ represents the phase angle.

#### 4.4. Wake Field

_{a}/V

_{A}, is significantly reduced in the oblique flow in front of the propeller disk at x/D = 0.237, as presented in Figure 19. Simultaneously, due to the uplift of the free surface, the influence of the rotation of the propeller on the air is weakened significantly, and the high-speed region above the free surface disappears. At the propeller disk, x/D = 0, the ventilation region on the suction side in the oblique flow is larger than that in the horizontal flow. At x/D = −0.237, it is more obvious that the positive and negative speeds alternate in oblique flow. In the oblique flow, more air is attached to the hub of the propeller, so that the pressure is lower than the results in the horizontal case.

_{t}/V

_{A}, upstream in the oblique flow is higher than that in the horizontal flow, especially near the shaft. At the propeller disk, there is a larger area of the high-speed region (red region), which is caused by the air along with the propeller. Since the free surface is uplifted by the air cavities, the velocity above the free surface is relatively low at x/D = −0.237 in the oblique flow.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Three-dimensional view of modeling and actual propellers. (

**a**) The 3D view of PSP-841B—

**left**is the actual propeller and

**right**is the 3D model; (

**b**) the 3D view of SPP-1—

**left**is the actual propeller and

**right**is the 3D model.

**Figure 8.**Free surface of SPP-1 in horizontal and oblique flow. (

**a**) The free surface of SPP-1 in horizontal (

**left**) and oblique (

**right**) flow at J = 0.8; (

**b**) the free surface of SPP-1 in horizontal (

**left**) and oblique (

**right**) flow at J = 1.4.

**Figure 13.**Forces/moments acting on SPP-1 in horizontal (HF) and oblique flow (OF). (

**a**) Forces acting on SPP-1 in horizontal (HF) and oblique flow (OF); (

**b**) moments acting on SPP-1 in horizontal (HF) and oblique flow (OF).

**Figure 15.**Pressure distribution on blade surfaces. (

**a**) The pressure distribution on pressure side; (

**b**) the pressure distribution on suction side.

**Figure 21.**Comparison of dimensionless velocity contours near Blade No. 1 at 0.4 R and J = 1.4 (

**left**: horizontal flow;

**right**: oblique flow).

Parameters | PSP-841B | SPP-1 |
---|---|---|

D (mm) | 250 | 633 |

H/D | 0.34 | 0.15 |

P/D at 0.7r | 1.24 | 1.60 |

EAR | 0.58 | 0.83 |

Blades Number | 4 | 4 |

Rotation Direction | right | left |

Grid Number (Million) | K_{t} | Error | 10 K_{q} | Error |
---|---|---|---|---|

EFD | 0.0511 | 0.1099 | ||

1.4 | 0.0572 | 12.1% | 0.1236 | 12.4% |

2.6 | 0.0560 | 9.5% | 0.1197 | 9.0% |

3.8 | 0.0551 | 7.8% | 0.1167 | 6.1% |

5.0 | 0.0546 | 6.9% | 0.1151 | 4.7% |

© 2019 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**

Ren, Z.; Hua, L.; Ji, P. Numerical Analysis on Hydrodynamic Characteristics of Surface Piercing Propellers in Oblique Flow. *Water* **2019**, *11*, 2015.
https://doi.org/10.3390/w11102015

**AMA Style**

Ren Z, Hua L, Ji P. Numerical Analysis on Hydrodynamic Characteristics of Surface Piercing Propellers in Oblique Flow. *Water*. 2019; 11(10):2015.
https://doi.org/10.3390/w11102015

**Chicago/Turabian Style**

Ren, Zhen, Lin Hua, and Penghui Ji. 2019. "Numerical Analysis on Hydrodynamic Characteristics of Surface Piercing Propellers in Oblique Flow" *Water* 11, no. 10: 2015.
https://doi.org/10.3390/w11102015