# Optimization of the Hydrodynamic Performance of a Double-Vane Otter Board Based on Orthogonal Experiments

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{x}and the lift coefficient C

_{y}of nine otter board models obtained in a wind tunnel experiment, and the lift–drag ratio K obtained by calculation. The lift–drag ratio for a working angle of attack of 30° was selected as the inspection index, and the experimental data were analyzed in an orthogonal design-direct analysis. Analysis of each factor revealed that the optimal level combination of factors was A

_{3}B

_{3}C

_{3}and that the decreasing order of the effects of the factors was A (aspect ratio) > B (gap ratio) > C (camber). The orthogonal experiment thus obtained an optimal otter board in terms of the aspect ratio (2.0), fore wing camber (0.16), and gap ratio (0.35), with the aspect ratio having the greatest effect on performance. The hydrodynamic performances of the otter board with the optimized structure and another otter board model were compared in numerical simulation, which verified the correctness of the analysis results. The experimental results provide a reference for the optimal design of the double-vane otter board.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Design and Manufacture of the Otter Board Model

^{2}. The structure and parameters of the model used in the present study are illustrated in Figure 1.

_{1}/L), the camber of the fore wing panel (h

_{B}/d

_{B}) and the aspect ratio (b/L). Each factor was set at three levels to determine its effect on performance. The selection of values refers to existing research and the application requirements of an otter board in mid-water trawling. The gap ratio was set at 0.25, 0.30, and 0.35, the wing panel camber was set at 0.12, 0.14, and 0.16, and the aspect ratio was set at 1.0, 1.5, and 2.0. An L

_{9}(3

^{4}) orthogonal experiment was designed as presented in Table 1 and models were constructed as presented in Table 2 to determine the best combination of factors [11].

^{2}. It follows that b ≤ 0.75 m and L·b ≤ 0.035 m

^{2}, and the chord length L of the nine otter board models is thus designed as L = 500 mm (b/L = 1.00), 400 mm (b/L = 1.50) and 300 mm (b/L = 2.00).

#### 2.2. Test Facility

^{2}. Figure 3 illustrates the experimental setup inside the wind tunnel. The model otter board was attached to a six-component strain-gage balance to measure forces in all directions.

#### 2.3. Test Method

#### 2.3.1. Parameter Definitions of the Test Model

^{−1}(at room temperature of 20 °C). Measurements were made at 2.5° intervals when the angle of attack was in the range of 0°–60° and at 5° intervals when the angle of attack was greater than 60°.

_{e}= VL/ν.

^{−1}and the coefficient of viscosity ν = 15 × 10

^{−6}m

^{2}·s

^{−1}. Equation (1) gives a Reynolds number R

_{e}of the nine otter boards under the above conditions between 0.57 × 10

^{6}and 0.94 × 10

^{6}, which indicates that this experimental design is in the automatic model area of the model test [14].

#### 2.3.2. Data Processing

_{x}, lift coefficient C

_{y}and lift–drag ratio K are expressed [15] as

_{x}= 2·ρ

^{−1}·S

^{−1}·V

^{−2}·F

_{x},

_{y}= 2·ρ

^{−1}·S

^{−1}·V

^{−2}·F

_{y},

_{y}/C

_{x},

_{y}and F

_{x}are the measured lift and drag forces (N), respectively, ρ (kg·m

^{−3}) is the air density, S is the projected area of the otter board (m

^{2}), and V is the actual velocity of incoming wind (m·s

^{−1}).

#### 2.3.3. Modeling and Meshing in Numerical Simulation

^{−3}), and ν is the kinematic viscosity.

^{−1}in the x-direction, the turbulence intensity was 2.86–3.05%, and the intensity and viscosity ratio was 772–1331. Table 4 gives the parameter settings used in the numerical simulation.

^{6}. Referring to Wang et al. [18], 4.5 × 10

^{6}elements are sufficient to ensure the accuracy of the numerical simulation. In subsequent numerical simulations of the present study, the number of cells was set at approximately 4.5 million. The solver Yplus around the otter board was in the range of 11.06–89.01. Figure 6 shows a cross-sectional view of the meshing around the otter board.

_{x}, lift coefficient C

_{y}and lift–drag ratio K were calculated.

## 3. Results

#### 3.1. Drag Coefficient, Lift Coefficient and Lift–Drag Ratio of the Otter Board

_{x}versus the angle of attack α of the incoming flow for the nine otter board models. The relationship between C

_{x}and α is largely linear but changes at an angle of attack of 40°–55°. In this angle range, there is a resistance-falling inflection zone except for otter board models 3 and 7.

_{y}versus the angle of attack α of the incoming flow for the nine otter board models. The maximum lift coefficient C

_{y}is highest for otter board model 6, being 2.221 (α = 37.5°). The maximum lift coefficient is lowest for otter board model 1, being 1.881 (α = 40°). The angle of attack corresponding to the maximum lift coefficient is called the critical angle of attack. The critical angle of attack ranges from 32.5° to 40° across the nine models is the largest for otter board model 3 and is the smallest for otter board models 7 and 8. The curves of the nine otter board models show the same trend of a rapid decline of the lift coefficient within an angle range of approximately 10° after the critical angle of attack.

#### 3.2. Orthogonal Analysis

_{j}, II

_{j}, and III

_{j}, the effects of the factors on the experimental results can be distinguished, and the levels can be optimally combined.

_{A}= 7.612, II

_{A}= 8.459 and III

_{A}= 8.927, III

_{A}> II

_{A}> I

_{A}, and it is seen that the optimal aspect ratio is 2.0. For factor B (the camber), I

_{B}= 8.323, II

_{B}= 8.349, and III

_{B}= 8.326, II

_{B}> III

_{B}> I

_{B}, and it is seen that the optimal camber of the fore wing is 0.14. For factor C (the gap ratio), I

_{C}= 8.141, II

_{C}= 8.313, and III

_{C}= 8.544, III

_{C}> II

_{C}> I

_{C}, and it is seen that the optimal gap ratio is 0.35. Without considering the interaction of the three factors, the optimal combination of levels is A

_{3}B

_{2}C

_{3}.

_{A}> R

_{C}> R

_{B}; i.e., the aspect ratio (factor A) is the most important factor, followed by the gap ratio (factor C) and finally the camber (factor B).

_{3}B

_{2}C

_{3}, with the aspect ratio strongly affecting the hydrodynamic performance of the otter board.

#### 3.3. Numerical Simulation Verification

_{x}, C

_{y}, and K between the numerical simulation and experimental results of otter board model 7 were approximately 8%, 5%, and 4%, respectively. The trends of the C

_{x}–α curve, C

_{y}–α curve, and K–α curve of otter board model 7 obtained in the numerical simulation are basically consistent with the results of the wind tunnel test. Figure 10 shows that the numerical C

_{y}of model 7 is lower than the test C

_{y}for most of the angle of attack range, and conversely, the simulated K value is higher than the test data for most of the angle of attack range.

_{y}–α curves of otter board models 7 and 10 in Figure 9 show that the maximum lift coefficient of otter board model 7 is 1.830 (α = 32.5°), whereas that of otter board model 10 is 1.902 (α = 35°). Therefore, the working angle of attack can be considered as 25°–30° (otter board model 7) and 27.5°–32.5° (otter board model 10). In the respective working ranges of the angle of attack, the lift coefficient and lift–drag ratio of otter board model 10 are higher than those of otter board model 7, showing better optimization.

## 4. Discussion

#### 4.1. Aspect Ratio

_{x}, lift coefficient C

_{y}and lift–drag ratio K of the nine otter board models at an angle of attack of 30° are plotted in Figure 11. Comparatively, an otter board with a larger aspect ratio (2.0) has a higher lift–drag ratio. Figure 9 shows that the three models with an aspect ratio of 2.0 have the top three maximum lift coefficients. The results of the orthogonal experiment reported in this paper show that the highest of the three levels of the aspect ratio factor is optimal.

#### 4.2. Camber

#### 4.3. Gap Ratio

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Structure and parameters of the otter board model. (

**a**) Front view; (

**b**) Configuration; (

**c**) Top view. P

_{A}: rear wing; P

_{B}: fore wing; L: chord; b: span; l

_{A}, l

_{B}: arc lengths of panels; L

_{1}: projection length of the gap; L

_{2}: projection length of l

_{B}; h

_{A}, h

_{B}: distances from the midpoint of the arc to the chord; d

_{A}, d

_{B}: chord lengths of panels; e: distance from the leading edge to the center pivot; β: front staggering angle (i.e., the angle formed by the fore wing chord and the line connecting with the left edge of both wings); γ: tail staggering angle (i.e., the angle formed by the line connecting with the right edge of both wings and a vertical line).

**Figure 2.**Nine otter board models: 1–9, corresponding to the numbers of nine otter board models in Table 3.

**Figure 3.**Otter board model in the wind tunnel: (

**a**) Left side view; (

**b**) Front view; (

**c**) Photograph taken on the right in the wind tunnel before testing; there was no camera when testing.

**Figure 4.**Parameter definitions of the test model in the wind tunnel. F

_{y}: lift; F

_{x}: drag; V: wind speed; α: angle of attack.

**Figure 12.**Flow state of otter board models 7 and 10 for an angle of attack of 30°: (

**a**) Flow state of model 7; (

**b**) Flow state of model 10.

Level | Factor | ||
---|---|---|---|

Aspect Ratio (A) | Camber (B) | Gap Ratio (C) | |

I | 1.0 | 0.12 | 0.25 |

II | 1.5 | 0.14 | 0.30 |

III | 2.0 | 0.16 | 0.35 |

No. | Level | Aspect Ratio | Level | Camber | Level | Gap Ratio |
---|---|---|---|---|---|---|

1 | I | 1.0 | I | 0.12 | I | 0.25 |

2 | I | 1.0 | II | 0.14 | II | 0.30 |

3 | I | 1.0 | III | 0.16 | III | 0.35 |

4 | II | 1.5 | I | 0.12 | II | 0.30 |

5 | II | 1.5 | II | 0.14 | III | 0.35 |

6 | II | 1.5 | III | 0.16 | I | 0.25 |

7 | III | 2.0 | I | 0.12 | III | 0.35 |

8 | III | 2.0 | II | 0.14 | I | 0.25 |

9 | III | 2.0 | III | 0.16 | II | 0.30 |

No. | L/m | b/m | A | S/m^{2} |
---|---|---|---|---|

1 | 0.50 | 0.50 | 1.0 | 0.25 |

2 | 0.50 | 0.50 | 1.0 | 0.25 |

3 | 0.50 | 0.50 | 1.0 | 0.25 |

4 | 0.40 | 0.60 | 1.5 | 0.24 |

5 | 0.40 | 0.60 | 1.5 | 0.24 |

6 | 0.40 | 0.60 | 1.5 | 0.24 |

7 | 0.30 | 0.60 | 2.0 | 0.18 |

8 | 0.30 | 0.60 | 2.0 | 0.18 |

9 | 0.30 | 0.60 | 2.0 | 0.18 |

^{2}/S, aspect ratio; S: L × b, the projected area of the otter board model.

Setting Items | Options |
---|---|

Simulation type | 3D, Steady |

Solver | CFX, Double precision, Pressure-based and implicit |

Turbulence model | k–ε EARSM model |

Pressure | Standard |

Inlet | Velocity inlet |

Outlet | Pressure outlet |

Otter board | No-slip wall |

Wall boundary condition | Scalable wall function |

No. | C_{x} | C_{y} | K | Nodes | Elements |
---|---|---|---|---|---|

Sim-1 | 0.532 | 1.718 | 3.232 | 2.4 × 105 | 1.3 × 106 |

Sim-2 | 0.542 | 1.735 | 3.204 | 3.1 × 105 | 1.7 × 106 |

Sim-3 | 0.559 | 1.742 | 3.115 | 4.4 × 105 | 2.4 × 106 |

Sim-4 | 0.566 | 1.766 | 3.122 | 8.5 × 105 | 4.5 × 106 |

Sim-5 | 0.565 | 1.760 | 3.115 | 1.4 × 106 | 7.6 × 106 |

* Exp | 0.600 | 1.843 | 3.072 |

No. | Level | Aspect Ratio | Level | Camber | Level | Gap Ratio | C_{x} | C_{y} | K |
---|---|---|---|---|---|---|---|---|---|

1 | I | 1.0 | I | 0.12 | I | 0.25 | 0.672 | 1.656 | 2.466 |

2 | I | 1.0 | II | 0.14 | II | 0.30 | 0.670 | 1.717 | 2.565 |

3 | I | 1.0 | III | 0.16 | III | 0.35 | 0.620 | 1.600 | 2.580 |

4 | II | 1.5 | I | 0.12 | II | 0.30 | 0.657 | 1.828 | 2.784 |

5 | II | 1.5 | II | 0.14 | III | 0.35 | 0.617 | 1.783 | 2.892 |

6 | II | 1.5 | III | 0.16 | I | 0.25 | 0.725 | 2.018 | 2.783 |

7 | III | 2.0 | I | 0.12 | III | 0.35 | 0.600 | 1.843 | 3.072 |

8 | III | 2.0 | II | 0.14 | I | 0.25 | 0.694 | 2.007 | 2.892 |

9 | III | 2.0 | III | 0.16 | II | 0.30 | 0.665 | 1.969 | 2.963 |

I_{j} | 7.612 | 8.323 | 8.141 | ||||||

II_{j} | 8.459 | 8.349 | 8.313 | ||||||

III_{j} | 8.927 | 8.326 | 8.544 | ||||||

R_{j} | 1.316 | 0.026 | 0.404 |

_{j}, II

_{j}, III

_{j}: sum of K values at the same level; R

_{j}: range; C

_{x}: drag coefficient; C

_{y}: lift coefficient; K: lift–drag ratio.

No. | L/m | b/m | Aspect Ratio | Camber | Gap Ratio |
---|---|---|---|---|---|

10 | 0.30 | 0.60 | 2.0 | 0.14 | 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Zhang, X.; Wan, R.; Xu, Q.; Qi, G.
Optimization of the Hydrodynamic Performance of a Double-Vane Otter Board Based on Orthogonal Experiments. *J. Mar. Sci. Eng.* **2022**, *10*, 1177.
https://doi.org/10.3390/jmse10091177

**AMA Style**

Wang L, Zhang X, Wan R, Xu Q, Qi G.
Optimization of the Hydrodynamic Performance of a Double-Vane Otter Board Based on Orthogonal Experiments. *Journal of Marine Science and Engineering*. 2022; 10(9):1177.
https://doi.org/10.3390/jmse10091177

**Chicago/Turabian Style**

Wang, Lei, Xun Zhang, Rong Wan, Qingchang Xu, and Guangrui Qi.
2022. "Optimization of the Hydrodynamic Performance of a Double-Vane Otter Board Based on Orthogonal Experiments" *Journal of Marine Science and Engineering* 10, no. 9: 1177.
https://doi.org/10.3390/jmse10091177