# Optimization and Design of a Flexible Droop-Nose Leading-Edge Morphing Wing Based on a Novel Black Widow Optimization Algorithm—Part I

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bibliographical Review

## 3. Optimization Framework Methodology

#### 3.1. CST Airfoil Parameterization

_{length}, it is possible to determine any other point on the arc. The Θ value is the same as the arc angle value. Therefore, since

#### 3.2. Black Widow Optimization (BWO)

#### 3.3. Aerodynamic Solver

#### 3.3.1. XFoil Solver

#### 3.3.2. Transition SST Model

#### 3.3.3. Mesh Generation

^{6}.

## 4. Results and Discussion

#### 4.1. Optimization of Cruise Phase

^{6}. The general optimization problem is presented as follows:

#### 4.1.1. Drag Minimization

^{6}.

#### 4.1.2. Endurance Maximization

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${c}_{l}$ | Lift force per unit span |

${C}_{{N}_{2}}^{{N}_{1}}$ | CST class function |

${c}_{d}$ | Drag force per unit span |

${C}_{L}$ | Lift coefficient |

${C}_{D}$ | Drag coefficient |

$c$ | Chord |

${c}_{p}$ | Specific fuel consumption |

${C}_{f}$ | Skin friction coefficient |

$D$ | Drag force |

$E$ | Endurance |

${E}_{a}$ | Aerodynamic Endurance Efficiency |

${K}_{i}^{n}$ | Binomial coefficient K |

$L$ | Lift force |

$M$ | Mach number |

$S$ | Wing surface |

$v$ | Aircraft speed |

${X}_{tu}$ | Translation variable in $x$ for the upper surface |

${X}_{tl}$ | Translation variable in $x$ for the lower surface |

$\rho $ | Air density |

Θ | Morphing deflection angle |

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**Figure 7.**Translation from the deflection arc (

**a**) and relative skin position to the deflection arc (

**b**).

**Figure 11.**Assessment of results obtained for the UAS-S45 optimized airfoil for drag minimization; (

**a**) Airfoil shape, and (

**b**) Cp distribution versus the chord.

**Figure 12.**Comparison of pressure variations for the UAS-S45 optimized airfoil for drag minimization. (

**a**) Reference airfoil, (

**b**) Optimized airfoil.

**Figure 13.**Comparison of the skin friction coefficient for the S45 optimized airfoil and for the reference airfoil for drag minimization.

**Figure 14.**Assessment of the (

**a**) velocity contours for the reference and optimized airfoil, and (

**b**) intermittency contours the reference and optimized airfoil for drag minimization.

**Figure 15.**Increase in the lift-to-drag ratio at the same drag coefficient for the optimized airfoil versus the reference airfoil.

**Figure 16.**Comparison of results for the S45 optimized airfoil for maximized aerodynamic endurance: (

**a**) Airfoil shape; (

**b**) Cp distribution; (

**c**) Pressure variation over the airfoil.

**Figure 17.**Assessment of the skin friction coefficient for the S45 optimized airfoil and for the reference airfoil for drag minimization.

**Figure 18.**Assessment of the (

**a**) velocity contours for the reference and optimized airfoil, and (

**b**) intermittency contours the reference and optimized airfoil for endurance maximization.

**Figure 19.**Comparison of fuel burn rate over flight time at different speeds for the reference and optimized configurations for the UAS-S45. (

**a**) Fuel burn for reference airfoil, (

**b**) Fuel burn for optimized min CD, (

**c**) Fuel burn for optimized CL/CD, (

**d**) Fuel burn for optimized CL

^{3/2}/CD.

**Figure 20.**Comparison of specific fuel consumption with flight time at different speeds for the reference and optimized UAS-S45. (

**a**) SFC for reference airfoil, (

**b**) SFC for optimized min CD, (

**c**) SFC for optimized CL/CD, (

**d**) SFC for optimized CL

^{3/2}/CD.

Angle of Attack (°) | Reference Airfoil | Optimized Airfoil | Relative Difference in ‘%’ | |
---|---|---|---|---|

${C}_{D}$ | 2° | 0.00755 | 0.00663 | −12 |

$\frac{{C}_{L}}{{C}_{D}}$ | 2° | 48.6887 | 56.1017 | 15 |

**Table 2.**Optimized drag results for the reference and optimized morphing airfoils for a certain length of a flexible morphing section for drag minimization.

Angle of Attack | Length of Flexible Section (m) | Reference Airfoil C_D | Optimized Airfoil C_D | Relative ‘%’ Difference |
---|---|---|---|---|

0 | 0.05 | 0.0091 | 0.00691 | −24.06 |

1 | 0.10 | 0.00801 | 0.00729 | −8.98 |

2 | 0.07 | 0.00755 | 0.00663 | −12.18 |

3 | 0.09 | 0.0093 | 0.00674 | −27.52 |

4 | 0.23 | 0.00976 | 0.00643 | −34.11 |

5 | 0.24 | 0.0102 | 0.00648 | −36.47 |

6 | 0.23 | 0.00912 | 0.00678 | −25.65 |

7 | 0.29 | 0.01016 | 0.0097 | −4.52 |

8 | 0.29 | 0.01261 | 0.01143 | −9.35 |

9 | 0.25 | 0.01251 | 0.01108 | −11.43 |

10 | 0.29 | 0.01386 | 0.0129 | −6.92 |

**Table 3.**Comparison of aerodynamic endurance and minimized drag of the reference and optimized airfoils.

Reference Airfoil | Optimized Airfoil | Relative Difference in ‘%’ | |
---|---|---|---|

$\frac{\left({C}_{L}^{\frac{3}{2}}\right)}{{C}_{D}}$ | 29.52 | 32.48 | 10 |

${C}_{D}$ | 0.00872 | 0.00796 | −8 |

**Table 4.**Optimized drag results for the reference and optimized morphing airfoils for a certain length of a flexible morphing section for maximization of endurance.

Angle of Attack (°) | Length of Flexible Section $\left(\frac{\mathit{l}}{\mathit{c}}\right)$ | Reference $\frac{{\mathit{C}}_{\mathit{L}}^{\frac{3}{2}}}{{\mathit{C}}_{\mathit{D}}}$ | Optimized $\frac{{\mathit{C}}_{\mathit{L}}^{\frac{3}{2}}}{{\mathit{C}}_{\mathit{D}}}$ | Improvement ‘%’ |
---|---|---|---|---|

0 | 0.19 | 5.86 | 9.06 | 54.51 |

1 | 0.06 | 15.28 | 19.54 | 27.86 |

2 | 0.12 | 29.52 | 32.48 | 10.05 |

3 | 0.06 | 34.68 | 49.78 | 43.55 |

4 | 0.24 | 44.63 | 65.67 | 47.13 |

5 | 0.18 | 55.30 | 93.42 | 68.94 |

6 | 0.21 | 78.94 | 114.83 | 45.45 |

7 | 0.19 | 81.90 | 130.02 | 58.75 |

8 | 0.22 | 78.81 | 142.62 | 80.96 |

9 | 0.23 | 91.34 | 154.41 | 69.05 |

10 | 0.20 | 95.11 | 160.50 | 68.74 |

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**MDPI and ACS Style**

Bashir, M.; Longtin-Martel, S.; Botez, R.M.; Wong, T. Optimization and Design of a Flexible Droop-Nose Leading-Edge Morphing Wing Based on a Novel Black Widow Optimization Algorithm—Part I. *Designs* **2022**, *6*, 10.
https://doi.org/10.3390/designs6010010

**AMA Style**

Bashir M, Longtin-Martel S, Botez RM, Wong T. Optimization and Design of a Flexible Droop-Nose Leading-Edge Morphing Wing Based on a Novel Black Widow Optimization Algorithm—Part I. *Designs*. 2022; 6(1):10.
https://doi.org/10.3390/designs6010010

**Chicago/Turabian Style**

Bashir, Musavir, Simon Longtin-Martel, Ruxandra Mihaela Botez, and Tony Wong. 2022. "Optimization and Design of a Flexible Droop-Nose Leading-Edge Morphing Wing Based on a Novel Black Widow Optimization Algorithm—Part I" *Designs* 6, no. 1: 10.
https://doi.org/10.3390/designs6010010