# Comparison of Constrained Parameterisation Strategies for Aerodynamic Optimisation of Morphing Leading Edge Airfoil

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## Abstract

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## 1. Introduction

## 2. Optimisation Framework

#### 2.1. Problem Formulation and Objectives Evaluation

#### 2.2. Parameterisation Strategy

## 3. Results and Discussion

#### 3.1. Optimisation without Flap

#### 3.2. Optimisation with Flap

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CAL | Constant Arc Length |

CFD | Computational Fluid Dynamics |

CP | Control Point |

DP | Design Point |

DV | Design Variable |

FRP | Fiber-reinforced Plastic |

MDO | Multi-Disciplinary Optimisation |

$\alpha $ | Angle of attack |

c | Airfoil chord |

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

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

${C}_{p}$ | Pressure coefficient |

$\kappa $ | Curvature |

L | Arc length |

$L/D$ | Lift to drag ratio |

$\theta $ | Displacement direction |

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**Figure 1.**Reference shapes considered in the optimisation. Morphing region extends up to 0.15 $x/c$. In black is a NACA 65(3)-218 airfoil, used as baseline in a first optimisation. In red is an intermediate baseline, used as a better starting point in the following optimisations (see Section 3.1). The original trailing edge of the NACA profile is drawn with a dashed line, the slotted flap with a solid line.

**Figure 3.**Best obtained relative percentage error in maximum lift coefficient for different farfield radii (

**a**). Validated lift and drag polars (

**b**).

**Figure 4.**Parameterisation variants and bounding box of control points. Control points with a black labels are fixed, moving control points and their corresponding bounding box are plotted in the same colour. (

**a**) Variant A: Spline edges are fixed and P8 is moved with the Constant Arc Length (CAL) procedure. (

**b**) Variant B: Lower edge P9 is moved horizontally with the CAL procedure, and the resulting gap is closed with a straight hatch. (

**c**) Variant C: The lower edge P9 slides along an optimiser-controlled direction $\theta $, and resulting gap is closed with a straight hatch.

**Figure 5.**Pareto frontiers obtained for case without flap, using three parameterisation variants. Objectives are normalized with respect to intermediate baseline. Selected points on Pareto corresponding to best improvement of ${C}_{l,max}$, $L/D$, and best compromise are also marked.

**Figure 6.**Pareto airfoils for three parameterisation variants, corresponding to marked points on Figure 5. Control polygon represented with a black dashed line, straight hatch with a solid magenta line. Blue: Variant A; red: Variant B; green: Variant C. (

**a**) Best $L/D$. (

**b**) Best compromise. (

**c**) Best ${C}_{l}$.

**Figure 7.**Pressure coefficient distribution at DP1 for baseline and best ${C}_{l}$ optimised airfoil, with three parameterisation variants.

**Figure 8.**Pareto airfoils for three parameterisation variants. Control polygon represented with a black dashed line, straight hatch with a solid magenta line. Blue: Variant A; red: Variant B; green: Variant C. (

**a**) Best $L/D$. (

**b**) Best compromise. (

**c**) Best ${C}_{l}$.

**Figure 9.**Pressure coefficient distribution at DP1 for baseline and optimised airfoil at left Pareto edge, with flap (

**a**). Contribution of pressure and viscous drag at DP1 in different zones for the same airfoils (

**b**).

**Figure 10.**Mach number contour and flow path visualization at DP1. (

**a**) NACA 65(3)-218. (

**b**) Intermediate baseline. (

**c**) Variant A—Best ${C}_{l}$. (

**d**) Variant C—Best ${C}_{l}$.

**Table 1.**Comparison of aerodynamic performance of optimised individuals with three parameterisation variants. ${C}_{d}$ = drag coefficient, ${C}_{l}$ = lift coefficient, $L/D$ = lift to drag ratio. Between parenthesis are the percentage relative variations, with respect to the intermediate baseline.

DP1 | DP2 | |||
---|---|---|---|---|

Airfoil | ${\mathit{C}}_{\mathit{d}}$ (rel. var. %) | ${\mathit{C}}_{\mathit{l}}$ (rel. var. %) | $\mathit{\alpha}$ | $\mathit{L}/\mathit{D}$ (rel. var. %) |

NACA 65(3)-218 | 0.04766 (−2.22) | 1.571 (−5.07) | 17.00 | 80.42 (−2.37) |

Interm. Baseline | 0.04874 | 1.655 | 18.00 | 82.37 |

Best L/D—var. A | 0.04722 (−3.11) | 1.669 (+0.82) | 18.00 | 83.17 (+0.97) |

Best L/D—var. B | 0.04663 (−4.33) | 1.687 (+1.95) | 18.00 | 83.74 (+1.66) |

Best L/D—var. C | 0.05400 (+10.80) | 1.705 (+3.00) | 19.00 | 83.85 (+1.79) |

Best Cl—var. A | 0.05390 (+10.58) | 1.685 (+1.82) | 19.00 | 82.65 (+0.33) |

Best Cl—var. B | 0.05280 (+8.33) | 1.712 (+3.43) | 19.00 | 83.32 (+1.15) |

Best Cl—var. C | 0.05252 (+7.75) | 1.722 (+4.07) | 19.00 | 83.52 (+1.39) |

**Table 2.**Comparison of aerodynamic performances of optimised individuals with two parameterisation variants, in the presence of a trailing edge flap. Between parenthesis are the percentages of relative variations, with respect to the intermediate baseline.

DP1 | DP2 | |||
---|---|---|---|---|

Airfoil | ${\mathit{C}}_{\mathit{d}}$ (rel. var. %) | ${\mathit{C}}_{\mathit{l}}$ (rel. var. %) | $\mathit{\alpha}$ | $\mathit{L}/\mathit{D}$ (rel. var. %) |

NACA 65(3)-218 | 0.04461 (0.51) | 2.250 (−4.64) | 14.00 | 82.30 (−1.79) |

Interm. Baseline | 0.04438 | 2.359 | 15.00 | 83.80 |

Best L/D—var. A | 0.04253 (−4.17) | 2.379 (+0.85) | 15.00 | 84.40 (+0.72) |

Best L/D—var. C | 0.04047 (−8.80) | 2.432 (+3.07) | 15.00 | 85.23 (+1.71) |

Best Cl—var. A | 0.04092 (−7.80) | 2.397 (+1.62) | 15.00 | 84.04 (+0.29) |

Best Cl—var. C | 0.03978(−10.36) | 2.438 (+3.36) | 15.00 | 84.99 (+1.42) |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Magrini, A.; Benini, E.; Ponza, R.; Wang, C.; Khodaparast, H.H.; Friswell, M.I.; Landersheim, V.; Laveuve, D.; Contell Asins, C.
Comparison of Constrained Parameterisation Strategies for Aerodynamic Optimisation of Morphing Leading Edge Airfoil. *Aerospace* **2019**, *6*, 31.
https://doi.org/10.3390/aerospace6030031

**AMA Style**

Magrini A, Benini E, Ponza R, Wang C, Khodaparast HH, Friswell MI, Landersheim V, Laveuve D, Contell Asins C.
Comparison of Constrained Parameterisation Strategies for Aerodynamic Optimisation of Morphing Leading Edge Airfoil. *Aerospace*. 2019; 6(3):31.
https://doi.org/10.3390/aerospace6030031

**Chicago/Turabian Style**

Magrini, Andrea, Ernesto Benini, Rita Ponza, Chen Wang, Hamed Haddad Khodaparast, Michael I. Friswell, Volker Landersheim, Dominik Laveuve, and Conchin Contell Asins.
2019. "Comparison of Constrained Parameterisation Strategies for Aerodynamic Optimisation of Morphing Leading Edge Airfoil" *Aerospace* 6, no. 3: 31.
https://doi.org/10.3390/aerospace6030031