# Optimization of Airfoils Using the Adjoint Approach and the Influence of Adjoint Turbulent Viscosity

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## Abstract

**:**

## 1. Introduction

## 2. Mesh Motion

## 3. The Adjoint Approach

#### 3.1. The General Principle

#### 3.2. Adjoint Equations and Gradient Calculation

#### 3.3. Details of the Present Implementation

#### 3.4. Projection of the Gradients of the Objective Function

#### 3.5. Verification of Gradients

#### 3.6. Inverse Design

## 4. Optimization of Airfoils

#### 4.1. NACA 0012 at $AoA={2}^{\circ}$

#### 4.2. NACA 0012 at $AoA={12}^{\circ}$

#### 4.3. DU 93-W-210 at $AoA={2}^{\circ}$

#### 4.4. DU 93-W-210 at $AoA={6}^{\circ}$

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Cells Around Airfoil | Cells in Radial Direction | Cells in Total | |
---|---|---|---|

coarse mesh | 400 | 130 | 52,000 |

medium mesh | 800 | 260 | 208,000 |

fine mesh | 1,600 | 520 | 832,000 |

**Table A2.**Relative differences ${\Delta}_{lift}$, ${\Delta}_{drag}$ in (%) of lift and drag coefficients compared to the coefficients resulting from the finest mesh.

$\mathit{A}\mathit{o}\mathit{A}={2}^{\circ}$ | $\mathit{A}\mathit{o}\mathit{A}={12}^{\circ}$ | |||
---|---|---|---|---|

${\Delta}_{\mathit{l}\mathit{i}\mathit{f}\mathit{t}}$ | ${\Delta}_{\mathit{d}\mathit{r}\mathit{a}\mathit{g}}$ | ${\Delta}_{\mathit{l}\mathit{i}\mathit{f}\mathit{t}}$ | ${\Delta}_{\mathit{d}\mathit{r}\mathit{a}\mathit{g}}$ | |

coarse mesh | $-0.1$ | $1.5$ | $-0.6$ | $5.0$ |

medium mesh | $0.4$ | $0.4$ | $0.1$ | $1.5$ |

fine mesh | $0.0$ | $0.0$ | $0.0$ | $0.0$ |

**Figure A1.**Validation of lift and drag coefficients of a NACA 0012 at $Re=2\times {10}^{6}$ (experimental results by Ladson [41]).

## Appendix B

**Figure A2.**Gradients obtained by finite differences (FDs) for lift objective (“gradient ${c}_{l}$”) and drag objective (“gradient ${c}_{d}$”). Different step sizes (StSz) are used for finite differencing.

## Appendix C

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**Figure 4.**Comparisons of gradients obtained by finite differences (FsD) and gradients via the adjoint approach for lift objective (“gradient ${c}_{l}$”) and drag objective (“gradient ${c}_{d}$”). Different step sizes (StSz) are used for finite differencing. Adjoint and frozen turbulence are used for gradients via the adjoint approach (labeled as “adj. turb” and “frz. turb”, respectively).

**Figure 5.**Objective functions as in Equation (14) and norm of the gradient for the lift objective and drag objective along the number of function evaluations.

**Figure 6.**Shapes of inverse design study for the lift objective and drag objective. For a better visualization of the inverse design, only every third point is plotted.

**Figure 7.**Objective functions for lift and drag as in Equation (14) along the number of function evaluations for the NACA 0012 at $AoA={2}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 8.**Gradients of the first optimization step for a decrease in drag by $3\%$ at $AoA={2}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”. The control points start from the trailing edge on the suction side, pass the leading edge, and end at the trailing edge on the pressure side.

**Figure 9.**Airfoil shapes using adjoint and frozen turbulence for the NACA 0012 at $AoA={2}^{\circ}$. The objective is a decrease in drag by $3\%$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 10.**Objective functions for lift and drag as in Equation (14) along the number of function evaluations for the NACA 0012 at $AoA={12}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 11.**Gradients of the first optimization step for an increase in lift by $2\%$ at $AoA={12}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”. The control points start from the trailing edge on the suction side, pass the leading edge, and end at the trailing edge on the pressure side.

**Figure 12.**Intermediate airfoil shape during an optimization of lift at $AoA={12}^{\circ}$ using frozen turbulence compared to the initial airfoil and the final shape using adjoint turbulence (all with 30 control points).

**Figure 13.**Airfoil shapes using adjoint and frozen turbulence for the lift and drag objective with NACA 0012 at $AoA={12}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 14.**Objective functions for lift and drag as in Equation (14) along the number of function evaluations for the DU 93-W-210 at $AoA={2}^{\circ}$. Frozen as well as adjoint turbulence is used. The number of control points is labeled as “XXcp”.

**Figure 15.**Airfoil shapes using adjoint and frozen turbulence for the lift and drag objective for DU 93-W-210 at $AoA={2}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 16.**Objective functions for lift and drag as in Equation (14) along the number of function evaluations for the DU 93-W-210 at $AoA={6}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 17.**Airfoil shapes using adjoint and frozen turbulence for the lift and drag objective for DU 93-W-210 at $AoA={6}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”.

**Figure 18.**Gradients of the first optimization step for lift and drag objective of the DU 93-W-210 at $AoA={6}^{\circ}$. Frozen as well as adjoint turbulence are used. The number of control points is labeled as “XXcp”. The control points start from the trailing edge on the suction side, pass the leading edge and end at the trailing edge on the pressure side.

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Schramm, M.; Stoevesandt, B.; Peinke, J.
Optimization of Airfoils Using the Adjoint Approach and the Influence of Adjoint Turbulent Viscosity. *Computation* **2018**, *6*, 5.
https://doi.org/10.3390/computation6010005

**AMA Style**

Schramm M, Stoevesandt B, Peinke J.
Optimization of Airfoils Using the Adjoint Approach and the Influence of Adjoint Turbulent Viscosity. *Computation*. 2018; 6(1):5.
https://doi.org/10.3390/computation6010005

**Chicago/Turabian Style**

Schramm, Matthias, Bernhard Stoevesandt, and Joachim Peinke.
2018. "Optimization of Airfoils Using the Adjoint Approach and the Influence of Adjoint Turbulent Viscosity" *Computation* 6, no. 1: 5.
https://doi.org/10.3390/computation6010005