# Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach

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## Abstract

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## 1. Introduction

## 2. Optimization Method

#### 2.1. Gradient Estimation by the Adjoint Approach

**u**and p being the flow velocity and pressure, respectively. The rate of strain tensor for any given vector x is defined as $D\left(\mathbf{x}\right)=\frac{1}{2}\left(\nabla x+{\left(\nabla x\right)}^{T}\right)$. This notation is used for the velocity $\mathbf{u}$ and adjoint velocity ${\mathbf{u}}_{\mathbf{a}}$ in the flow and adjoint field, respectively. The viscosity $\nu $ is a sum of the molecular and turbulent viscosity.

#### 2.2. Gradient Projection and Evaluation

## 3. Simulation and Optimization Set-Ups

## 4. Results

#### 4.1. Flow and Adjoint Field

#### 4.2. Convergence Behavior of the Optimizations

#### 4.3. Additional Twist Computed by the Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Numerical grid. The flow is coming from the left in (

**a**). In (

**b**), the o-type mesh around a cut through the blade is shown.

**Figure 3.**Magnitude (Mag) of the velocity U and adjoint velocity $Ua$ around the blade at a cut at 3 m of the span.

**Figure 4.**Field visualisation based on the Q-criterion in terms of vorticity. The full rotor for the flow field in (

**a**) and for the adjoint field in (

**b**,

**c**) is shown. For good comparability, the adjoint field is shown in (

**b**) in the same direction, as the flow field, i.e., downstream. As it moves upstream, it is secondly shown in the direction of the propagation of the adjoint field in (

**c**).

**Figure 5.**Development of the normalised thrust force over the optimization steps for design parameters within varying parts of the blade. Long: 2 m–4.5 m radial position, middle: 3 m–4.5 m radial position and short: 3.75 m–4.75 m radial position. The aimed values and the limits of the allowed range are shown with black and grey lines, respectively.

**Figure 6.**Development of the normalised thrust force over the optimization steps for the long blade part. The aimed force is $900\pm 1.5\%$ N and the initial step sizes are ${10}^{-2}$ (red dots, StSz1) and $2\times {10}^{-2}$ (blue squares, StSz2).

**Figure 7.**Development of the normalized thrust force over the optimization steps for design parameters within varying deformable parts of the blade. Long: ${r}_{1}$ = 2 m–4.5 m radial position, middle: ${r}_{2}$ = 3 m–4.5 m radial position and short: ${r}_{3}$ = 3.75 m–4.75 m radial position.

**Figure 8.**Spanwise production of thrust of the original NREL Phase VI blade from 1 to $4.75$ m in radial direction.

**Table 1.**Optimization set-ups for different parts of the blade, aimed values and convergence interval around the aimed value.

Case | ${\mathit{F}}_{{\mathit{T}}^{*}}$ | Convergence Interval | Radial Part r | Blade Part | Length | |
---|---|---|---|---|---|---|

(N) | % | (N) | (m) | % | (m) | |

1 | 900 | 1.5 | 886.5–913.5 | 2–4.5 | 50 | 2.5 |

2 | 3–4.5 | 30 | 1.5 | |||

3 | 3.75–4.75 | 20 | 1 | |||

4 | 2.5 | 877.5–922.55 | 2–4.5 | 50 | 2.5 | |

5 | 3–4.5 | 30 | 1.5 | |||

6 | 3.75–4.75 | 20 | 1 | |||

7 | 925 | 1.5 | 911.125–938.875 | 2–4.5 | 50 | 2.5 |

8 | 3–4.5 | 30 | 1.5 | |||

9 | 3.75–4.75 | 20 | 1 | |||

10 | 2.5 | 901.875–948.125 | 2–4.5 | 50 | 2.5 | |

11 | 3–4.5 | 30 | 1.5 | |||

12 | 3.75–4.75 | 20 | 1 |

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**MDPI and ACS Style**

Vorspel, L.; Stoevesandt, B.; Peinke, J.
Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach. *Appl. Sci.* **2018**, *8*, 1112.
https://doi.org/10.3390/app8071112

**AMA Style**

Vorspel L, Stoevesandt B, Peinke J.
Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach. *Applied Sciences*. 2018; 8(7):1112.
https://doi.org/10.3390/app8071112

**Chicago/Turabian Style**

Vorspel, Lena, Bernhard Stoevesandt, and Joachim Peinke.
2018. "Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach" *Applied Sciences* 8, no. 7: 1112.
https://doi.org/10.3390/app8071112