# Aerodynamic Optimization of Airfoil Profiles for Small Horizontal Axis Wind Turbines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Overview of the Optimization Procedure

#### 2.2. Mathematical and Numerical Modelling of the Aerodynamics

^{−5}.

#### 2.3. Flow Solution Domain and Boundary Conditions

#### 2.4. Generation of Computational Grid for Flow Simulation

^{+}[37] is everywhere smaller than 1 for the next-to-wall cells and at least 3 cells remain within the region y

^{+}< 5. The structure of a typical grid is displayed in Figure 3. From case-to-case, changes in the grid are confined to the O-grid surrounding the airfoil. The variations in the airfoil profile cause a grid modification only in this region and different BA can be obtained by the respective rotations of the O-grid connected to the airfoil.

#### 2.5. Parametrization of the Airfoil Shape

_{i}) and two tangent vectors (with lengths l

_{ij}and directions α

_{i}) on both its ends. With this function, the geometry of the airfoils has been generated by using four piecewise Hermite curves (HC

_{i}), two representing the suction and two the pressure side, as shown in Figure 4.

_{2}, P

_{3}, P

_{4}by ensuring that the end-point of one curve is the same as the starting point of another and the local tangent vectors belonging to the curves that are joined at this point have the same direction for ensuring C

^{1}continuity. The mathematical expression of the Hermite curve i (HC

_{i}), bounded by points P

_{i}and P

_{j}(with j = I + 1), is provided below (Figure 4).

_{i}, the variable t is a parameter varying along the curve between 0 (at P

_{i}) and 1 (at P

_{j}) and

_{1}and P

_{5}that are connected by a tiny straight line. In addition to the coordinates of the five points, the directions of the vector couples as well as their lengths describe the shape of the airfoil. The number of degrees of freedom can be reduced by introducing certain constrains on these parameters, which will be addressed below.

#### 2.6. Response Surface Methodology and the Optimization Algorithm

## 3. Results

#### 3.1. Grid Independence

#### 3.2. Validation

#### 3.3. Influence of the Domain Size, Blade to Blade Interaction

_{L}with decreasing L/C is the reverse of what is implied by the classical, one-dimensional linear cascade theory, which predicts a decreasing C

_{L}with decreasing L/C. This is because that the mentioned theory assumes a perfect flow deflection (which is reasonable for turbomachinery of gas and steam turbines with rather small L/C values), whereas in the present case with relatively large L/C values and, thus, non-ideal flow deflection on the average, a decrease of L/C improves the deflection of the passage flow, leading to an increase in C

_{L}.

#### 3.4. Optimization

_{1}, P

_{5}) and the associated vectors of the related Hermite curves (HC

_{1}, HC

_{4}) are fixed. The lengths of the vectors that are connected to P

_{2}, P

_{4}and head towards the trailing edge (l

_{21}, l

_{44}) are additionally assumed to be constant. The point defining the leading edge (P

_{3}) is also fixed in its position (Figure 4). Furthermore, the control points on the suction (P

_{2}) and pressure (P

_{4}) sides (Figure 4) are not allowed to move parallel to the chord but only in the perpendicular direction. An additional constraint was that the distance between P

_{2}and P

_{4}should not undershoot a certain value, to provide “too thin” blades, for structural reasons. Given these constraints and the continuity of the curves, the remaining degrees of freedom that define the profile shape are nine in total, that is, the vertical positions of the points P

_{2}and P

_{4}, the directions (α

_{2}, α

_{3}, α

_{4}) and the lengths of the vector pairs (each pair have a common direction) at the points P

_{2}, P

_{3}and P

_{4}(l

_{22}, l

_{32}, l

_{33}, l

_{43}). The schematic of the resulting complete optimization loop is presented in Figure 8.

_{T}and the coefficient of thrust standard deviation C

_{σ}that are obtained nondimensionalizing the variables by the dynamic pressure 1/2ρV

^{2}and the chord C.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C | chord length |

C_{D} | drag coefficient |

C_{L} | lift coefficient |

C_{T} | average thrust coefficient |

C_{σ} | coefficient of thrust standard deviation |

L | circumferential domain size (distance between periodic boundaries) |

Re | Reynolds number |

T | thrust |

V | magnitude of approach velocity at Re = 100,000 |

y^{+} | distance to wall non-dimensionalized by wall shear stress, viscosity and density |

ρ | density |

σ | standard deviation of thrust |

AoA | Angle of Attack |

BA | Blade Angle |

BiMADS | Biobjective Mesh Adaptive Direct Search |

BEM | Blade Element Momentum |

CFD | Computational Fluid Dynamics |

FA | Flow Angle |

FVM | Finite Volume Method |

GA | Genetic Algorithms |

HAWT | Horizontal Axis Wind Turbines |

LES | Large Eddy Simulations |

NGN | Number of Grid Nodes |

OCCD | Orthogonal Central Composite Design |

RANS | Reynolds Averaged Numerical Simulations |

RSM | Response Surface Methodology |

SST | Shear Stress Transport |

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**Figure 6.**Comparison of predicted lift and drag coefficients with the measured values for the profile NREL-S822 of Selig et al. [51] (Re = 400,000).

**Figure 7.**Variation of the predicted lift coefficient with circumferential domain size, for Re = 400,000, BA = 5°, FA = 15° (AoA = 10°).

**Figure 9.**Predicted Pareto fronts of feasible solutions of optimization. Full circles: L/C = 3, Empty circles: L/C = 6.

**Figure 12.**Predicted distribution of dimensionless velocity magnitude around the airfoil for Re = 100,000, BA = 5°, FA = 15° (AoA = 10°), for L/C = 3, (

**a**) Original profile; (

**b**) Point 1; (

**c**) Point 2; (

**d**) Point 3 (Figure 9).

**Figure 13.**Predicted distribution of dimensionless velocity magnitude around the airfoil for Re = 100,000, BA = 5°, FA = 15° (AoA = 10°), for L/C = 6, (

**a**) Original profile; (

**b**) Point 1; (

**c**) Point 2; (

**d**) Point 3 (Figure 9).

Normalized Scale Parameters | |||||
---|---|---|---|---|---|

Variable | −1.414 | −1 | 0 | 1 | 1.414 |

Re/10^{3} | 90 | 92.93 | 100 | 107.07 | 110 |

FA (°) | 10 | 11.46 | 15 | 18.54 | 20 |

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**MDPI and ACS Style**

Benim, A.C.; Diederich, M.; Pfeiffelmann, B.
Aerodynamic Optimization of Airfoil Profiles for Small Horizontal Axis Wind Turbines. *Computation* **2018**, *6*, 34.
https://doi.org/10.3390/computation6020034

**AMA Style**

Benim AC, Diederich M, Pfeiffelmann B.
Aerodynamic Optimization of Airfoil Profiles for Small Horizontal Axis Wind Turbines. *Computation*. 2018; 6(2):34.
https://doi.org/10.3390/computation6020034

**Chicago/Turabian Style**

Benim, Ali Cemal, Michael Diederich, and Björn Pfeiffelmann.
2018. "Aerodynamic Optimization of Airfoil Profiles for Small Horizontal Axis Wind Turbines" *Computation* 6, no. 2: 34.
https://doi.org/10.3390/computation6020034