# Numerical Study on the Aerodynamic Characteristics of the NACA 0018 Airfoil at Low Reynolds Number for Darrieus Wind Turbines Using the Transition SST Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{N}model proposed by Smith and Gamberoni [52], proved difficult to implement in modern general-purpose CFD codes and were based on global flow parameters [51]. In the case of numerical analysis of the airfoil in a homogeneous flow, this approach does not have to give unreasonable results for the location of the transition [2]. A much greater challenge for a transition turbulence model is the correct determination of the transition characteristics in the boundary layer of the airfoil in a flow of variable turbulence intensity, e.g., in the case of a blade of the Darrieus wind turbine rotor [53]. This, however, requires the turbulence model to use local turbulence parameters to analyze the transition characteristics. Quite extensive research on the characteristics of the laminar separation bubble on the NACA 0021 airfoil and in the range of low Reynolds numbers was carried out by Choudhry et al. using the Transition SST and $\kappa -{\kappa}_{L}-\omega $ approaches [2]. Choudhry et al. reported that, due to the generation of additional turbulence, the Transition SST predicted earlier reattachment as compared to the experiments and the $\kappa -{\kappa}_{L}-\omega $. Melani et al. [19] compiled many experimental and numerical aerodynamic characteristics of the NACA 0018 airfoil at low and moderate Reynolds numbers. The data sets collected in their work suggest that the lower the Reynolds number, the larger the differences in the results of the aerodynamic forces, especially for Reynolds numbers in the range from 40 k to 160 k. Melani at al. [19] also presented the lift force characteristic for the Reynolds number of 150 k obtained using the Transition SST approach, these data are also available in [54]. The results indicate a high agreement with the experiments in the range of angles of attack up to 6 degrees. To the best of the authors’ knowledge, [19] is the only work in which the NACA 0018 airfoil performance was analyzed in a similar range of Reynolds numbers using the Transition SST turbulence approach.

## 2. Numerical Methods

_{99}, as l = 0.4 δ

_{99}. The thickness of the boundary layer was estimated based on velocity profile close to the airfoil surface for the angle of attack of 0 deg. For all simulations presented in this paper, the turbulence length scale was established to be 0.001 m. The intermittency at the inlet was equal to 1. The effect of turbulence intensity and intermittency on the aerodynamic characteristics of the airfoil was not investigated in these studies. The center of the first cell over the airfoil surface is located at a nondimensional height of ${y}^{+}\approx 1$. The mesh, presented in Figure 2b,c, consisted of 700,400 elements. The number of mesh points on the airfoil edges was 830. Originally, the NACA 0018 airfoil had a blunt trailing edge, but in these studies it was assumed that the trailing edge of the airfoil was sharp.

## 3. Simulation Results and Validation

#### 3.1. Lift and Drag Coefficients

#### 3.2. Pressure Distributions

#### 3.3. Laminar Separation Bubble Characteristics and Mean Velocity Distributions in the Near-Wall Region

## 4. Vertical Axis Wind Turbine Test Case

- CFD polar calculated using the SST model
- CFD polar calculated using the transitional SST model
- Polar obtained from the measurement of Timmer
- Polar data calculated using XFOIL with the critical amplification factor of N = 9.0.

## 5. Conclusions

- This paper continues the research carried out by Królak and presented in his thesis [72]. Królak used the same geometric model, the Reynolds-averaged Navier–Stokes (RANS) technique, and the Transition SST turbulence model. The results of the aerodynamic force coefficients and the pressure distributions were burdened with considerable numerical errors. In particular, there were considerable nonphysical oscillations around the transition location. This article shows that the numerical calculations of the NACA airfoil should be carried out in a transient mode. Langtry reported similarly in many of the analyzed test cases [51].
- The analysis of the aerodynamic properties of the NACA 0018 airfoil was possible only in the transient mode, which, however, significantly increased the computational effort and increased the simulation time of a single case.
- The use of the Transition SST model made it possible to find two regions on the ${\overline{C}}_{L}$ characteristic that were characterized by two aerodynamic derivatives $d{\overline{C}}_{L}/d\alpha $ in the range up to the critical angle of attack, instead of one derivative, as predicted by the two-equation k-ω SST turbulence model. The first region was found up to an angle of attack of 6 degrees, and the second up to 11 degrees.
- The values of the aerodynamic derivatives corresponded quite well with the experimental data; however, above the angle of attack equal to 6 degrees, the results of the lift coefficient were underestimated compared to the experimental data. This is probably due to the 3D effects, which were not included in these numerical studies.
- The Transition SST approach predicted the minimum drag coefficient by far the most accurately in comparison to the experimental results.
- The Transition SST model relatively accurately estimated the location and size of the laminar separation bubble on the suction surface of the airfoil. The average relative error in the localization of the separation point for the entire range of the investigated angles of attack was 22% compared to the experiment of Gerakopulos [21]. The reattachment point was estimated much more precisely; the mean relative error was 5.5%. This gives there was a mean relative error of 22% in estimating the length of the laminar separation bubble. The underestimation of the separation point of the laminar boundary layer in the CFD analysis compared to the experiment may be the result of using a two-dimensional numerical model that neglected the evolution of vortex structures in the direction of the span [66]. Another likely reason is that the model was not calibrated for this particular issue. The original formulation of the Transition SST model was calibrated for the small separation bubbles visible in the machines [2].
- The use of the more expensive turbulence model, i.e., the Transition SST model, to calculate the airfoil characteristics of the NACA 0018, however, significantly improved the normal force distribution of the Darrieus wind turbine rotor calculated using simplified aerodynamic methods.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$c$ | chord length |

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

${\overline{C}}_{D}$ | averaged drag coefficient |

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

${C}_{FN}$ | normal force coefficient |

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

${\overline{C}}_{L}$ | averaged lift coefficient |

${C}_{FT}$ | tangential force coefficient |

${C}_{P}$ | static pressure coefficient |

$FN$ | normal force |

$FT$ | tangential force |

R | reattachment point; rotor radius of the Darrieus wind turbine rotor |

${R}_{CD}$ | the radius of the front edge of the domain size |

S | separation point |

SD | standard deviation |

T | transition point |

${U}_{0}$ | free stream wind velocity (in NACA 0018 simulations) |

${V}_{\infty}$ | free stream wind velocity (in VAWT calculations) |

$\alpha $ | angle of attack |

$\mu $ | dynamic viscosity |

${\nu}_{\infty}$ | kinematic viscosity |

$\rho $ | air density |

$\omega $ | wind turbine rotational speed; vorticity |

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**Figure 1.**Laminar separation bubble—reproduced from [2].

**Figure 2.**The NACA 0018 airfoil and the definition of the angle of attack (

**a**); C-type structured mesh around the airfoil and the boundary conditions (

**b**); the mesh in the vicinity of the leading edge and the trailing edge of the NACA 0018 airfoil (

**c**).

**Figure 4.**Characteristics of the aerodynamic force coefficients of the NACA 0018 airfoil as a function of the simulation time: (

**a**,

**c**) Drag coefficients; (

**b**,

**d**) Lift coefficients.

**Figure 5.**Averaged values of the aerodynamic force coefficients and their standard deviation at 6. degrees. (

**a**) Mean drag coefficient; (

**b**) Mean lift coefficient.

**Figure 11.**Skin friction coefficient distributions over NACA 0018 airfoil at an angle of attack of 6 degrees. Location of separation (S), transition (T) and reattachment (R).

**Figure 13.**Variation of transition location with the angle of attack. Validation of the CFD results using the Transition SST turbulence model using the experiment of Gerakopulos et al. [21].

**Figure 14.**Near-wall distributions of mean velocity on the airfoil suction side at the angle of attack of 6 degrees; comparison of the numerical results (blue dots) with the experiment (red line) [4].

**Figure 15.**Near-wall distributions of mean velocity on the airfoil pressure side at the angle of attack of 6 degrees; the comparison of the numerical results (blue dots) with the experiment (red line) [4].

**Figure 16.**Instantaneous non-dimensional vorticity contours for four angles of attack. Location of separation (S), transition (T) and reattachment (R).

**Figure 18.**Normal blade load as a function of azimuthal position; comparison with the experiment [67].

Case | Number of Mesh Points on the Airfoil | Total Number of Mesh Elements |
---|---|---|

Case 1 | 208 | 175,440 |

Case 2 | 416 | 350,880 |

Case 3 | 830 | 700,400 |

Case 4 | 1660 | 1,400,800 |

Case 5 | 3320 | 2,767,600 |

Case | ${\mathit{R}}_{\mathit{C}\mathit{D}}$ | Total Number of Mesh Elements | ${\overline{\mathit{C}}}_{\mathit{L}}$ | ${\overline{\mathit{C}}}_{\mathit{D}}$ | $\left|\mathbf{\Delta}{\overline{\mathit{C}}}_{\mathit{L}}\right|\left[\mathit{\%}\right]$ | $\left|\mathbf{\Delta}{\overline{\mathit{C}}}_{\mathit{D}}\right|\left[\mathit{\%}\right]$ |
---|---|---|---|---|---|---|

Case 1 | 3.75c | 432,600 | 0.7457 | 0.0227 | ||

Case 2 | 7.5c | 700,400 | 0.7425 | 0.0220 | 0.43 | 3.28 |

Case 3 | 15c | 1,486,800 | 0.7412 | 0.0218 | 0.18 | 0.82 |

Angle of Attack [deg] | ${\overline{\mathit{C}}}_{\mathit{D}}\text{}\left(\mathbf{SD}\right)$ | ${\overline{\mathit{C}}}_{\mathit{L}}\text{}\left(\mathbf{SD}\right)$ |
---|---|---|

0 | 0.0168 (1.31 × 10^{−4}) | 0.0004 (8.40 × 10^{−3}) |

1 | 0.0170 (1.32 × 10^{−4}) | 0.0960 (4.60 × 10^{−3}) |

2 | 0.0178 (6.17 × 10^{−5}) | 0.2120 (2.10 × 10^{−3}) |

3 | 0.0187 (1.81 × 10^{−5}) | 0.3500 (2.86 × 10^{−4}) |

4 | 0.0197 (9.61 × 10^{−6}) | 0.4959 (1.74 × 10^{−4}) |

5 | 0.0208 (1.19 × 10^{−5}) | 0.6339 (1.99 × 10^{−4}) |

6 | 0.0220 (7.12 × 10^{−6}) | 0.7425 (3.71 × 10^{−5}) |

7 | 0.0229 (1.28 × 10^{−5}) | 0.7750 (7.21 × 10^{−5}) |

8 | 0.0246 (1.09 × 10^{−5}) | 0.7833 (1.17 × 10^{−4}) |

9 | 0.0270 (3.71 × 10^{−6}) | 0.7984 (8.18 × 10^{−5}) |

10 | 0.0306 (2.28 × 10^{−6}) | 0.8279 (5.24 × 10^{−5}) |

11 | 0.0357 (3.02 × 10^{−6}) | 0.8606 (8.28 × 10^{−5}) |

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**MDPI and ACS Style**

Rogowski, K.; Królak, G.; Bangga, G. Numerical Study on the Aerodynamic Characteristics of the NACA 0018 Airfoil at Low Reynolds Number for Darrieus Wind Turbines Using the Transition SST Model. *Processes* **2021**, *9*, 477.
https://doi.org/10.3390/pr9030477

**AMA Style**

Rogowski K, Królak G, Bangga G. Numerical Study on the Aerodynamic Characteristics of the NACA 0018 Airfoil at Low Reynolds Number for Darrieus Wind Turbines Using the Transition SST Model. *Processes*. 2021; 9(3):477.
https://doi.org/10.3390/pr9030477

**Chicago/Turabian Style**

Rogowski, Krzysztof, Grzegorz Królak, and Galih Bangga. 2021. "Numerical Study on the Aerodynamic Characteristics of the NACA 0018 Airfoil at Low Reynolds Number for Darrieus Wind Turbines Using the Transition SST Model" *Processes* 9, no. 3: 477.
https://doi.org/10.3390/pr9030477