# Distinct Turbulent Regions in the Wake of a Wind Turbine and Their Inflow-Dependent Locations: The Creation of a Wake Map

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

## 3. Results

#### 3.1. Mean Velocity

#### 3.2. Variance

#### 3.3. Turbulence Intensity

#### 3.4. Energy Spectral Density and Decay Exponent in the Inertial Sub-Range

#### 3.5. Integral Length and Taylor Length

#### 3.6. Castaing Parameter

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

cf. | confer (Latin)—compare with |

TI | turbulence intensity |

## References

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**Figure 1.**Sketch of the evolution of the turbulent wake of a wind turbine (adapted from [7] by the author).

**Figure 2.**Experimental setup showing the wind tunnel with its inlet nozzle on the left and the positions of the model wind turbines together with the measurement positions marked by dots. Altered from [30].

**Figure 3.**Downstream evolution of the mean velocity $u/{u}_{0}$ that is normalized by the inflow velocity ${u}_{0}$: (

**a**) shows an interpolated contour plot of $u/{u}_{0}$ downstream of turbine 1. The horizontal red dotted line indicates the position of the blade tip and the vertical red dotted line the position of turbine 2. The dashed red curve gives an approximation of the wake boundary at $u/{u}_{0}=0.95$. (

**b**–

**d**) show the centerline evolution of $u/{u}_{0}$ over $X/D$ with logarithmic axes for the wakes of turbine 1, turbine 2 mid, and turbine 2 side.

**Figure 4.**Downstream evolution of the variance ${\sigma}^{2}$ at the centerline for the wakes of turbine 1, turbine 2 mid and turbine 2 side (

**a**–

**c**) with logarithmic axes. The straight vertical line indicates the local maximum, the dashed line the maximum curvature, and the dash-dotted line mirrors the distance between the first two lines.

**Figure 5.**Downstream evolution of the turbulence intensity: The interpolated contour plot (

**a**) shows the evolution of the turbulence intensity downstream of turbine 1. The horizontal red dotted line indicates the position of the blade tip and the vertical red dotted line the position of turbine 2. The dashed red curve gives an approximation of the wake boundary at $u/{u}_{0}=0.95$. In (

**b**–

**d**), the centerline evolution of the turbulence intensity is plotted with logarithmic axes for the three configurations. In addition, the decay region of the turbulence intensity is fitted, and the fit regions are indicated by the dotted and the dash-dotted line.

**Figure 6.**Downstream evolution of the pre-multiplied energy spectral density $E\left(f\right)\xb7f$ over $f\xb7D/{u}_{0}$ for two different radial positions ($Y/D=0$ (sub-plots (

**a**,

**c**,

**e**), and $Y/D=-0.52$ (sub-plots (

**b**,

**d**,

**f**)) downstream of the three wake scenarios turbine 1 (

**a**,

**b**), turbine 2 mid (

**c**,

**d**), and turbine 2 side (

**e**,

**f**). An increasing downstream position is indicated by a brighter color. In addition, the decay according to $E\left(f\right)\propto {f}^{-5/3}$ is indicated (blue line), and the spectrum of the inflow generated by turbine 1 is plotted for turbine 2 mid and side at the position of the rotor (dark red dash-dotted lines) and at the last measurement position (red dashed lines).

**Figure 7.**Comparison of the energy spectra in the far wake for the three scenarios turbine 1 (dark violet), turbine 2 mid (turquoise) and turbine 2 side (green), and the radial positions $Y/D=0$ (straight line), $Y/D=-0.21$ (dashed line), and $Y/D=-0.52$ (dotted line). In (

**a**), the energy spectra are normalized by the energy at $f=50\phantom{\rule{0.222222em}{0ex}}\mathrm{Hz}$ to collapse the inertial sub-range and thus emphasize the similarity and in (

**b**) the spectra are shown.

**Figure 8.**Decay exponent $\kappa $ of the energy spectral density in the inertial sub-range for the three scenarios in (

**a**–

**c**). The exponent is fitted according to $E\left(f\right)\propto {f}^{\kappa}$, and $\kappa =-5/3$ is indicated in the dashed grey horizontal line. The vertical lines indicate as before the downstream region of the maximum variance.

**Figure 9.**Downstream evolution of the integral length scale L: An interpolated contour plot shows the evolution of the whole measurement section downstream of turbine 1 (

**a**). The dotted red horizontal line shows the position of the blade tip, the dotted red vertical line the position of turbine 2, and the dashed red line gives an approximation of the wake boundary at $u/{u}_{0}=0.95$. The centerline evolution is plotted with a logarithmic x-axis for all three wake scenarios in (

**b**–

**d**).

**Figure 10.**Downstream evolution of the Taylor length ${\lambda}_{tay}$ at the centerline with logarithmic axes for the three wakes in (

**a**–

**c**).

**Figure 11.**Evolution of the Castaing parameter ${\lambda}^{2}$ downstream of turbine 1 as interpolated contour plot (

**a**) and downstream of turbine 1, turbine 2 mid, and turbine 2 side at the centerline (

**b**–

**d**).

**Figure 12.**Evolution of the Castaing parameter ${\lambda}^{2}$ over time scale $\tau $ for turbine 1 (

**a**), turbine 2 mid (

**b**), and turbine 2 side (

**c**) at the respective last measurement position ($X/D=12.62$ or $X/D=8.66$) at the centerline. The vertical line indicates $\tau \simeq L$ and the dashed line indicates $\tau \simeq D/2$. In addition, a fit according to ${\lambda}^{2}\propto 0.25\xb7ln\mathrm{F}\left(\tau \right)$ with $\mathrm{F}\left(\tau \right)\propto {\tau}^{-4/9\xb7\mu}$. The respective values for the intermittency parameter $\mu $ are written in the legend.

**Figure 13.**Wake Map: The four downstream regions near wake, transition region, decay region, and far wake with their respective properties are indicated. The extension of the regions depends on the inflow conditions. The wake also has a wake core with indications of homogeneous isotropic turbulence in the far wake, and a ring with high intermittency at scales $\tau \simeq D$ surrounding the wake. Figure adapted from [30].

**Table 1.**Values and downstream positions of the velocity minimum ${(u/{u}_{0})}_{min}$, the minimum variance ${\sigma}_{min}^{2}$, the maximum variance ${\sigma}_{max}^{2}$, the minimum turbulence intensity $T{I}_{min}$, and the maximum turbulence intensity $T{I}_{CLpeak}$ for all scenarios.

Turbine 1 | Turbine 2 Mid | Turbine 2 Side | |
---|---|---|---|

${(u/{u}_{0})}_{min}$ | 0.27 | 0.15 | 0.24 |

$X/D\left({(u/{u}_{0})}_{min}\right)$ | 4.52 | 1.41 | 1.76 |

${\sigma}_{min}^{2}$ | 0.06 | 0.11 | 0.36 |

$X/D\left({\sigma}_{min}^{2}\right)$ | 3.66 | 1.07 | 1.41 |

${\sigma}_{max}^{2}$ | 0.40 | 0.62 | 0.85 |

$X/D\left({\sigma}_{max}^{2}\right)$ | 8.83 | 2.37 | 2.97 |

$T{I}_{min}$ | 0.12 | 0.24 | 0.27 |

$X/D\left(T{I}_{min}\right)$ | 3.66 | 0.90 | 1.07 |

$T{I}_{CLpeak}$ | 0.21 | 0.49 | 0.39 |

$X/D\left(T{I}_{CLpeak}\right)$ | 6.24 | 1.76 | 2.45 |

**Table 2.**Decay exponents ${\beta}_{1}$ and ${\beta}_{2}$ for the turbulence intensity for the three scenarios.

Turbine 1 | Turbine 2 Mid | Turbine 2 Side | |
---|---|---|---|

${\beta}_{1}$ | −1.021 | −1.325 | −1.113 |

${\beta}_{2}$ | −1.149 | −0.812 | −0.815 |

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**MDPI and ACS Style**

Neunaber, I.; Hölling, M.; Stevens, R.J.A.M.; Schepers, G.; Peinke, J.
Distinct Turbulent Regions in the Wake of a Wind Turbine and Their Inflow-Dependent Locations: The Creation of a Wake Map. *Energies* **2020**, *13*, 5392.
https://doi.org/10.3390/en13205392

**AMA Style**

Neunaber I, Hölling M, Stevens RJAM, Schepers G, Peinke J.
Distinct Turbulent Regions in the Wake of a Wind Turbine and Their Inflow-Dependent Locations: The Creation of a Wake Map. *Energies*. 2020; 13(20):5392.
https://doi.org/10.3390/en13205392

**Chicago/Turabian Style**

Neunaber, Ingrid, Michael Hölling, Richard J. A. M. Stevens, Gerard Schepers, and Joachim Peinke.
2020. "Distinct Turbulent Regions in the Wake of a Wind Turbine and Their Inflow-Dependent Locations: The Creation of a Wake Map" *Energies* 13, no. 20: 5392.
https://doi.org/10.3390/en13205392