# Wind Turbine Wake Characterization from Temporally Disjunct 3-D Measurements

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Field Scans

- Data are filtered by range gate number (G), signal-to-noise ratio ($SNR$) and radial velocity magnitude (${U}_{r}$). The criteria are:$$\begin{array}{c}2\le G\le 45\\ 0.01\phantom{\rule{3.33333pt}{0ex}}(-20\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB})<SNR\le 10\phantom{\rule{3.33333pt}{0ex}}\left(10\phantom{\rule{3.33333pt}{0ex}}\mathrm{dB}\right)\\ {U}_{r}<30\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}^{-1}\end{array}$$
- The location vector in Cartesian coordinates in a fixed frame of reference for each point is obtained as:$$\overrightarrow{{d}_{F}}=({x}_{F},{y}_{F},{z}_{F})=(r\mathrm{cos}\varphi \mathrm{sin}\theta ,r\mathrm{cos}\varphi \mathrm{cos}\theta ,r\mathrm{sin}\varphi )$$
- For each 3D scanned volume, the highest elevation sector scan ($\varphi ={19}^{\xb0}$) is selected to determine the mean wind direction $\langle \beta \rangle $ (angled brackets refer to values averaged over the entire 3D scan). Within this scan, all points sampled at least 0.25 D above the rotor are considered, which includes five range gates. An estimate of the horizontal wind components ${u}_{F}$ and ${v}_{F}$ is obtained for each range gate within this sector scan sub-sample by solving the linear system:$$\left(\right)open="["\; close="]">\begin{array}{cc}\mathrm{cos}{19}^{\xb0}\mathrm{sin}{\theta}_{1}& \mathrm{cos}{19}^{\xb0}\mathrm{cos}{\theta}_{1}\\ \vdots & \vdots \\ \mathrm{cos}{19}^{\xb0}\mathrm{sin}{\theta}_{46}& \mathrm{cos}{19}^{\xb0}\mathrm{cos}{\theta}_{46}\end{array}=\left(\right)open="["\; close="]">\begin{array}{c}{U}_{{r}_{1}}\\ \vdots \\ {U}_{{r}_{46}}\end{array}$$The values of (${u}_{F},{v}_{F}$) are then averaged across the five range gates within the sub-sample to yield the mean wind direction estimate $\langle \beta \rangle $ for the scanned volume. The robustness of this method for the present experiment is reflected in the time series shown in Figure 3, where the minimum and maximum $\langle \beta \rangle $ estimates for each 3D scan are compared with sonic anemometer measurements and the turbine nacelle position. The LiDAR estimates follow the sonic values more closely than the turbine data, indicating that the assumptions of low veer and negligible vertical velocity are acceptable for the atmospheric conditions observed during the experiment.
- Assuming a constant mean wind direction $\langle \beta \rangle $ for each 3D scan, the horizontal wind speed at each point ${U}_{F}({x}_{F},{y}_{F},{z}_{F})$ is estimated from the radial velocity ${U}_{r}$ as:$${U}_{F}=-\frac{{U}_{r}}{\mathrm{cos}\varphi \mathrm{cos}\mathrm{\Delta}\theta}$$
- All points in the scan are rotated so that the coordinate system is aligned with the mean wind direction. Throughout the manuscript, the quantities in this streamwise frame of reference are given without the F subscript as: $\overrightarrow{U}=(u,v)$, $\overrightarrow{d}=(x,y)$ where x and y are the cross-stream and streamwise directions, respectively.
- Vertical planes of data are obtained at a given distance downstream of the turbine by selecting all of the sampled points whose streamwise coordinate y falls within the desired distance plus or minus some specified buffer $\mathrm{\Delta}y$, taken to be the range gate width of 30 m. In this analysis, the scanning LiDAR was deployed at the base of the turbine (Figure 2), and we apply an assumption of no yaw error. Therefore, the analysis only considers 3D scans for which the LiDAR-estimated $\langle \beta \rangle $ is within ${15}^{\xb0}$ of the turbine nacelle position, resulting in 80 sector scan stacks. The value of ${15}^{\xb0}$ was chosen as a threshold because while it is small (half of the wind industry standard ${30}^{\xb0}$ sectors when performing azimuthal analyses), it still allows for an offset between the turbine and the nacelle, which is necessary given the uncertainties inherent in both datasets (e.g., inaccuracies in the wind direction estimate from the LiDAR and in the recorded nacelle position) and the potential presence of yaw misalignment. As indicated by the sonic time series (Figure 3), the large differences between the LiDAR and the nacelle datasets reflect not an inability of the LiDAR to estimate the wind direction, but rather the necessarily delayed response of the nacelle to wind direction changes or high uncertainty in the turbine measurements.
- In order to quantify the potential contribution of each vertical slice to wind turbine wake characterization, it is important to consider how much of the area of interest is covered by the sampled points and how dense this coverage is. To do that, we define two indices. The first one is the scanning geometry coverage ($SGC$), which is calculated as:$$SGC=\frac{A}{{A}_{ref}}$$$$SGD=\frac{n}{{n}_{ref}(\mathrm{\Delta}x,\mathrm{\Delta}z)}$$

#### 2.2. Synthetic Scans

#### 2.3. Wake Identification

#### 2.4. Wake Characterization

## 3. Results

#### 3.1. Difference between Scan, Mean and Snapshot

#### 3.2. Difference between Wake Characteristics from Scan, Mean and Snapshot

#### 3.3. Field Wakes’ Characterization

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Scanning geometry used during the experiment, with example retrieved radial velocity measurements ${U}_{r}$ (shading). The instrument location is at the origin, and the coordinates are aligned with the north-south (N-S) and east-west (E-W) directions. Note that the vertical axis is exaggerated for readability.

**Figure 2.**Map of the experiment site (NAD1983, UTM Zone 20 N) showing the locations of the turbine (hollow circle), LiDAR (blue square), the meteorological mast with the sonic anemometer (red triangle), elevation contours and the azimuthal span of sector scans from 3–8 D downstream of the instrument, when the wind direction is aligned with the center of the arcs.

**Figure 3.**Wind direction as estimated from LiDAR measurements at least 0.25 D above the rotor (red line, lower limit = ${\beta}_{min}$ and upper limit = ${\beta}_{max}$), the turbine nacelle position as measured and provided by the turbine operators (black line) and the wind direction at 60 m (gray line) measured with a sonic anemometer (${\beta}_{sonic}$) and averaged over 2 min for readability.

**Figure 4.**Scanning geometry density ($SGD$,

**top**) and coverage ($SGC$,

**bottom**) indices calculated for vertical planes at discrete downstream distances of the turbine from 3 D–8 D, as a function of the absolute value of the wind direction offset ($\left(\right)open="|"\; close="|">\mathrm{\Delta}\beta $) for measurements with the constant scanning geometry described in the text.

**Figure 5.**Schematic of the metrics used to characterize wakes: center (circle), width (horizontal gray solid line), height (vertical gray solid line) and orientation (clockwise angle from vertical α) for a coordinate system looking downstream in y for a wake in the $xz$-plane.

**Figure 6.**Mean wind speed profiles ${\overline{U}}_{L}$ (dashed red) from the large-eddy simulation (LES) over the time it takes to complete a full scan, instantaneous profiles ${U}_{L}$ (solid black) at the first second of the scan and synthetic scan profiles given as mean ± one standard deviation (shaded gray). (

**a**) Vertical profiles; (

**b**) horizontal profiles.

**Figure 7.**Velocity deficit for a synthetic 3D scan (shaded dots) and linear interpolation onto a regular grid (shaded contours) in the vertical (z) and cross-stream (x) directions at discrete distances downstream of the turbine from 3 D–8 D. Wake center from 12-min mean fields (+), from instantaneous fields at the first second of the scan (Δ) and from synthetic scan points (○). Free stream is estimated from downstream data. The wind direction offset for this scan is $\mathrm{\Delta}\beta \sim {7}^{\xb0}$. Percent values given in each frame are $SGC$ (top) and $SGD$ (bottom) in (%).

**Figure 8.**Absolute differences $\left|\mathrm{\Delta}\right|$ in wake metrics for the LES synthetic scans (derived using the disjunct sampling equivalent to field measurements) relative to the metrics derived using the 12-min mean LES dataset (black) and the instantaneous fields from LES at the first second of the scan (gray). The different symbols refer to the free stream profile ${U}_{\infty}\left(z\right)$ used to derive $vd$ and characterize the virtual scan wakes as described in the text. Variables and units of absolute difference are given for each subplot. For all subplots, the horizontal axis is the distance downstream of the turbine.

**Figure 9.**Velocity deficit for a 3D scan from field measurements (shaded dots) and linear interpolation onto a regular grid (shaded contours) in the vertical (z) and cross-stream (x) directions at discrete distances downstream of the turbine from 3 D–8 D. Wake shape (black dots) and center (circle). Wind direction offset for this scan is $\mathrm{\Delta}\beta \sim {7}^{\xb0}$. Percent values given in each frame are $SGC$ (top) and $SGD$ (bottom) in (%).

**Figure 10.**Time series of wake center position in the vertical (

**top**) and horizontal (

**bottom**) directions as estimated from the field measurements, for the intermediate downstream distances between 4 and 7 D. The dashed line marks the hub position (${x}_{H}$, ${z}_{H}$) = (0, 0).

**Figure 11.**Temporal mean (markers) and standard deviation (whiskers) of the spatially-averaged velocity deficit ($vd$) estimated from field measurements. Considering 59 3D scans in the 22-h period. Similarity prediction given by the solid black line.

**Table 1.**Normalized rms differences ${\mathrm{\Delta}}_{rms}$ (%) between synthetic scan wind speeds ${\stackrel{\u02da}{U}}_{L}$, the instantaneous values at the first second of the scan ${U}_{L}$ and the mean over the time it takes to complete the scan ${\overline{U}}_{L}$. Averages are over 80 3D scans, chosen to mimic the observations.

3 D | 4 D | 5 D | 6 D | 7 D | 8 D | |
---|---|---|---|---|---|---|

${\stackrel{\u02da}{U}}_{L}$ and ${U}_{L}$ | 13.2 | 13.2 | 13.5 | 12.3 | 11.5 | 9.9 |

${\stackrel{\u02da}{U}}_{L}$ and ${\overline{U}}_{L}$ | 9.9 | 9.7 | 8.5 | 8.7 | 8.0 | 7.6 |

**Table 2.**Mean wake characteristics obtained from the field measurements and including 59 3D scans. The wake center is given as a vector in the cross-stream ($\widehat{x}$) and vertical directions ($\widehat{z}$) for a coordinate system centered at the turbine hub.

Unit | 3 D | 4 D | 5 D | 6 D | 7 D | 8 D | |
---|---|---|---|---|---|---|---|

center | D$\widehat{x}$, D$\widehat{z}$ | 0.13, 0.08 | 0.18, 0.12 | 0.16, 0.16 | 0.15, 0.20 | 0.10, 0.25 | 0.08, 0.18 |

orientation | ° | 15 | 4 | 15 | 16 | 25 | 6 |

height | D | 0.8 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9 |

width | D | 1.5 | 1.4 | 1.3 | 1.5 | 1.4 | 1.6 |

$vd$ mean | - | 0.22 | 0.19 | 0.15 | 0.13 | 0.12 | 0.11 |

$vd$ SD | - | 0.11 | 0.10 | 0.08 | 0.07 | 0.06 | 0.06 |

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**MDPI and ACS Style**

Doubrawa, P.; Barthelmie, R.J.; Wang, H.; Pryor, S.C.; Churchfield, M.J.
Wind Turbine Wake Characterization from Temporally Disjunct 3-D Measurements. *Remote Sens.* **2016**, *8*, 939.
https://doi.org/10.3390/rs8110939

**AMA Style**

Doubrawa P, Barthelmie RJ, Wang H, Pryor SC, Churchfield MJ.
Wind Turbine Wake Characterization from Temporally Disjunct 3-D Measurements. *Remote Sensing*. 2016; 8(11):939.
https://doi.org/10.3390/rs8110939

**Chicago/Turabian Style**

Doubrawa, Paula, Rebecca J. Barthelmie, Hui Wang, S. C. Pryor, and Matthew J. Churchfield.
2016. "Wind Turbine Wake Characterization from Temporally Disjunct 3-D Measurements" *Remote Sensing* 8, no. 11: 939.
https://doi.org/10.3390/rs8110939