2.1 Experimental methods

The experiments were performed in the wind tunnel of the NTNU in Trondheim, Norway. The closed-loop wind tunnel has a closed test section of 2.71 m × 1.81 m × 11.15 m (width × height × length). The inlet to the test section was equipped with a turbulence grid having a solidity of 35 % and a mesh size of 0.24 m. Further details on the grid are described by Bartl and Sætran (2017).

Two different model wind turbines were used that vary in geometry, blade design and direction of rotation. Those deliberate distinctions allow for an isolation of general effects of wake properties. The turbines will be denoted NTNU and ForWind. Table 1 summarizes the main features and differences of both turbines, further details are described by Schottler et al. (2017b). Figure 1 shows technical drawings. As can be seen, the ForWind turbine was placed on four cylindrical poles to lift the rotor above the wind tunnel boundary layer to a hub height of 820 mm above the wind tunnel floor.

Table 1Summary of main turbine characteristics. The tip speed ratio (TSR) is based on the free-stream velocity u_ref at hub height. The Reynolds number at the blade tip, Re_tip, is based on the chord length at the blade tip and the effective velocity during turbine operation. For the ForWind turbine, 0.96 R was chosen as the radial position to account for the rounded blade tips. The blockage corresponds to the ratio of the rotor's swept area to the wind tunnel's cross-sectional area. The direction of rotation refers to observing the rotor from upstream, with (c)cw meaning (counter)clockwise. The thrust coefficients were measured at γ=0^∘ and corrected for thrust on the tower and support structure.

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Figure 1Technical drawings of the NTNU turbine (a) and the ForWind turbine (b).

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One turbine at a time was placed on a turning table allowing for yaw misalignment, denoted by the angle γ, which is positive for a clockwise rotation of the rotor when observed from above as sketched in Fig. 2.

Figure 2Sketch of the setup, top view. D denotes the respective rotor diameter as listed in Table 1.

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For the NTNU turbine, the reference velocity measured in the empty wind tunnel was $u_{\mathrm{ref},\mathrm{NTNU}}=\mathrm{10}$ m s⁻¹ at a turbulence intensity of TI $={\mathit{\sigma }}_u/\langle u\rangle =\mathrm{0.1}$. For the ForWind turbine, the inflow velocity was $u_{\mathrm{ref},\mathrm{ForWind}}=$7.5 m s⁻¹ and TI =0.05. In both cases, u(t) was homogeneous within ±6 % and the TI within ±3 % on a vertical line at the turbine's position.

In this study we consider two-dimensional cuts through the wake, normal to the main flow direction at a downstream distance of $x/D=\mathrm{6}$ for both turbines as illustrated in Fig. 2. Data were acquired using a Dantec FiberFlow two-component laser Doppler anemometer (LDA) system, recording the u and v component of the flow. The accuracy is stated to be 0.04 % by the manufacturer. During turbine operation, the LDA system was traversed in the y–z plane, normal to the main flow direction. Each measured plane consists of 357 points, 21 in the z direction ranging from −D to +D and 17 points in the y direction ranging from −0.8 D to 0.8 D; see Fig. 3. The resulting distance separating two points of measurement is thus 0.1 D. For one location, 5×10⁴ samples were recorded, resulting in time series of varying lengths of approximately 30 s. As can be seen, the NTNU turbine has a slimmer tower and nacelle relative to its rotor diameter than the ForWind turbine.

Figure 3Non-dimensional measurement grid behind the rotor for γ=0^∘. The respective contours of the turbines are shown in black (ForWind) and red (NTNU). For the NTNU turbine the wind tunnel walls are located at $z/D=\pm \mathrm{1.5}$ and $y/D=\pm \mathrm{1.0}$; for the ForWind turbine they are located at $z/D=\pm \mathrm{2.34}$ and $y/D=\pm \mathrm{1.56}$.

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The grid of physically measured values was interpolated to a grid of $\mathrm{401}\times \mathrm{321}\approx$ 129 000 points for further analyses. The distance between the interpolated grid points is thereby reduced to 0.005 D. Natural neighbor interpolation is used, resulting in a smoother approximation of the distribution of data points (Amidror, 2002).

2.2 Wake center detection

In order to quantify the lateral wake position, we compute the power of a potential downstream turbine as described by Schottler et al. (2017b). A similar approach was shown by Vollmer et al. (2016). We define the potential power of a downstream turbine to be

Figure 4Illustration of the wake center detection method. The hub of a potential downstream turbine is located at the red ×. $\langle u_i\left(t\right){\rangle }_{A_i,t}$ is the spatially and temporarily averaged u component of the velocity. The potential power P^* is calculated for each ring segment and then added up. This procedure is repeated for 50 horizontal hub locations ×, while the position resulting in the lowest value of P^* is interpreted as the wake center.

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The rotor area is divided into 10 ring segments. A_i is the area of the ith ring segment, and $\langle u_i\left(t\right){\rangle }_{A_i,t}$ denotes the temporally and spatially averaged velocity in mean flow direction within the area A_i. P^* is estimated for 50 different hub locations in the range $-\mathrm{0.5}D\le z\le \mathrm{0.5}D$, at hub height. We define the horizontal wake center as the z position resulting in the minimum of P^*. The procedure is illustrated in Fig. 4.

Figure 5〈u〉∕u_ref at γ=0^∘ for the NTNU turbine (a) and ForWind turbine (b). The white lines indicate the contours of the respective turbine. Values exceeding $\langle u\rangle /u_{\mathrm{ref}}=\mathrm{1.1}$ are masked.

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2.3 Examined quantities

TKE is defined by the fluctuations of the three velocity components as

where u^′(t) is the fluctuation around the mean of u(t) so that

For briefness, we write 〈u〉 instead of 〈u(t)〉. As the third flow component w was not recorded, we assume $\langle w^{\prime }(t{)}^{\mathrm{2}}\rangle \approx \langle v^{\prime }(t{)}^{\mathrm{2}}\rangle$ so that Eq. (2) becomes

which will be used in further analyses. This approximation is discussed in Sect. 4 and supported by Fig. 15. For a thorough analysis of the wake turbulence, we examine velocity changes during a time lag τ and refer to them as velocity increments:

Investigating their probability density function (PDF) allows for scale-dependent analyses of turbulent flows, including all higher-order moments of u_τ, and hence all structure functions of order n, $S_{\mathit{\tau }}^n=\langle u_{\mathit{\tau }}^n\rangle$ of a velocity time series (Frisch, 1995). The impact of certain properties of velocity increment PDFs on wind turbines is to date a widely discussed topic in wind energy research (Berg et al., 2016; Milan et al., 2013; Mücke et al., 2011; Schottler et al., 2017c). For more details, we refer the reader to Morales et al. (2012) or Schottler et al. (2017c). Following Chillà et al. (1996), the shape parameter

is used to quantify the shape of the distribution p(u_τ). F(u_τ) is the flatness of the time series of velocity increments:

Equation (6) becomes zero for a Gaussian distribution; larger values correspond to broader, more heavy-tailed PDFs.

λ² is of practical relevance as it provides an analytical expression for the shape of p(u_τ). A discussion about the interpretation is given in Sect. 4. In this analysis, we compute λ² for timescales τ that relate to the rotor diameter D of the respective turbine. Using Taylor's hypothesis of frozen turbulence (Mathieu and Scott, 2000), the length scale r=D is converted to the timescales τ:

whereas 〈u〉 refers to the respective time series, resulting in varying values of τ within a wake. In order to compute u_τ(t) using Eq. (5), evenly spaced data are needed. The procedure applied to uniformly resample the non-uniform LDA data is described in Appendix A. The approach results in a constant sampling rate for each wake.

3.1 The non-yawed wakes

At first, we investigate wakes without yaw misalignment, γ=0^∘. Figure 5 shows the contour plots of the velocity component in mean flow direction 〈u〉∕u_ref for both turbines 6D downstream.

The velocity deficits behind both turbines show a circular shape as expected, exceeding the rotor area, indicating a slight wake expansion. For both wakes, the minimum velocity is $\langle u\rangle /u_{\mathrm{ref}}=\mathrm{0.64}$. Besides those general similarities, some differences are apparent. Both graphs show the tower wake, which is more strongly pronounced for the ForWind turbine. This can be explained by the larger tower diameter relative to the rotor diameter as shown in Fig. 3. Similarly, the four poles the ForWind turbine is placed on (cf. Fig. 1) are likely to enhance this effect. Figure 5 also reveals that the wake behind the ForWind turbine is slightly displaced vertically towards the ground. This effect can be linked to the tower wake, creating an uneven vertical transport of momentum as recently demonstrated by Pierella and Saetran (2017). Next, the NTNU wake shows areas of velocities exceeding $\langle u\rangle /u_{\mathrm{ref}}=\mathrm{1.1}$ at the edges of the velocity deficit, especially in the corners of the contour plot. Very likely, this is a blockage effect as the measurement plane is significantly larger for the NTNU turbine, having a higher blockage ratio (13 % for the NTNU rotor, 5.4 % for the ForWind rotor). As suggested by Chen and Liou (2011), blockage effects are expected for a cross-sectional blockage ratio exceeding 10 % when using model wind turbines, which is confirmed here for wake velocities. In order to better compare both contour plots, values exceeding $\langle u\rangle /u_{\mathrm{ref}}=\mathrm{1.1}$ are masked. To further analyze the wake flows, Fig. 6 shows the contour plots of the TKE behind both turbines.

Figure 6Turbulent kinetic energy (TKE) in m² s⁻² according to Eq. (4) for γ=0^∘. (a) NTNU turbine, (b) ForWind turbine.

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The contours of the TKE appear as circular shape, slightly larger than the rotor area. Behind the NTNU rotor, an outer ring of high TKE values appears more pronounced than in the center region. This observation is significantly less distinct for the ForWind turbine. The differences of the pronounced ring arise most likely from the different blade geometries. The airfoil of the NTNU turbine (NREL S826) has higher lift coefficients for the relevant angles of attack and Reynolds numbers than the ForWind rotor (SD7003 airfoil). A comparison of both airfoils is given by Schottler et al. (2017b). As a result, larger pressure differences between suction and pressure side of the blades are expected, resulting in more pronounced tip vortices shed from the NTNU rotor. Although those are already decayed at $x/D=\mathrm{6}$ (Eriksen and Krogstad, 2017), the tip vortices are likely to be the origin for a pronounced TKE at blade tip locations behind the NTNU rotor. Further increasing in complexity and completeness of the wakes' stochastic description, Fig. 7 shows the contour plots of the shape parameter λ² behind both turbines.

Figure 7λ² for both turbines at γ=0^∘. The timescales τ correspond to the length scale of the rotor diameter; cf. Eq. (8). The red markings × and ∘ show measurement positions for which p(u_τ) were calculated as shown in Fig. 8. (a) NTNU turbine, (b) ForWind turbine. Note the different scaling.

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The length scale τ is related to the rotor diameter D of the respective turbine. The scale is transferred from space to time using Taylor's hypothesis; cf. Eq. (8). In both cases, the contours of λ² show a circular ring, whose diameter is significantly larger than the rotor diameter.

In order to quantify the qualitative shapes of the contours shown in Fig. 7, Fig. 8 shows the increment PDFs of the respective time series, p(u_τ), at the positions indicated by the red marks ($\circ /\times$) in Fig. 7. u_τ is normalized by the standard deviation, σ_τ, for better visual comparison.

Figure 8p(u_τ) of the time series at two measurement positions ($y=\mathrm{0},z=D/\mathrm{2}$; $y=\mathrm{0},z=D$), corresponding to the red marks in Fig. 7. (a) NTNU turbine, (b) ForWind turbine, both at γ=0^∘. The timescales τ are related to the length scales of rotor diameters by Taylor's hypothesis using Eq. (8). For $z/D=\mathrm{1}$ (red curve) the Castaing distribution is shown with ${\mathit{\lambda }}_{\mathrm{NTNU}}^{\mathrm{2}}=\mathrm{0.046}$ and ${\mathit{\lambda }}_{\mathrm{ForWind}}^{\mathrm{2}}=\mathrm{0.17}$ (Castaing et al., 1990). A Gaussian fit is added to guide the eye.

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As shown in black, the positions behind the rotor tips, where λ²≈0, reveal increment PDFs very close to a Gaussian distribution, which holds for both turbines. For z=D, which lies within the ring of large λ² values, p(u_τ) strongly deviates from a Gaussian, showing a heavy-tailed distribution. Figure 8 further shows p(u_τ) based on the model proposed by Castaing et al. (1990). Those distributions were evaluated based on the λ² values computed by Eq. (6) at z=D, visualizing exemplarily how well the distributions' shapes are grasped by λ². Our results show that, depending on the examined quantity, different radial wake regions are of interest. To compare the varying spatial extensions of the three quantities' significant areas, Fig. 9 shows diagonal cuts through the respective contour plots for the non-yawed cases along the line y=z.

Figure 9Diagonal cuts on the line y=z through the contour plots for γ=0^∘. Values are normalized to their respective maximum. The vertical dotted lines mark μ±1σ_u (black) and μ±2σ_u (red) of a Gaussian fit through the velocity deficit shown in blue.

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The area of pronounced TKE approximately coincides with the rotor area. The notable peaks are separated by ≈0.86 D (NTNU) and ≈0.77 D (ForWind) and are significantly less pronounced behind the ForWind rotor as previously described. Clearly, the λ² peaks span a much larger distance: approximately 1.7 D (NTNU) and 2.0 D (ForWind). At their location, the velocity deficit has recovered to ≥90 % of the free-stream velocity in all cases. Thus, for a thorough description of wind turbine wakes, a much larger radial area is of interest as compared to a description restricted to mean values and the turbulent kinetic energy as often done in literature and wake models. An approximation of the lateral extension of high TKE and λ² values based on a Gaussian fit through the velocity deficit is given by μ±1σ_u and μ±2σ_u, respectively, with μ being the mean value and σ_u the standard deviation of the fit. For illustration, the dotted lines in Fig. 9 mark the respective locations. It is shown that the radial areas of TKE and λ² can be related in this way to the velocity deficit. To get a feeling of the impact on potential downstream turbines, Fig. 10 compares p(u_τ) in absolute terms at a free-stream position ($y/D=\mathrm{0.8}$, $z/D=\mathrm{1}$) and at a position featuring high λ² values ($y/D=\mathrm{0}$, $z/D=\mathrm{1}$), exemplary for the ForWind turbine. It becomes clear that velocity increments exceeding 3 m s⁻¹ occur much more frequently within the ring of high λ² values than in the free stream. Hereby we show that this radial position of the wake features significantly different flows than the free stream. To compare more visually, Fig. 11 shows the corresponding time series u_τ(t). Clearly, the spiky signature of extreme events becomes obvious in Fig. 11b, confirming that no free-stream condition is reached at $z/D=\mathrm{1}$.

Figure 10p(u_τ) of the free stream at $y/D=\mathrm{0.8}$, $z/D=\mathrm{1}$ and at $y/D=\mathrm{0}$, $z/D=\mathrm{1}$, exemplary for the ForWind turbine.

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Figure 11Time series of increments u_τ(t) for the positions $y/D=\mathrm{0.8}$, $z/D=\mathrm{1}$ (free stream, a) and $y/D=\mathrm{0}$, $z/D=\mathrm{1}$ (a). The standard deviations σ_τ are indicated in red.

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3.2 Wakes during yaw misalignment

During a yaw misalignment of $\mathit{\gamma }=\pm \mathrm{30}$^∘, the velocity deficits behind both rotors are deflected and deformed as shown in Fig. 12 by the contours of the main flow component 〈u〉∕u_ref.

Figure 12〈u〉∕u_ref during yaw misalignment. (a, b) $\mathit{\gamma }=-\mathrm{30}$^∘, (c, d) γ=30^∘. (a, c) NTNU turbine, (b, d) ForWind turbine. The solid white lines indicate the contours of the respective turbine, while the dashed lines denote the rotor area without yaw misalignment. The red × marks the position of minimum measured velocity 〈u〉. Values exceeding 1.1 are masked for better comparison.

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The wakes are deflected sideways behind both turbines, whereas the lateral direction is dependent on the yaw angle's sign. This is expected due to a lateral thrust component of the rotor as a result of yaw misalignment, which has been observed and described in numerous studies, including Medici and Alfredsson (2006), Jiménez et al. (2010), Vollmer et al. (2016) and Trujillo et al. (2016). The deflection of the velocity deficit is quantified using the approach described in Sect. 2.2; the results are listed in Table 2, including the resulting wake skew angles.

Table 2Wake center location as computed by the approach described in Sect. 2.2 with corresponding skew angles.

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As Table 2 shows, the skew angles behind the ForWind turbine are equal apart from their sign for both directions of yaw misalignment. The NTNU rotor, however, shows slightly different deflection angles for γ=30^∘ and $\mathit{\gamma }=-\mathrm{30}$^∘, which is likely caused by blockage effects, which play a more significant role for the NTNU rotor due to the larger blockage ratio. This can also be seen in Fig. 12, where speed-up effects are visible in the corners. In Schottler et al. (2017b), where the same setup was used¹, the skew angle for the NTNU rotor decreased from $x/D=\mathrm{3}$ to $x/D=\mathrm{6}$, which is a further indication for wall effects due to blockage, especially during yaw misalignment. Furthermore, both values show smaller angles as for the ForWind turbine.

In Fig. 12, minimum 〈u〉 values are marked, showing a vertical transport of momentum in all cases. For γ=30^∘, the wake is moved upwards behind the NTNU turbines and downwards behind the ForWind rotor. Directions are reversed for $\mathit{\gamma }=-\mathrm{30}$^∘. Similar observations have been made by Bastankhah and Porté-Agel (2016). The vertical transport is related to an interaction of a wake's rotation with the tower shadow/ground. Our results isolate this effect, as the direction of vertical transport is opposite comparing both turbines, having an opposite direction of rotation. The fact that the vertical transport is stronger behind the ForWind rotor further supports this explanation as the tower wake is more pronounced due to the larger tower diameter and the structure the turbine is placed on. A deformation of the velocity deficit to a curled “kidney” shape is observed for both turbines during yaw misalignment, whereas it is slightly more pronounced behind the ForWind turbine. The curled shape behind a wind turbine model in yaw has previously been observed by Howland et al. (2016) using a drag disc of 30 mm diameter and by Bastankhah and Porté-Agel (2016) using a rotating turbine model of 150 mm diameter. Figure 12 confirms these findings on two further scales. For a better comparison of the curled shape of the velocity deficit during yaw misalignment, we apply the following parametrization, exemplarily shown in Fig. 13a for the ForWind turbine at γ=30^∘: data points of horizontal cuts through the wake, $\langle u{\rangle }_{y=\mathrm{const}.}$, are fitted by a polynomial. The procedure is repeated for values of y ranging from −0.4 D to 0.4 D. The positions of the polynomials' minima (green marks) are fitted by a quadratic function (red line). Figure 13b shows the comparison of both turbines for $\mathit{\gamma }=\pm \mathrm{30}$^∘.

Figure 13(a) Example of parameterizing the curled shape of the velocity deficit. The green markings show minimal velocities of a polynomial function used to fit the interpolated data points in a horizontal line; y is a constant. The red dashed line shows a quadratic fit based on the green markings. (b) Visualization of the curled shapes of the velocity deficits. For both turbines, the cases $\mathit{\gamma }=\pm \mathrm{30}$^∘ are shown. Dashed lines show a visualization of the wakes tilt, connecting the respective intersections of the curves.

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As already seen in Fig. 12, the wakes behind the ForWind turbine are deflected further and the curled shape is more strongly pronounced, which can be attributed to blockage effects. Figure 13b also shows that the wakes behind both turbines are slightly tilted. Looking at the black curves (ForWind turbine), an asymmetry can be noticed as the curves are tilted towards the left, while the red curves are tilted towards the right. This is illustrated by the gray dashed lines in Fig. 13b which connect the points of intersection for $\mathit{\gamma }=\pm \mathrm{30}$^∘. Not shown in detail here, the same effect was observed for different inflow conditions and other downstream distances, using the same setup and methods as in this study. Similar asymmetries have been observed by Bastankhah and Porté-Agel (2016) for positive and negative yaw angles, which is explained by an interaction of a wake's rotation with the tower wake and the ground. By using turbines of opposite rotation direction, we can attribute the asymmetries in vertical transport and tilt in opposite direction for $\mathit{\gamma }=\pm \mathrm{30}$^∘ to the rotation of rotor and wake.

Adding TKE and λ² contours during yaw misalignment, Fig. 14 shows all three examined quantities, exemplary at a yaw misalignment of $\mathit{\gamma }=-\mathrm{30}$^∘, for both turbines. The shapes of the TKE contours are deformed similarly to 〈u〉. A curled shape evolves, and the differences between both turbines as described for $\mathit{\gamma }=\mathrm{0}{}^{\circ }$ are still notable during yaw misalignment.

Similarly, the circular rings of high λ² values are deformed to a curled shape at $\mathit{\gamma }=\pm \mathrm{30}$^∘. Thus, the general effect of heavy-tailed increment PDFs surrounding the velocity deficit in a wake is stable against yaw misalignment and the resulting inflow variations at the rotor blades. Further, this finding is confirmed in large-eddy simulations (LES) performed at the Universidad de la República, Uruguay, shown in Appendix B.

It is found to be a general effect as it is observed for all wakes considered, independent of yaw misalignment or turbine design.

Figure 14〈u〉∕u_ref (a, d), TKE (b, e) and λ² (c, f) for $\mathit{\gamma }=-\mathrm{30}$^∘ behind the NTNU turbine (a–c) and the ForWind turbine (d–f). The timescale for λ² corresponds to the length scale of the rotor diameter. The red marks show the approximation of the respective parameter's radial extension based on μ±1σ_u (TKE, b, e) and μ±2σ_u (λ², a, d) as described in Sect. 3.1.

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Figure 15(a) TKE k (cf. Eq. 2), (b) TKE k^* (cf. Eq. 4) and (c) 〈v^′(t)²〉 vs. 〈w^′(t)²〉 for all measurement positions. R_P is Pearson's correlation coefficient. Data are based on a wake 6D behind the NTNU turbine and γ=30^∘.

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The red markings in Fig. 14 show the approximation of the radial extension of the TKE and λ² based on μ±1σ_u and μ±2σ_u. μ and σ_u correspond to Gaussian fits of the velocity deficits at various horizontal cuts (y is a constant) from $y/D=-\mathrm{0.5}$ to $y/D=\mathrm{0.5}$, with μ being the fit's mean value and σ_u the standard deviation. It is shown that the method results in quite good first-order approximations, even during yaw misalignment.