4.1 Power output, instantaneous and mean flow field

In this section, we start by comparing the row- and time-averaged power and the row-averaged root-mean-square (r.m.s.) of power fluctuations between the two cases, which are calculated by first averaging over time and then over each row. Figure 3(a) compares the calculated row- and time-averaged power $\langle P\rangle _{r}$ for both cases, normalised by the power of the first row, where $\langle \cdot \rangle _{r}$ indicates row averaging, with the field data obtained from Barthelmie et al. (Reference Barthelmie, Rathmann, Frandsen, Hansen, Politis, Prospathopoulos, Rados, Cabezón, Schlez and Phillips2007, Reference Barthelmie, Hansen, Frandsen, Rathmann, Schepers, Schlez, Phillips, Rados, Zervos and Politis2009). Consistent with the measurements, the first row extracts the most power from the incoming wind and the following rows extract only approximately 60 % of the power of the first row. Both AS-RN and AS-R simulations yield very similar average power per row indicating that the presence of the nacelle model does not have a significant effect on the average power extracted by the wind farm. Figure 3(b) shows the row-averaged r.m.s. of power fluctuations $\langle P_{rms}\rangle _{r}$ per row for each simulation. A general trend that the r.m.s. power increases towards an asymptotic maximum of 37 % of the average power is observed for both simulations. However, some differences are observed in that $\langle P_{rms}\rangle _{r}$ for the AS-R case exhibit more spatial variability compared to the AS-RN case. The r.m.s. of the fluctuations of the power produced by the entire wind farm is also found to be 3.93 % and 3.33 % for AS-RN and AS-R cases, respectively.

Figure 3.

(a) Row- and time-averaged power and (b) row-averaged r.m.s. of power fluctuations per row from measurements (Barthelmie et al. Reference Barthelmie, Rathmann, Frandsen, Hansen, Politis, Prospathopoulos, Rados, Cabezón, Schlez and Phillips2007) (circles) and AS-RN (squares) and AS-R (triangles) cases. It is noted that (b) shows the row-averaged r.m.s. of the power fluctuations from a single turbine instead of the r.m.s. of the fluctuations of the power from a row of turbines.

Row- and time-averaged downwind velocity contours, $\langle U\rangle _{r}$ , where the upper case letter indicates time averaging and $\langle \cdot \rangle _{r}$ indicates row averaging, are shown in figure 4 to illustrate the influence of nacelle model on turbine wake recovery in the first row (a) and the last row (b), respectively. In the AS-RN simulation we see the development of inner wakes that forms directly behind the nacelles, and the outer wake that forms over the extent of the entire rotor disk. We refer to Kang et al. (Reference Kang, Yang and Sotiropoulos2014) and Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016) for more details on the interaction between the inner wake and the outer wake. In both rows, the effect of nacelles in the near wakes is observed as a strong inner wake that forms behind the nacelles in the AS-RN simulation. While on the other hand, the AS-R simulation reveals that there is a strong unphysical centreline jet due to the absence of the nacelles. However, in both rows, there is little difference in the average downwind velocity in the AS-RN and AS-R simulations at downwind distances $x/D>3$ . This is consistent with the mean turbine power results that showed little difference between the AS-RN and AS-R cases (see figure 3 a). Figure 4 also shows the differences in the mean wake between the first and last rows. It is seen that the wake recovery is slower and the wake width is slightly narrower in the first row than in the last row.

Figure 4.

Contours of the row- and time-averaged downwind velocity, $\langle U\rangle _{r}$ , for (a) the first row and (b) the last row in the horizontal plane at the hub height. The top and bottom figures are for cases AS-RN and AS-R, respectively.

Figure 5.

Contours of the row-averaged TKE, $\langle k\rangle _{r}$ , for (a) the first row and (b) the last row at the hub height. The top and bottom figures are for cases AS-RN and AS-R, respectively.

Row-averaged turbulence kinetic energy (TKE) contours, $\langle k\rangle _{r}$ , are shown in figure 5, which reveal the influence of nacelles on the variation of TKE for turbine wakes in the first row (a) and the last row (b), respectively. It is seen that the nacelles have a significant effect on the TKE at both near and far wake locations, where the TKE from the simulation with the nacelle model (AS-RN) is higher than the simulation without the nacelle model (AS-R) – this result is consistent with the earlier findings of Kang et al. (Reference Kang, Yang and Sotiropoulos2014) for the wake of single turbine. The presence of the nacelles in the AS-RN case produces higher TKE right behind the nacelles in the inner wakes while the TKE is lower in the same location for the AS-R simulation where the unphysical jet is formed. As observed in Kang et al. (Reference Kang, Yang and Sotiropoulos2014) and Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016), the inner wake expands outwards intercepting the outer wake (rotor tip shear layer). At $x/D=2$ , the inner wake interacts with the outer wake and produces high levels of TKE. The simulation with the nacelles, AS-RN, produces higher levels of TKE after $x/D=2$ compared with the AS-R simulation indicating that the nacelles have an effect on the turbulence intensity far downwind of the turbine. However, the location of the maximum TKE in the wake for both simulations is around $x/D=3$ . Significant differences in the TKE levels are observed in the last row where the TKE in the downwind wake is higher due to the higher levels of incoming TKE in the upwind turbine wakes.

Figure 6.

Contours of the row-averaged Fourier power spectral density of the downwind component of the velocity fluctuations pre-multiplied by frequency, $\langle f\unicode[STIX]{x1D6F7}\rangle _{r}$ , as a function of the normalised downwind distance, $x/D$ , and Strouhal number, $St=fD/U_{h_{1}}$ , at hub height $y=y_{h}$ for the AS-RN simulation. Vertical blue lines delineate the axial location of each wind turbine. Annotations refer to the region of high PSD that is associated with wake meandering and the circles mark the maximum PSD in the wake meandering region in each corresponding turbine.

4.2 Wake meandering statistics

Motivated by the differences in the TKE distributions of the AS-RN and AS-R turbine wakes, we proceed in this section to investigate the statistics of wake meandering. First, we assess the frequency of wake meandering within the wind farm through the Fourier power spectrum analysis of the downwind component of the velocity fluctuations pre-multiplied by frequency, $f\unicode[STIX]{x1D6F7}$ . Figure 6 shows the contours of the power spectral density (PSD) for the AS-RN simulation as a function of the downwind distance $x/D$ and Strouhal number $St=fD/U_{h_{1}}$ , where $U_{h_{1}}$ is the average free-stream downwind velocity at hub height for the first turbine row. The PSD is calculated by extracting time series of data at the hub height and at the time interval of $T/11$ for the duration of the simulation at every grid node in the downwind direction (1561 nodes) through the centre of the column of wind turbines. Finally, the PSD is averaged over the turbine wakes in each row (the same downwind distance from each turbine in each row). Downwind of each turbine there are several regions of high PSD that occur at Strouhal numbers greater than unity. These are attributed to the rotor rotating frequencies that persist downwind (this is not very obvious in the near wake where the nacelle wake dominates and for which the dominant frequency is much lower). At Strouhal numbers less than unity there is a region of high PSD that develops approximately $3D$ downwind of each turbine and persists for several rotor diameters until the next downwind turbine. This region behind each turbine spanning, $0.1<St<0.3$ , is the signature of wake meandering in spectral space. In figure 6 circles mark the maximum PSD in the wake meandering region, which shows a decrease in the meandering associated Strouhal number from the first row to the third row. Moreover, we see that the downwind extent of the wake meandering region and the strength of the PSD associated with meandering frequency increases from the first row to the fifth row suggesting that wake meandering experiences an increase in energy in the first several rows of the wind farm. This energy increase is consistent with the increase in TKE and power fluctuations described above.

To assess the effect of nacelle-induced coherent structures on wake meandering dynamics, we show in figure 7 the contours of the pre-multiplied PSD in the $St-x$ space for the AS-R simulation. Comparing figures 6 and 7 we can see that in both simulations the peak wake meandering frequencies in each row are in the same range of $0.1<St<0.3$ . The peak Strouhal number decreases from the first row to the third row in the AS-RN case, while the peak Strouhal number remains nearly constant. The most significant difference between the AS-RN and AS-R simulations is the magnitude of the PSD in the wake meandering region. The contour levels in the wake meandering region in figure 7 are clearly lower than those in figure 6, indicating that the amount of energy in wake meandering is considerably less in the AS-R simulation. This further suggests that the helical coherent structures induced by the nacelles energise the wake meandering.

Figure 7.

See figure 6 for caption. The AS-R case is shown here.

To further elucidate the statistics of the wake meandering within the wind farm, we construct the meander profiles behind all the turbines for each simulation. Similar to the previous work of Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016), the three-dimensional spatio-temporal meander profiles are obtained from instantaneous velocity fields by employing a three-step process. We recognise that a temporally and spatially evolving flow field $\boldsymbol{u}(\boldsymbol{x},t)$ (bold indicates a vector quantity) can be rewritten in terms of its triple decomposition as follows:

(4.1) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{U}(\boldsymbol{x})+\boldsymbol{u}_{c}(\boldsymbol{x},t)+\boldsymbol{u}_{i}(\boldsymbol{x},t),\end{eqnarray}$$

where $\boldsymbol{U}(\boldsymbol{x})$ is time averaged, a strictly spatial term, $\boldsymbol{u}_{c}(\boldsymbol{x},t)$ is the coherent term and $\boldsymbol{u}_{i}(\boldsymbol{x},t)$ is the incoherent term (Hussain Reference Hussain1986). In order to obtain a flow field, a filtering procedure is employed to remove random fluctuations from the flow. A coherent time scale $\unicode[STIX]{x1D70F}_{c}$ for wake meandering is calculated in the wake of each turbine (as described in Foti et al. Reference Foti, Yang, Guala and Sotiropoulos2016) and is found to be similar to the turbines numerically studied by Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016, Reference Foti, Yang and Sotiropoulos2018b ), $\unicode[STIX]{x1D70F}_{c}=0.7T$ . The filtered velocity field:

(4.2) $$\begin{eqnarray}\widetilde{\boldsymbol{u}}(\boldsymbol{x},t)=\boldsymbol{U}(\boldsymbol{x})+\boldsymbol{u}_{c}(\boldsymbol{x},t)\end{eqnarray}$$

is first calculated using finite-time interval average, employing fluctuation analysis and the central limit theorem (Chrisohoides & Sotiropoulos Reference Chrisohoides and Sotiropoulos2003), over the coherent time scale of the velocity field throughout the duration of the simulation as follows:

(4.3) $$\begin{eqnarray}\widetilde{\boldsymbol{u}}(\boldsymbol{x},t)=\frac{1}{\unicode[STIX]{x1D70F}_{c}}\int _{t-\unicode[STIX]{x1D70F}_{c}/2}^{t+\unicode[STIX]{x1D70F}_{c}/2}\boldsymbol{u}(\boldsymbol{x},t^{\prime })\,\text{d}t^{\prime }.\end{eqnarray}$$

Next, the downwind velocity minima, which track the location of the wake meandering centreline, are obtained in the wake of each turbine. A bounding box in the radial direction around each turbine is employed to limit the velocity minima search to only the wake of that turbine. Finally, a continuous meander profile is created through a low-bypass spatial filtering with a filter width of $D/2$ . For details the reader is referred to Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016).

Figure 8 shows three different views of the downwind velocity and the meander profiles projected onto the horizontal plane behind each turbine for the AS-RN simulation at one time instance. Turbine–wake interactions are observed as the wakes from upwind turbines directly interact with downwind turbines. Figure 8 clearly shows that wakes undergo meandering, which may cause turbine power fluctuations due to its inherent unsteadiness. In fact, we see that the power fluctuations as shown in figure 3(b) increase with downwind distance, which provides evidence that upwind turbine wakes affect the power fluctuation levels of downwind turbines. The effects of upwind turbine wakes seem to become saturated around row 6 where the power fluctuations begin to reach a near steady value. The full-view figure shows the overall pattern of the meander profiles at this time instance. From the zoomed-in figures, we can see that the meander profiles remain fairly straight in the first row, but fluctuate wildly for rows deep in the wind farm. From the zoomed-in figures, we can also observe how the meander profiles affect the downwind turbines: (i) the downwind turbine experiences higher incoming velocity when the incoming meander profile (at the velocity minima) is displaced laterally outside the location of the downwind turbine rotor; (ii) the downwind turbine experiences a low incoming velocity when the meander profile directly intersects the downwind turbine. Over a period of time, we can reason that the inherent unsteadiness of the wake meandering from upwind turbines increases the turbulence intensity and causes power fluctuations and dynamic loads for the downwind turbines. In what follows, we proceed to use the meander profiles to investigate how the presence of the nacelles affects the dynamics of wake meandering.

Figure 8.

Computed meander profiles (black line) of the instantaneous downwind velocity at hub height of AS-RN simulation (zoomed-in figures on (b) and (c)). The turbine blade locations are shown in green.

First, we show how the nacelle-induced unstable hub vortex contributes to the coherent and incoherent motions in the wake by comparing in figure 9 the probability density functions (PDFs) of the coherent and incoherent velocity magnitude. The coherent and incoherent velocity magnitude is selected only along the paths of the meander profile behind each turbine for the first row and the last row. As seen, the probability is larger for high coherent velocity magnitude for the AS-RN case compared to the AS-R case. This quantitatively indicates that additional energy is augmented to coherent structures in wake meandering by the unstable hub vortex. The PDFs of the incoherent velocities, on the other hand, are very similar for the simulations with and without the nacelle model.

Figure 9.

The PDF of the coherent $f(|u_{c}|/U_{h})$ and incoherent $f(|u_{i}|/U_{h})$ velocity magnitude normalised by the hub height velocity $U_{h}$ using the velocity along the meander profile at each time instance for both the AS-RN (solid) and AS-R (dashed) cases. The PDFs of the coherent velocity magnitude for the AS-RN and AS-R cases are a black line and a red line, respectively. The PDFs of the incoherent velocity magnitude for the AS-RN and AS-R cases are a blue line and cyan line, respectively.

Amplitudes and wavelengths of the meander profiles, derived by projecting the three-dimensional meandering profiles onto the two-dimensional hub height plane to obtain a wave-like form, are used to interpret the coherent motions and the dynamics of the instantaneous wake. The amplitudes of a meander profile are determined by the distance from the centre of the wave-like meander profile to the spanwise extrema. Similarly, wavelengths of the meander profile are captured by calculating the distance between two successive maxima or minima. In Foti et al. (Reference Foti, Yang and Sotiropoulos2018b ) the average amplitude of the meander profile for a single turbine was shown to be scaled by a characteristic length scale $L_{T}$ , which is given by

(4.4) $$\begin{eqnarray}L_{T}=(T/\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2})^{1/2}/f_{m},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the air density, $R$ is the rotor radius and $T$ is the thrust on each turbine, which is defined as $T=\unicode[STIX]{x1D70C}C_{T}\unicode[STIX]{x03C0}R^{2}U_{h}^{2}/2$ and depends on the thrust coefficient $C_{T}$ . The meandering frequency $f_{m}=St_{m}D/U_{h}$ , where $St_{m}$ is the Strouhal number of wake meandering, is calculated directly from the power density spectra of each wind turbine wake as shown in figures 6 and 7 for both the AS-RN and AS-R cases, respectively, using the incoming velocity associated with each turbine. It is important to note that the Strouhal number range used in figures 6 and 7 is based on the average free-stream velocity for the first turbine row. The dominant Strouhal numbers for wake meandering $St_{m}$ are in the range of 0.17–0.25. This range of Strouhal numbers is consistent with a number of studies (Okulov & Sørensen Reference Okulov and Sørensen2007; Medici & Alfredsson Reference Medici and Alfredsson2008; Chamorro et al. Reference Chamorro, Hill, Morton, Ellis, Arndt and Sotiropoulos2013; Okulov et al. Reference Okulov, Naumov, Mikkelsen, Kabardin and Sørensen2014; Howard et al. Reference Howard, Singh, Sotiropoulos and Guala2015; Foti et al. Reference Foti, Yang, Guala and Sotiropoulos2016). Figure 10(a) shows the row-averaged amplitude normalised by $L_{T}$ for each row of the wind farm in AS-RN and AS-R. With this scaling, we observe trends where the AS-RN case produces larger amplitudes compared to the AS-R case. The higher amplitude is attributed to the unstable hub vortex that forms behind the nacelle in the inner wake and interacts with the outer wake as explained in Kang et al. (Reference Kang, Yang and Sotiropoulos2014) and Foti et al. (Reference Foti, Yang, Guala and Sotiropoulos2016). The manifestation of the increased wake meandering amplitude in the turbulence statistics is the higher TKE in the wake at $x/D=3$ for the AS-RN case compared to the AS-R case. However, the peak amplitudes do not increase as the row number increases, suggesting that within a wind turbine array the $L_{T}$ meandering length scale can account for some wind farm interactions. Comparing the average amplitudes in figure 10(a) to single turbine cases described in Foti et al. (Reference Foti, Yang and Sotiropoulos2018b ), we find that the average amplitudes for each row have similar magnitudes. The row-averaged wavelength normalised by the rotor diameter $D$ is shown in figure 10(b). The wavelengths of wake meandering increase to over the size of the diameter. Larger wavelengths for the simulation with the nacelle model, AS-RN, compared to the AS-R case are observed. For each row around the distance of $4D$ behind the turbine the wavelength starts to decrease, which can be partly attributed to the blockage effect of the downwind turbine. The last three rows have a larger wavelength, especially the final row, where there is no downwind turbine to affect the downwind growth of the meandering. However, we observe a slight decrease in wavelength in the final row, as well.

Figure 10.

Comparison of (a) average wake meandering amplitude $\langle \overline{A}\rangle _{r}$ normalised by the characteristic length scale $L_{T}$ (4.4) and (b) average wavelength $\langle \overline{\unicode[STIX]{x1D706}}\rangle _{r}/D$ of wake meandering normalised by the rotor diameter $D$ between AS-RN (solid line) and AS-R (dashed line) simulations.

4.3 Analysis of wake expansion using wake meandering statistics

In this section, we explore the expansion and area of influence of the turbine wakes in the wind farm using wake meandering statistics in order to further elucidate the effects of wake meandering on the dynamics of the wind farm and the role that nacelle-induced coherent structures play in determining such dynamics. By understanding the dynamics of the expansion of wake meandering, we hope to gain insights that will enable strategies for control power fluctuations and turbine loading and optimising wind farm layout. Figure 11 shows the row-averaged ratio of the average meander amplitude to the average meander wavelength $\langle \overline{A}/\overline{\unicode[STIX]{x1D706}}\rangle _{r}$ , also called the $A{-}\unicode[STIX]{x1D706}$ ratio, for each row in both the AS-RN and AS-R cases. The ratio of the amplitude to the wavelength is employed to demonstrate and quantify the dynamic growth of the wake meandering and the expansion of the wake. The quantity is assumed to be a surrogate for the ratio of the transverse to downwind growth of wake meandering. As shown in figure 11, the profiles of the $A{-}\unicode[STIX]{x1D706}$ ratio in each row are nearly linear in the region $3<x/D<7$ downwind of each turbine monotonically increasing with distance from the turbine. However, the linearity of the $A{-}\unicode[STIX]{x1D706}$ ratio is not readily observable from figure 10(a,b) as both the average amplitude and wavelength cease to grow after $x/D>6$ for each turbine row, which is partly attributed to the presence of the next turbine row. The $A{-}\unicode[STIX]{x1D706}$ ratio indicates that with increasing distance from the turbine, the meander amplitude increases with respect to the wavelength, and that the wake meandering area of influence, i.e. the area bounded by average maximum spanwise extent of the meander profiles behind the turbine, also expands. Figure 11 also reveals some differences between rows, such as the maximum $A{-}\unicode[STIX]{x1D706}$ ratio achieved behind each turbine. The sixth row has the maximum $A{-}\unicode[STIX]{x1D706}$ ratio, which corresponds to the row of saturation of the power fluctuations and TKE in the wake. Moreover, the slope of the $A{-}\unicode[STIX]{x1D706}$ ratio is lower for the AS-R case compared to the AS-RN case.

Figure 11.

Ratio of average wake meandering amplitude to average wake meandering wavelength $\langle \overline{A}/\overline{\unicode[STIX]{x1D706}}\rangle _{r}$ in AS-RN (solid) and AS-R (dashed) cases.

The $A{-}\unicode[STIX]{x1D706}$ ratio can be further investigated through linear regression over the linear range of the profile to obtain two parameters: the slope $c_{1}$ and intercept $c_{2}$ . The linearity of the $A{-}\unicode[STIX]{x1D706}$ ratio leads to the following equation (we remove $_{r}$ for clarity):

(4.5) $$\begin{eqnarray}\frac{\langle \overline{A}\rangle (x^{\ast })}{D}=c_{1}x^{\ast }\frac{\langle \overline{\unicode[STIX]{x1D706}}\rangle (x^{\ast })}{D}+c_{2}\frac{\langle \overline{\unicode[STIX]{x1D706}}\rangle (x^{\ast })}{D},\end{eqnarray}$$

where $x^{\ast }=(x-x_{t})$ is the downwind distance from the turbine location $x_{t}$ , and $\overline{A}$ and $\overline{\unicode[STIX]{x1D706}}$ are both functions of $x^{\ast }$ . The row-averaged slope of the $A{-}\unicode[STIX]{x1D706}$ ratio $c_{1}$ is shown in figure 12 to quantify how the $A{-}\unicode[STIX]{x1D706}$ ratio grows for each row. Overall, the slopes of the $A{-}\unicode[STIX]{x1D706}$ ratio from the AS-RN case are higher than those from the AS-R case simulation. This further suggests that the nacelle-induced coherent structures affect wake meandering and augment the turbulence intensity, which increases the expansion or spanwise extent of the wake meandering. While the first rows between the two test cases show a significant difference, there is generally a 10 to 50 % difference between the wake meandering expansion.

Figure 12.

Coefficient $\langle c_{1}\rangle _{r}$ (4.5), slope of average wake meandering amplitude to average wake meandering wavelength averaged over each row, in the AS-RN (solid) and AS-R (dashed) cases.

The growth rate of wake meander is defined by taking the derivative of the amplitude in (4.5) with respect to the downwind distance $x^{\ast }$ from the turbine as shown in the following (we remove $_{r}$ for clarity):

(4.6) $$\begin{eqnarray}\frac{\text{d}}{\text{d}x^{\ast }}\frac{\langle \overline{A}\rangle (x^{\ast })}{D}=c_{1}\frac{\langle \overline{\unicode[STIX]{x1D706}}\rangle (x^{\ast })}{D}+c_{1}x^{\ast }\frac{\text{d}}{\text{d}x^{\ast }}\frac{\langle \overline{\unicode[STIX]{x1D706}}\rangle (x^{\ast })}{D}+c_{2}\frac{\text{d}}{\text{d}x^{\ast }}\frac{\langle \overline{\unicode[STIX]{x1D706}}\rangle (x^{\ast })}{D}.\end{eqnarray}$$

The growth rate of wake meandering is comprised of three terms: the mean wavelength, the product of the downwind distance and gradient of the mean wavelength and the gradient of the mean wavelength. Figure 13(a) shows the downwind variations of wake meander growth rate. While the wake meander growth rate is positive, indicating meander growth throughout the wind farm, in general the wake meander growth rate decreases with the downwind distance. The positive wake meander growth rate also indicates that the turbine downwind area influenced by the wake meandering increases with downwind distance. The turbines in the downwind row, as suggested above, create a blockage stunting the expansion of the wake meandering. Moreover, the presence of the nacelles effectively increases the growth of the amplitude and wake meandering.

The individual terms of the growth rate of wake meandering in (4.6) are shown in figure 13(b) in order to assess which term has the largest effect and how to relate the growth rate to physical observations in the wake. It is seen that the wavelength term (the first term on the right-hand side of (4.6)) has the largest effect at all turbine downwind locations. The effect of this term is relatively constant throughout the wake and shows a significant dependence on the nacelle model. This term indicates that a portion of the growth rate of wake meandering is directly proportional to the wavelength, which has been shown to scale with the diameter of the wind turbine (Foti et al. Reference Foti, Yang and Sotiropoulos2018b ). The last two terms in (4.6) show the influence of the spatial gradient of the wavelength on wake meandering expansion. The third term on the right-hand side of (4.6) is observed to have large positive effect in the near wake where significant wake expansion happens, decreasing with downwind distance and having negative effect for further downwind locations. The second term follows a similar trend as the third term, however, with a significantly less influence on the magnitude. Overall, we see that the effect of wavelength (the first term in (4.6)) determines the mean spatial (throughout the wake) growth rate and the effect of the gradient of the wavelength (the third term in (4.6)) determines how the growth rate changes in the downwind direction.

Figure 13.

(a) The downwind evolution of the slope of the average amplitude averaged over each row. (b) The downwind evolution of individual contributions to the slope of the amplitude averaged over each row in AS-RN (solid line, filled marker) and AS-R (dashed line, unfilled marker) cases. See (4.6) for definition of various terms.

4.4 Analysis of the wake expansion using velocity deficits

In the previous section we quantified the wake expansion and the wake meandering influenced area in terms of the growth rate of wake meandering amplitudes. In this section we explore three other metrics to quantify wake expansions for comparison to our wake meandering statistics. Figure 14 shows several row- and time-averaged velocity deficit profiles for the sixth row plotted on the horizontal plane at hub height. The profiles indicate that there is a difference in the velocity deficit between the AS-RN and AS-R cases suggesting that the nacelles-induced coherent structures affect the wake recovery, even in the mean sense. The profiles also show that the expansion of the wakes are slightly different. In what follows, we discuss the metrics for calculating wake growth rates to quantify wake expansion.

Figure 14.

Profiles of the row- and time-averaged windwise velocity deficit $1-\langle U\rangle _{r}/U_{h}$ for row 6 turbines on the hub height plane at $x/D=1$ , 2, 3, 4 and 5.

Some previous studies have sought methods to define and track the wakes behind wind turbines (mostly in the mean sense) by using quantities such as mid-point of the wake (Jiménez, Crespo & Migoya Reference Jiménez, Crespo and Migoya2010), centre of mass (Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016) and Gaussian modelling (Trujillo et al. Reference Trujillo, Bingöl, Larsen, Mann and Kühn2011; Fleming et al. Reference Fleming, Gebraad, Lee, van Wingerden, Johnson, Churchfield, Michalakes, Spalart and Moriarty2014). The last method employs fitting a Gaussian-like profile to the wake profiles as a function of downwind distance. We use a similar method in order to quantify the expansion of the wake. However, as with the construction of wake meandering, the time-resolved instantaneous flow field is too noisy to perform an accurate fit. For that, we employ the previously discussed finite-time interval averaging of the wake over the wake coherent time scale to acquire the filtered flow field $\widetilde{\boldsymbol{u}}$ as defined by (4.2) to identify coherent wake features.

Figure 15.

Downwind evolution of the average of the wake radius parameter $\langle \unicode[STIX]{x1D70E}_{f}\rangle _{r}$ normalised by $D$ for each row for the AS-RN (solid) and AS-R (dashed) cases.

By filtering the flow field, we are able to remove the incoherent velocity fluctuations from the wake while still retaining the qualities of the meandering wake, such as the off-centre location of the velocity minima. The filtered windwise velocity deficit, or turbine-added filtered velocity, which is computed by subtracting the filtered windwise velocity one diameter upwind of the turbine on the hub height plane, is fitted to the following equation:

(4.7) $$\begin{eqnarray}\widetilde{u}(x,z)-\widetilde{u}(-D,z)=\frac{B}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{f}}\exp \left(-\frac{(x-\unicode[STIX]{x1D707}_{f})^{2}}{2\unicode[STIX]{x1D70E}_{f}^{2}}\right),\end{eqnarray}$$

where $B$ is the amplitude of the Gaussian curve, $\unicode[STIX]{x1D70E}_{f}$ is the filtered wake radius parameter and $\unicode[STIX]{x1D707}_{f}$ is the centre of the Gaussian curve. The optimal solution to these three parameters is sought with a curve fitting procedure. The curve fit is applied at downwind locations behind each turbine, where $\unicode[STIX]{x1D707}_{f}$ is given an initial guess of the velocity minimum location from the previous section. In this analysis, we intend to use the wake radius trends provided by the $\unicode[STIX]{x1D70E}_{f}$ , the fitted filtered wake radius parameter, to quantify the wake radius at time scales and length scales comparable to the wake meandering time and length scales. In this sense, we argue that the Gaussian curve fit can be applied to extract such trends.

Figure 15 shows the wake radius parameter $\langle \unicode[STIX]{x1D70E}_{f}\rangle _{r}$ averaged with respect to its downwind location and averaged over each row. We scale the wake radius parameter by 1.75 to include 42 % of the data on one side of the mean and in order to correlate it with the defined wake radius at the turbine rotor plane. First, we draw attention to the near wake region where the wake radius parameter is approximately $D/2$ , the turbine blade tip location. In order to quantify the wake with the Gaussian fit, we multiply the wake radius parameter by 1.75 such that close to the turbine, the wake radius $D/2$ corresponds with the value of the wake radius parameter. This is in agreement with the idea of employing Gaussian statistics to capture the wake because in Gaussian statistics, $\unicode[STIX]{x1D70E}_{f}$ is essentially the standard deviation that comprises only 34 % of the data on one side of the mean (centre of the wake).

As shown in figure 15, the downwind evolution of the wake radius parameter increases, nearly linearly, throughout the wake. Around $x/D>6$ downwind of each turbine, the growth rate decreases due to the next row of turbines. The maximum wake radius achieved in each wake is dependent on the row. The first row has the smallest wake radius while the growth of the wake increases with the number of rows. For the interior rows, the wake radius increases to approximately 80–90 % of the diameter. Figure 15 also reveals a difference between the wake radius for the AS-RN and AS-R cases. The case without the nacelles shows a marked increase in the wake radius parameter in the far wake locations and across different rows while is lower in the near wake. This is counter to the observation on the downwind variation of wake meander statistics (e.g. figure 10 a), where the meander amplitudes and wavelengths are higher for the AS-RN case.

We also apply the Gaussian curve fitting method to the time-averaged flow field $U$ to extract an average wake radius parameter $\unicode[STIX]{x1D70E}_{a}$ to provide details on wake expansion in the mean sense. The expansion rate of the wake, $\unicode[STIX]{x1D6FC}_{f}$ and $\unicode[STIX]{x1D6FC}_{a}$ based on $\unicode[STIX]{x1D70E}_{f}$ and $\unicode[STIX]{x1D70E}_{a}$ , respectively, can be obtained from the slope of the downwind evolution of the wake width by taking the windwise gradient with respect to the distance from the turbine. Due to the near linearity at $x/D>3$ for $\langle \unicode[STIX]{x1D70E}_{f}\rangle _{r}$ and at $1<x/D<5$ for $\langle \unicode[STIX]{x1D70E}_{a}\rangle _{r}$ (see figure 15), a single wake expansion rate per turbine can be obtained. Figure 16 shows the wake expansion for both the filtered and average wake radius parameter for both simulated cases. The filtered wake expansion is shown to increase from the first row to the second row and remain relatively constant through the rest of the turbine rows. The wake expansion calculated from the filtered velocity deficits from the AS-RN simulation is less than that from the AS-R case. The wake expansion calculated from the time-averaged velocity deficits is more dependent on the row number compared to the filtered wake expansion. For both simulation cases, the average wake expansion increases from the first row to the fourth row. Then, there is a decrease to a more constant wake expansion after row 6. This result is also consistent with the saturation of wake energy and the onset of a fully developed wind farm regime. The average wake expansion does not show as pronounced difference between the AS-RN and AS-R simulations, but towards the last few rows the wake expansion for the AS-R simulation is higher. We compare these results with the wake meandering expansion rate shown in figure 12 or 13 and see that the expansion magnitude is relatively close. However, the trends of the case with the nacelle model and without the nacelle model are reversed.

Before we discuss the difference between the wake expansion and wake meandering expansion, we consider the wake width and a wake growth rate defined by the turbine-added mean velocity. The wake width using the turbine-added mean velocity is found where $U(x,z)-U(-D,z)=-0.01U_{h}$ . The wake growth rate $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x0394}U}$ is then defined as the downwind derivative of the computed wake radius. The growth rate is nearly constant between $2<x/D<5$ . The row-averaged growth rates are also shown in figure 16 for each simulation case. The mean velocity wake growth rate increases from the first to the second row and then decreases to a near constant throughout the other rows in the wind farm. In row numbers greater than 3, the turbine-added mean velocity wake recovers before the subsequent row and the growth rate decreases to zeros as detailed in figure 16 for the last row.

Figure 16.

The row-averaged wake growth rates $\unicode[STIX]{x1D6FC}_{f}$ , $\unicode[STIX]{x1D6FC}_{a}$ and $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x0394}U}$ computed using the Gaussian curve fitting on the filtered average velocity deficit profiles, the Gaussian curve fitting on the average velocity deficit profiles and the mean velocity deficit respectively. AS-RN $\unicode[STIX]{x1D6FC}_{f}$ (black square), AS-R $\unicode[STIX]{x1D6FC}_{f}$ (black circle), AS-RN $\unicode[STIX]{x1D6FC}_{a}$ (red plus), AS-R $\unicode[STIX]{x1D6FC}_{a}$ (red x), AS-RN $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x0394}U}$ (blue triangle) and AS-R $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x0394}U}$ (blue diamond).

Finally, we compare in figure 17 wake influenced regions computed using the approaches discussed earlier for the first and last row for both simulation cases for the purpose of assessing the role of the nacelle-induced coherent structure in the spanwise expansion of the wake and provide general insights into wind turbine layouts in wind farms. The wake influenced area is defined as the area of the wake between the upper and lower half lines defined by the different wake expansion computation techniques. The growth rate of wake meandering, only quantified from $x/D>3$ , has an origin at the centreline of the turbine with the other expansion coefficients defined to have an origin at the turbine tip. These origins are based on how the expansion is defined. The boundary line of the filter wake expansion and average wake expansion are defined by the filtered and average wake radius parameter, respectively, with respect to the downwind distance. The boundary lines for the mean velocity wake growth is determined by the spanwise location where the mean velocity deficit approaches zero.

As shown in figure 17, in the first row for the AS-RN case, the wake area determined by the mean velocity deficit growth rate extends from the turbine blade tip and linearly grows towards the second row. Both the filtered and average wake expansion wake boundaries behave in a similar manner, starting at the turbine blade tip and linearly extending towards the second row. On the other hand, the wake meandering expansion boundaries include the smallest area by increasing linearly and slowly reducing slope towards the second row. We can visualise the significant difference in the wake meandering expansion for the AS-RN and AS-R simulations where the AS-RN wake meandering expansion wake influenced area is larger than that for the AS-R simulation. Instantaneous meander profiles shown in the figure corroborate that wake meandering transverse extent grows with downwind distance faster in the AS-RN case.

Figure 17.

The row-averaged wake influenced area based on mean velocity deficit (black, solid), Gaussian fitting to the average wake (green, dotted), Gaussian fitting to the filtered wake (blue, dash-dotted) and the wake meandering area $\text{d}(A/D)/\text{d}x^{\ast }$ (red, dashed). Several meander profiles (magenta, dash-dotted) are included for (a,c) the first row and (b,d) the last row. Also plotted are contours of the row-averaged mean downwind velocity normalised by the first row incoming hub height velocity $\langle U\rangle _{r}/U_{h_{1}}$ . (a,b) and (c,d) are the cases AS-RN and AS-R, respectively.

While the wake influenced areas are similar for three metrics in the first row, the results for the last row, which are statistically similar for rows 6–10, show that the definition of the wake can greatly affect the computed wake area. The contrasting results are mainly because of the definitions of the wake influenced area in terms of using time-averaged metrics. By using time-averaged velocity statistics (average wake expansion growth rate and mean velocity deficit growth rate), the wake influence area begins to increase from the turbine blade tip and then decreases because the velocity deficit recovers in the mean sense. Moreover, the mean velocity deficit and mean turbine-added velocity in rows 6–10 are lower and higher, respectively, for the AS-R case compared to the same row in the AS-RN case. This causes higher mean growth rates for the AS-R case as demonstrated in figure 16. The lower mean velocity deficit, especially in rows 6–10, in the AS-R case is caused because the upwind wakes recover slower due to lower turbulence mixing and turbulence intensity as shown through the TKE in figure 5. Overall, the higher turbulence intensity in the wakes caused by the nacelle-induced coherent structures enables a faster wake recovery. Recognising this, mean statistics may not be the most suitable metrics to use in order to compare the wake expansions where the wind farm begins to become fully developed.

Figure 18.

The half-width $\langle R\rangle _{r}/D$ normalised by the diameter of the row-averaged wake influenced area based on $\text{d}(A/D)/\text{d}x^{\ast }$ (red) and Gaussian fit to the average wake (green circle) for the AS-RN (solid) and AS-R (dashed) cases.

However, both the wake meandering expansion, and to a lesser extent the filtered wake expansion, reveal that the wake has a much larger influenced area when considering statistics at the time scale of wake meandering. The finite-time interval averaging used to extract the coherent velocity for the wake meander profiles integrate the dynamics and instantaneous behaviour of wake meandering indicating the inherent unsteadiness of the wake, which cannot be quantified from the time-averaged wake widths. Quantitatively, figure 18 shows the comparison of the half-width of the wake meandering and average wake expansion influenced areas for rows in both simulations. The differences between the AS-RN and AS-R simulations are clearly visualised with the wake meandering region being larger in the AS-RN simulation. There are subtle differences in the filtered wake expansion between the two simulations where the half-width is slightly higher for the AS-RN case between 1–4 $D$ downwind. On the other hand in figure 18, the average wake expansion growth rate for the AS-R case in downwind rows is larger than those for the same rows in the AS-RN, which corroborate our observations in figure 17. Overall, by comparing metrics that use the instantaneous behaviour, the dynamics of the wake is elucidated, confirming that the enhanced turbulence intensity caused by the nacelle-induced coherent structure affects the meandering region of the wake. The statistics of the wakes for rows 6–10 discussed in the previous two sections reveal that the wind farm starts to exhibit statistically consistent results for each of those rows. This suggests that starting near or at row 6 the onset of a fully developed state is attained within the wind farm boundary layer.