3.1. Blade pitch

First, the effect of changes in blade pitch ($\beta$) on the near wake is investigated. Most utility-scale wind turbine controllers increase the pitch of the blades when the incoming wind is above the rated wind speed to reduce the angle of attack of the airfoils and decrease the thrust force of the rotor as a way to regulate the structural loading. To examine the effect of this process on the wake, a time series of pitch changes at different rates, similar to those observed for a utility-scale turbine, is implemented in the OpenFAST turbine controller. The pitch first increases at each rate for 10 s, then decreases at the same rate for 10 s, as shown in figure 4(a) along with the resulting changes in wake expansion angle. The wake expansion varies inversely with changes in blade pitch due to the aforementioned reduction in thrust caused by lowering the angle of attack. The same relationship is observed in the field experiment from Abraham & Hong (Reference Abraham and Hong2020). Interestingly, the correlation is not linear, rather some hysteresis occurs in the wake response. This hysteresis can also be observed in the field data, as evidenced by the fact that there are multiple different values of wake expansion for a given blade pitch angle during a sample sequence of 800 s (figure 4b). The same characteristic looping behaviour is observed in the relationship between blade pitch and wake expansion from the field data as that observed in the simulation results, although the loops are not as regular as those from the simulations due to the many different factors influencing wake behaviour in the field. Because individual variables are difficult to isolate in the field, the cause of this observation could not be confirmed with certainty. In the simulation, the cause of such hysteretic response becomes clear when comparing the wake expansion with the tip speed ratio, $\lambda ={\omega D}/{2u_\infty }$ where $\omega$ is the angular velocity of the rotor, which changes in response to changes in blade pitch. Figure 4(c) shows the strong correlation between $\lambda$ and $\varphi$, which suggests that the wake is actually responding to the change in rotor speed rather than the change in blade pitch itself. A similar trend is observed in the experimental data, conditionally sampled for periods where the wind is above the rated speed and the blade pitch is changing, although with a wider range of $\lambda$ due to larger changes in $u_\infty$ and much more variability due to the dynamic atmospheric wind speed and direction (figure 4d). Note that the results of the simulation are repeatable because of the use of uniform inflow, while the experimental data are not deterministic due to the multitude of uncontrolled stochastic variables in the field. Still, the trends observed in the experimental data are robust, as described in more detail in Abraham & Hong (Reference Abraham and Hong2020). The relationship between $\lambda$ and $\varphi$ is stronger than that between $\beta$ and $\varphi$ because the reduction in lift force on the blades due to the increase in pitch causes the rotor to slow down, which causes the wake expansion to decrease. This reduction in rotor speed lags slightly behind the increase in blade pitch due to the inertia of the flow. Schepers (Reference Schepers2007) described the effect of flow inertia to explain the observed lag in turbine response to a step increase in blade pitch. He attributed the lag to the time it takes to accelerate the flow after a decrease in axial induction. Ebrahimi & Sekandari (Reference Ebrahimi and Sekandari2018) observed a similar delay in thrust change after a step change in wind speed. A linear regression fit is applied to the relationship between $\lambda$ and $\varphi$ from the experiment, revealing a shallower slope than that of the simulation data (figure 4d). This difference in slope is attributed to the smoothing applied to the experimental wake boundary data to account for turbulent fluctuations, which tends to obscure more extreme values. The experimental data also include many additional sources of uncertainty, including concurrent changes in multiple parameters that may interact, and the measurement of incoming velocity at the turbine nacelle, which is within the induction zone of the turbine (Li et al. Reference Li, Abraham, Li and Hong2020). However, it is remarkable that the overall trends from the experiment and simulations still match well in spite of these discrepancies, indicating that the large-scale wake behaviours dominate over the effects of smaller-scale turbulence.

Figure 4.

(a) Prescribed changes in blade pitch (dashed black line) and the resulting changes in wake expansion (solid red line). (b) Instantaneous wake expansion angle vs blade pitch from the simulation (red squares) and a sample sequence of 800 s from the experiment from Abraham & Hong (Reference Abraham and Hong2020) (blue line). (c) Time series of tip speed ratio changes (dashed black line) caused by changes in blade pitch and resulting wake expansion (solid red line). (d) Instantaneous wake expansion angle vs tip speed ratio from the simulation (red squares) and from periods of the experimental dataset from Abraham & Hong (Reference Abraham and Hong2020) where the wind is above the rated speed and the blade pitch is changing (blue crosses). A linear regression fit to the experimental data is included to facilitate comparison with the simulation data.

To quantify the time scale of the wake expansion response, changes in blade pitch angle at different rates ($\beta '$) are simulated. Blade pitch is varied from $1^{\circ }$ (the default pitch angle of the Eolos turbine when operating below the rated wind speed) to $7^{\circ }$ at several different rates between 0.1$^{\circ }$ and 4$^{\circ }$ s$^{-1}$, and from $7^{\circ }$ to $1^{\circ }$ at rates between $-0.1^{\circ }$ and $-4^{\circ }$ s$^{-1}$. Such step changes are used to replicate the blade pitching behaviour of an operational turbine, which cannot change the blade pitch instantaneously. Figure 5(a) shows a sample time sequence of the change in wake expansion in response to an increase of blade pitch at a constant rate. The wake begins expanding when the pitch begins changing, then asymptotically approaches its maximum change in expansion, $\varphi _{{max}}$. To facilitate physical interpretation of the results, values are normalized using the time scale $\tau _0=D/U_\infty$ and angles are converted from degrees to radians. Figure 5(b) shows that $\varphi _{{max}}$ is independent of pitch rate, and that with a $6^{\circ }$ (0.10 rad) change in blade pitch, the wake expansion always changes by ${\sim }16^{\circ }$ (0.28 rad). On the other hand, the duration of the wake expansion change is strongly dependent on pitch rate. To quantify this effect, a wake expansion time scale ($\tau _{\varphi }$) is defined as in exponential decay, i.e. as the time for the change in wake expansion to reach $\varphi _{\tau }=(1-1/e)\varphi _{{max}}$. Berger & Kühn (Reference Berger and Kühn2018) similarly observed exponential decay of wake velocity in response to a step change in rotor thrust. Figure 5(c) shows that $\tau _{\varphi }$ is significantly higher at slow pitch rates ($\tau _{\varphi }/\tau _0\sim 5$) compared with fast pitch rates ($\tau _{\varphi }/\tau _0\sim 1$). Figure 5(d) further elucidates this trend by scaling the wake expansion time scale by the pitch manoeuvre duration ($t_{end}-t_{start}$). This scaling reveals that $\tau _{\varphi }/(t_{end}-t_{start})$ is linearly dependent on $\beta '\tau _0$, highlighting that as the pitch rate increases, the lag in wake response increases, indicating an increased deviation from quasi-steady behaviour. Some asymmetry is observed between increasing and decreasing blade pitch, where the time scales tend to be larger for increasing pitch. This trend is consistent with the explanation provided by Schepers (Reference Schepers2007) for the difference between an upward and downward pitching step: for an increase in blade pitch, the axial induction is initially large, leading to lower wake velocity and advection time than for a decrease in blade pitch, where the wake velocity is initially higher. To provide a sense of the typical blade pitch rate of a utility-scale wind turbine operating in the field, the mean pitch rate implemented by the Eolos turbine during the experiment described in Abraham & Hong (Reference Abraham and Hong2020) was 0.4$^{\circ }$ s$^{-1}$ ($\beta '\tau _0=0.04$), with a maximum and minimum rate of 2.4$^{\circ }$ s$^{-1}$ ($\beta '\tau _0=0.26$) and 0.2$^{\circ }$ s$^{-1}$ ($\beta '\tau _0=0.02$), respectively. These findings have important implications for the design of advanced wind farm control algorithms that use blade pitch to regulate turbine axial induction (e.g. Yılmaz & Meyers Reference Yılmaz and Meyers2018), which cannot expect the wake to react immediately to changes in blade pitch.

Figure 5.

(a) Sample change in blade pitch angle at a constant rate (dashed black line) and the resulting wake expansion response (solid red line). The wake expansion time scale, $\tau _{\varphi }$, and maximum change in wake expansion, $\varphi _{{max}}$, are defined in grey. (b) Relationship between normalized blade pitch rate and maximum wake expansion change. (c) Relationship between normalized pitch rate and normalized wake expansion time scale. (d) Relationship between normalized pitch rate and wake expansion time scale normalized by the duration of the blade pitch manoeuvre. The grey dashed lines show linear regression fits to the data for $\beta '\tau _0 < 0$ and $\beta '\tau _0 > 0$.

3.2. Wind direction

In Abraham & Hong (Reference Abraham and Hong2020), a clear correlation was observed between instantaneous incoming wind direction and spanwise wake deflection, although it was in the opposite direction of that described in several previous studies conducted under steady conditions (e.g. Jiménez, Crespo & Migoya Reference Jiménez, Crespo and Migoya2010; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016; Shapiro, Gayme & Meneveau Reference Shapiro, Gayme and Meneveau2018). Here, the mechanism behind such conflicting results is investigated in more detail. First, the spanwise component of the wind is varied sinusoidally with a 20 s period while the rotor direction is held constant, providing the turbine with dynamic yaw misalignment ($\gamma$) similar to the conditions experienced in the field due to the stochasticity of the atmospheric flow (figure 6(a) and supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.737). The resulting instantaneous wake deflection is shown to follow the trend observed in the field (figure 6b). Note that, as in the wake expansion case, the slope of the wake deflection response to yaw misalignment angle is shallower for the experimental data than the simulation data. Once again this discrepancy is attributed to the turbulence and uncertainties of the experimental data. Additionally, as will be described in more detail in the following sentences, the wake deflection response is dependent on the time scale of wind direction changes. Atmospheric wind experienced by turbines in the field fluctuates across a range of time scales, whereas the simulated wind direction changes with a constant period.

Figure 6.

(a) Prescribed changes in wind direction with a 20 s period (dashed black line) and the resulting changes in spanwise wake deflection (solid red line). (b) Instantaneous wake deflection angle vs wind direction for the experiment from Abraham & Hong (Reference Abraham and Hong2020) (blue crosses) and the simulation (red squares). A linear regression fit to the experimental data is included to facilitate comparison with the simulation data. (c) Prescribed changes in wind direction with a 50 s period (dashed black line) and the resulting changes in spanwise wake deflection (solid red line). (d) Analytical solution for the steady wake deflection angle from Jiménez et al. (Reference Jiménez, Crespo and Migoya2010) (grey line) compared with the instantaneous wake deflection vs wind direction from the simulation (red squares).

The period of wind direction changes is increased to 50 s, and a different trend is observed compared with the 20 s period (figure 6(c) and supplementary movie 2). With these slower wind direction changes, the wake deflection more closely follows the analytical relationship defined in Jiménez et al. (Reference Jiménez, Crespo and Migoya2010) for steady yaw error, although with some hysteresis (figure 6d). This hysteresis is caused by a lag in the wake response to changes in yaw error. These results suggest the existence of a characteristic wake response time scale during which the wake deflection transitions from the transient opposite deflection response to the steady analytical response. The dependencies of this time scale will be explored in the following section using step changes in $\gamma$ at different rates. Note that the wake deflection response is asymmetrical, with a larger deflection on the positive side than the negative side, likely caused by wake rotation (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). The turbine modelled in the current study rotates in the counterclockwise direction. For a clockwise rotating turbine, the negative deflection may be larger than the positive deflection.

3.3. Rotor yaw

Using changing wind direction to investigate yaw error includes the coupling of two effects: yaw misalignment induced wake deflection and changes in the direction of wake advection by the surrounding flow. Although this coupling is representative of the wake response in the field under stochastic atmospheric flow, these effects must be separated to gain further insight into the flow modulation caused by the turbine. Therefore, the same yaw misalignment sequence as that shown in figure 6(c) is implemented by changing the rotor yaw angle while keeping the wind direction constant (figure 7a). In this scenario, the lag in wake response is longer than previously observed for the changing wind direction scenario with the same period (50 s). This discrepancy is attributed to the fact that, when the wind direction changes, the surrounding flow advects the wake in the spanwise direction, pushing it towards the steady yaw response. Additionally, the streamwise velocity must be kept constant to ensure constant mass flow through the simulation domain, so spanwise velocity is varied in the wind direction simulation. This increase in spanwise velocity causes the wind velocity magnitude to increase. Jiménez et al. (Reference Jiménez, Crespo and Migoya2010) showed that the spanwise force exerted by the turbine on the wake is dependent on the square of the inflow velocity magnitude under steady conditions. Because the velocity magnitude is larger in the wind direction simulation during periods of yaw misalignment than in the rotor yaw simulation, the force pushing the wake towards steady-state deflection is stronger in the wind direction simulation. Therefore, the dynamic wake deflection approaches steady-state behaviour more quickly than in the rotor yaw simulation. Although the time scales of the wake response in the two cases are different, the general trends observed are consistent, including the magnitude of the inverse wake deflection. To isolate the effect of dynamic wake modulation from that of wake advection by ambient wind, and to facilitate the extension of these findings to wake control applications, the following analysis uses rotor yaw angle changes to investigate dynamic spanwise wake deflection in more detail.

Figure 7.

(a) Prescribed sinusoidal changes in rotor yaw angle with a period of 50 s, matching the changes in wind direction shown in figure 6(c), (dashed black line) and the resulting changes in spanwise wake deflection (solid red line). (b) Instantaneous wake deflection angle vs yaw misalignment angle.

To quantify the inverse wake deflection magnitude and time scale more precisely, changes in rotor direction with constant yaw rates are investigated. The yaw angle is varied from $0^{\circ }$ to $20^{\circ }$ at several different rates ($\gamma '$) between 0.1$^{\circ }$ and 4$^{\circ }$ s$^{-1}$, and from $20^{\circ }$ to $0^{\circ }$ at rates between $-0.1^{\circ }$ and $-4^{\circ }$ s$^{-1}$. As with blade pitch, step changes are used to replicate the yawing behaviour of an operational turbine, which cannot change the rotor angle instantaneously. Figure 8(a) shows a sample time sequence of wake deflection with a constant yaw rate. Under this condition, the wake first deflects in the direction opposite of the yaw change, then deflects in the same direction as the yaw change for the remaining duration of the rotor motion, consistent with the results of previous studies (Leishman Reference Leishman2002; Raach et al. Reference Raach, van Wingerden, Boersma, Schlipf and Cheng2017, Reference Raach, Boersma, Doekemeijer, van Wingerden and Cheng2018). Results are once again normalized by $\tau _0$ and angles are converted to radians. Figure 8(b) quantifies the magnitude of the maximum inverse wake deflection, $\xi _{{max}}$, showing a clear dependence on yaw rate. This deflection is not insignificant; at the largest yaw rates, it reaches 75 % of the rotor misalignment angle. However, the duration of this deflection decreases with increasing yaw rate magnitude. The inverse wake deflection time scale ($\tau _\xi$) is defined in the same way as the wake expansion time scale, i.e. as the time for the wake deflection to reach $\xi _{\tau }=(1-1/e)\xi _{{max}}$. The value of $\tau _\xi /\tau _0$ generally increases with decreasing yaw rate magnitude, although the peak occurs at a normalized yaw rate $\gamma '\tau _0 > 0$. This asymmetry is also likely caused by wake rotation, and may be reversed in the case of a clockwise rotating turbine. Note that $\tau _\xi$ ranges from $0.23\tau _0$ to $0.54\tau _0$ for the yaw rates investigated here, and a yaw rate of 0.5$^{\circ }$ s$^{-1}$ ($\gamma '\tau _0=0.08$, typical of e.g. the Eolos turbine) results in $\tau _\xi =0.53\tau _0$ when the yaw is increasing and $\tau _\xi =0.35\tau _0$ when it is decreasing. These values show that the wake takes a significant amount of time to stabilize in response to changes in yaw angle, which has important implications for the design of yaw-based wind farm optimization algorithms. This crucial point will be discussed further in § 4. When an inverse wake deflection rate is defined as $\xi _{{max}}\tau _0/\tau _\xi$, another interesting trend is revealed. This quantity is linearly dependent on $\gamma '\tau _0$, with a nearly one-to-one relationship (the linear regression slope is $-1.2$ and intersects the ordinate axis at $\xi _{{max}}\tau _0/\tau _\xi =0.01$). This relationship clearly demonstrates the tradeoff between inverse wake deflection magnitude and duration under different yaw rates. The physical explanation for this relationship will be further elucidated in § 3.4. Note that these yaw changes are implemented under turbine operational conditions that are otherwise optimized by the turbine controller (i.e. blade pitch is set to the value determined to maximize power based on the wind speed, as it would be for an operational utility-scale turbine). The effect of yaw changes under sub-optimal operating conditions has not been investigated. However, we expect a reduction in thrust coefficient to increase the inverse wake deflection time scale, as Jiménez et al. (Reference Jiménez, Crespo and Migoya2010) showed that the steady wake deflection response weakens with reduced thrust coefficient.

Figure 8.

(a) Sample change in yaw angle at a constant rate (dashed black line) and the resulting wake deflection response (solid red line). The wake deflection time scale, $\tau _{\xi }$, and maximum inverse wake deflection, $\xi _{{max}}$, are defined in grey. (b) Relationship between normalized yaw rate and maximum inverse wake deflection. (c) Relationship between normalized yaw rate and normalized inverse wake deflection time scale. (d) Relationship between yaw rate and $\xi _{{max}}\tau _0/\tau _{\xi }$. The dashed grey line shows the linear regression fit to the data.

3.4. Vorticity analysis during dynamic yaw misalignment

Further insight into the mechanism behind these wake behaviours can be gained by looking at the vorticity in the wake. Previous studies have shown that streamwise vorticity plays an important role in the behaviour of the yawed wake, leading to wake deflection and the curled wake shape that develops downstream (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016; Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016; Shapiro et al. Reference Shapiro, Gayme and Meneveau2018; Martínez-Tossas et al. Reference Martínez-Tossas, Annoni, Fleming and Churchfield2019; Zong & Porté-Agel Reference Zong and Porté-Agel2020). When streamwise vortices with opposing signs form on the top and bottom halves of the wake, they induce a spanwise velocity. Based on these studies, the following analysis focuses on the streamwise component of the vorticity vector. In the current study, after the turbine yaws, negative streamwise vorticity ($\omega _x$) is observed in the top half of the wake and positive streamwise vorticity is observed in the bottom half (see supplementary movie 3 – stills in Appendix B), consistent with the steady yawed wake behaviour described in Zong & Porté-Agel (Reference Zong and Porté-Agel2020). More interestingly, during the positive yaw manoeuvre, a short period of increased vorticity is observed in the top-left quadrant (I) of the wake and a period of decreased vorticity is observed in the bottom-right quadrant (IV) of the wake (figure 9(a) and supplementary movie 3 – stills in Appendix B). The maximum streamwise vorticity in sections of each of the four quadrants of the wake located at $z=\pm 0.3D$, $0.2D < y < 0.5D$ or $-0.5D < y < -0.2D$, and $0.5D < x < 0.6D$ compared with the theoretical vorticity trend excluding the transient effect is plotted over time in figure 9(b), demonstrating the observed vorticity changes. Without the transient effect, the streamwise vorticity would be expected to transition smoothly from its initial value to the reduced value in the top half of the wake (quadrants I and II) and the increased value in the bottom half (quadrants III and IV). However, the transient effect shifts the vorticity in the opposite directions in quadrants I and IV. This transient vorticity shift can be explained by the angle of the vortices shed from the turbine blade tips during the yaw manoeuvre, where one side of the rotor is moving upstream and the other is moving downstream. As explained in Zong & Porté-Agel (Reference Zong and Porté-Agel2020), the angle of the blade tip vortex trajectory determines the streamwise component of the vorticity, which dictates the deflection direction of the wake. When the rotor is turning, the side moving upstream (the left side in figure 9) is experiencing a larger streamwise velocity relative to the flow, increasing the streamwise component of the vorticity on that side (supplementary movie 3). On the side moving downstream, the relative velocity is reduced, along with the streamwise vorticity component. As the yaw angle increases, the steady yaw angle effect described in Zong & Porté-Agel (Reference Zong and Porté-Agel2020) dominates over the transient effect, leading to the expected steady-state vorticity distribution. In quadrants I and IV the transient effect is in the opposite direction of the steady effect, while the two effects are in the same direction in quadrants II and III. Therefore quadrants I and IV contribute to the opposite wake deflection while II and III do not. When the yawing manoeuvre ends and the two sides of the rotor experience the same relative streamwise velocity, the transient effect disappears altogether.

Figure 9.

(a) Isocontour of streamwise vorticity with cut planes in each quadrant. The isocontour value is $\omega _x = 0.1$ s$^{-1}$. Note that the turbine tower and nacelle are included in the simulated turbine model, although only the rotor is shown here for clarity. The yellow arrow near the hub indicates the direction of the yaw. The yellow boxes labelled with roman numerals indicate the regions plotted in (b), which shows the maximum streamwise vorticity component within each region over time compared with the theoretical vorticity trend excluding the transient effect. The plotted regions are located at $z=\pm 0.3D$, $0.2D< y<0.5D$ or $-0.5D< y<-0.2D$, and $0.5D< x<0.6D$.

Although the value of the streamwise vorticity remains positive in all four quadrants during the transient wake response, the difference in vorticity between the top and bottom of the wake is enough to induce a positive spanwise velocity and corresponding inverse wake deflection. Both the spatially averaged spanwise velocity ($v_{{avg}}$, figure 10a) and the maximum spanwise velocity ($v_{{max}}$, figure 10b) within the rotor area at a $y$–$z$ plane located $0.5D$ downstream show a peak just after the yawing manoeuvre begins, indicating a transient inverse wake deflection response. Subsequently, the spanwise velocity decreases due to the steady yaw response. As the streamwise vorticity is responsible for the negative spanwise velocity and wake deflection during the steady yaw response (Zong & Porté-Agel Reference Zong and Porté-Agel2020), so our results indicate the opposite streamwise vorticity shift shown in figure 9 causes the positive spanwise velocity and wake deflection that are characteristic of the transient inverse wake deflection response. Note that the magnitude of the transient spanwise velocity peak increases with downstream distance, as does the inverse vorticity shift. The values at $x=0.5D$ are presented to highlight the velocity and vorticity trends that develop directly after the flow passes through the rotor.

Figure 10.

(a) Spatially averaged and (b), maximum spanwise velocity ($v$) over the rotor area in the $y$–$z$ plane $0.5D$ downstream of the turbine during a yawing manoeuvre.

The magnitude of the transient vorticity shift is directly dependent on yaw rate, as shown in figure 11. This linear relationship sheds light on the clear dependence of the inverse wake deflection rate, $\xi _{max}\tau _0/\tau _\xi$, on yaw rate shown in figure 8. The inverse wake deflection time scale $\tau _\xi$ depends on yaw rate because higher yaw rates execute a yawing manoeuvre of the same magnitude more quickly, but the magnitude of the inverse wake deflection $\xi _{max}$ is also affected by yaw rate because the strength of the vorticity generated at the rotor is determined by yaw rate. This trend is in contrast to that observed for blade pitch in figure 5, because the magnitude of the maximum wake expansion change $\varphi _{max}$ is unaffected by pitch rate. The transient wake expansion response to dynamic blade pitch changes is simply a slow approach to steady-state behaviour, whereas the wake deflection response to yaw changes occurs in the opposite direction of the steady-state behaviour.

Figure 11.

Transient vorticity shift dependence on yaw rate, including (a) the definition of the normalized vorticity shift magnitude, $\varDelta \omega _{x,{max}}\tau _0$, and (b) a plot of $\varDelta \omega _{x,{max}}\tau _0$ vs normalized yaw rate, $\gamma '\tau _0$. The dashed grey trendline shows the linear regression of the data in (b). Note that fewer yaw rates are shown here than in figure 8 because higher temporal frequency outputs were required from the simulations to fully resolve the vorticity peak.

The vertical component of vorticity also changes when the rotor is yawing, with the left side producing stronger vorticity than the right side, as with the streamwise component. This imbalance in vertical vorticity may also induce a positive spanwise velocity, contributing to the observed spanwise wake deflection, although it would likely be much smaller than that generated by the streamwise component of vorticity (Leweke, Le Dizès & Williamson Reference Leweke, Le Dizès and Williamson2016). Previous field-scale flow visualization studies show strong, clear tip vortices within the near wake of utility-scale wind turbines (Hong et al. Reference Hong, Toloui, Chamorro, Guala, Howard, Riley, Tucker and Sotiropoulos2014; Yang et al. Reference Yang, Hong, Barone and Sotiropoulos2016; Abraham et al. Reference Abraham, Dasari and Hong2019; Dasari et al. Reference Dasari, Wu, Liu and Hong2019; Abraham & Hong Reference Abraham and Hong2020), so turbulence and shear are not expected to significantly inhibit the aforementioned mechanism in the field. Indeed, the same inverse wake deflection described above was observed in the field data from Abraham & Hong (Reference Abraham and Hong2020). That being said, figure 6 shows that the magnitude of the resulting wake deflection is damped by the increased mixing caused by these atmospheric effects.

Based on the analysis of the spanwise wake deflection, one might expect a similar trend in the vertical direction due to the differences in streamwise vorticity between the left and right sides of the wake. However, no significant transient vertical wake shift is observed during the yawing manoeuvre. This lack of vertical deflection is attributed to the large number of competing effects acting on the wake in the vertical direction. First, the interaction between wake rotation and the ground has been shown to substantially influence the vertical displacement of the wake centre during yaw (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016). Furthermore, Kleusberg, Schlatter & Henningson (Reference Kleusberg, Schlatter and Henningson2020) showed that the angle of attack of the blades varies between the top and bottom of the rotor when the turbine is yawed, introducing an additional source of vertical asymmetry. Finally, differences in spanwise vorticity between the top and bottom of the rotor may contribute to changes in the vertical velocity field of the wake. The combination of these effects, along with the spanwise asymmetry in streamwise vorticity observed in figure 9, balance such that no significant transient vertical wake deflection is observed.

3.5. Downstream propagation

In order to apply the findings of the current study to the development of control strategies, the extension of these near-wake time scales to the far wake must be investigated. Qualitatively, it is clear that these dynamic wake modulation behaviours persist downstream. In the wake expansion case, an interesting phenomenon is observed as the wake advects downstream after a change in blade pitch. When the blade pitch increases, the velocity deficit decreases due to a reduction in lift on the blades, so the upstream part of the wake moves faster than the downstream part, leading to a bulge at the point of blade pitch change (figure 12a). When the blade pitch decreases, the opposite occurs and the wake pinches off (figure 12b). These same wake width changes have been exploited to enhance wake mixing in previous studies investigating dynamic induction control of wind farms (Munters & Meyers Reference Munters and Meyers2018a,Reference Munters and Meyersb; Yılmaz & Meyers Reference Yılmaz and Meyers2018; Frederik et al. Reference Frederik, Weber, Cacciola, Campagnolo, Croce, Bottasso and van Wingerden2020). When the rotor yaw changes, the disturbance due to inverse wake deflection is also observed to persist and strengthen as it moves into the far wake (figure 12c,d).

Figure 12.

Sample timesteps of the wake streamwise velocity at hub height from the top (positive-$z$) view 50 s after (a) an increase and (b) a decrease in blade pitch. The resulting changes in wake expansion have advected several times the rotor diameter downstream, indicated by the white arrows. Sample timesteps of the same quantity (c) 10 s and (d) 60 s after a change in rotor yaw. The position and orientation of the turbine rotor is marked by white lines and the location of the resulting inverse wake deflection at each timestep is indicated by white arrows.

Quantitative investigation of dynamic wake modulation propagation downstream is also provided, although we note the limitations of this analysis. In the far wake, coherent structures have broken down and atmospheric turbulence is much more dominant than in the near wake (Zhan, Letizia & Iungo Reference Zhan, Letizia and Iungo2020). Turbulence is not included in the current study to facilitate the observation of deterministic near-wake modulation, as described in § 2.1. The downside of this simplification is that it limits the ability to draw realistic conclusions about far-wake propagation. Therefore, the following analysis adds primarily theoretical value for idealized conditions. First, the wake expansion response to changes in blade pitch is calculated at $5D$ downstream of the rotor and compared with that observed at $0.18D$. The $5D$ distance is chosen because it is well within the far wake (typically considered to begin at $2-4D$ downstream, Göçmen et al. Reference Göçmen, van der Laan, Réthoré, Diaz, Larsen and Ott2016), but still within the higher-resolution portion of the simulation mesh (figure 1a). Furthermore, Santhanagopalan, Rotea & Iungo (Reference Santhanagopalan, Rotea and Iungo2018) found $5D$ to be the optimal spacing between the first and second row of turbines under most conditions. To facilitate this far-wake comparison, $\delta _{\varphi ,{max}}/D$, where $\delta _{\varphi }$ is the deviation of the wake diameter from $D$, is used to represent the amount of wake expansion rather than $\varphi _{{max}}$. This metric is selected because its calculation does not depend on the downstream distance, unlike $\varphi$ (see figure 2), and therefore demonstrates how the magnitude of a perturbation changes as it propagates downstream. As shown in figure 13(a), the maximum wake expansion does not change significantly between $0.18D$ and $5D$. The maximum inverse wake deflection, on the other hand, decreases in magnitude as the perturbation is advected downstream (figure 13b). The wake deflection metric used, $\delta _{\xi ,{max}}/D$, is defined using the spanwise deviation of the wake centre (figure 2), with the wake centre location determined by a Gaussian fit to the streamwise wake velocity profile as described in § 2.3. Although figure 13(b) shows a significant reduction in inverse wake deflection at $5D$, figure 12(d) clearly shows qualitatively that it does, in fact, persist into the far wake. This discrepancy between the qualitative and quantitative observations is attributed to the significant distortion of the wake cross-section in the far wake during periods of yaw misalignment (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016; Howland et al. Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016; Shapiro et al. Reference Shapiro, Gayme and Meneveau2018; Martínez-Tossas et al. Reference Martínez-Tossas, Annoni, Fleming and Churchfield2019; Zong & Porté-Agel Reference Zong and Porté-Agel2020).

Figure 13.

(a) Relationship between blade pitch rate and maximum wake expansion change at $0.18D$ and $5D$ downstream of the turbine. (b) Relationship between yaw rate and maximum inverse wake deflection at $0.18D$ and $5D$ downstream. (c) The effect of pitch rate on the wake expansion time scale at $0.18D$ and $5D$ downstream. (d) The effect of yaw rate on the inverse wake deflection time scale at $0.18D$ and $5D$ downstream.

Wake expansion and deflection time scales are also calculated at $5D$ downstream and compared with those at $0.18D$, after accounting for the time it takes for the wake to be advected this distance by the ambient air. Figure 13(c) shows that the wake expansion time scale increases significantly as the perturbation caused by changing blade pitch propagates downstream. This increase in time scale is caused by the phenomenon depicted in figures 12(a) and 12(b). When the blade pitch increases, the maximum reduction in wake diameter occurs after the bulge caused by the velocity differences in the upwind and downwind portions of the wake passes. This bulge increases the time between the start of the pitch change manoeuvre and $\delta _{\varphi ,\tau }$, the point used to determine $\tau _{\varphi }$ (figure 14). When the blade pitch decreases, the narrowing of the wake causes the same lag in wake expansion time scale. On the other hand, the deflection time scale at $5D$ downstream does not change significantly from that at $0.18D$ downstream (figure 13d). Some increased scatter is observed, especially at the lower yaw rates, attributed to the relatively small inverse wake deflection at these yaw rates and the significant wake distortion discussed above, both of which lead to increased uncertainty in the definition of the wake deflection time scale.

Figure 14.

Sample change in blade pitch angle at a constant rate (dashed black line) and the resulting wake expansion response at $5D$ downstream (solid red line), with the wake expansion time scale, $\tau _{\varphi }$, and maximum change in wake expansion, $\delta _{\varphi ,\tau }$, defined in grey.