Fluid-Dynamic Mechanisms Underlying Wind Turbine Wake Control with Strouhal-Timed Actuation

Abstract

A reduction in wake effects in large wind farms through wake-aware control has considerable potential to improve farm efficiency. This work examines the success of several emerging, empirically derived control methods that modify wind turbine wakes (i.e., the pulse method, helix method, and related methods) based on Strouhal numbers on the $\mathcal{O}\left(0.3\right)$. Drawing on previous work in the literature for jet and bluff-body flows, the analyses leverage the normal-mode representation of wake instabilities to characterize the large-scale wake meandering observed in actuated wakes. Idealized large-eddy simulations (LES) using an actuator-line representation of the turbine blades indicate that the $n=0$ and $\pm 1$ modes, which correspond to the pulse and helix forcing strategies, respectively, have faster initial growth rates than higher-order modes, suggesting these lower-order modes are more appropriate for wake control. Exciting these lower-order modes with periodic pitching of the blades produces increased modal growth, higher entrainment into the wake, and faster wake recovery. Modal energy gain and the entrainment rate both increase with streamwise distance from the rotor until the intermediate wake. This suggests that the wake meandering dynamics, which share close ties with the relatively well-characterized meandering dynamics in jet and bluff-body flows, are an essential component of the success of wind turbine wake control methods. A spatial linear stability analysis is also performed on the wake flows and yields insights on the modal evolution. In the context of the normal-mode representation of wake instabilities, these findings represent the first literature examining the characteristics of the wake meandering stemming from intentional Strouhal-timed wake actuation, and they help guide the ongoing work to understand the fluid-dynamic origins of the success of the pulse, helix, and related methods.

1. Introduction

The wake of a wind turbine presents complications for nearby turbines, depending on the atmospheric conditions, turbine characteristics, and turbine siting. The nascent field of wind farm flow control seeks to reduce the deleterious effects of the wake momentum deficit by leveraging the turbine as a flow actuator though the intelligent scheduling of either the blade pitch, rotor speed, or nacelle yaw [1,2]. Wind farm flow control approaches fall into three categories: wake reduction (i.e., reducing the energy extraction of upstream turbines to increase the net power in the wind farm; this is commonly known as turbine derating), wake steering (i.e., deflecting the wakes of upstream turbines around downstream turbines), and wake mixing (i.e., actuating the wake periodically to increase mixing with the surrounding ambient flow). Here, we focus on the lattermost of these.

1.1. Overview of Wind Turbine Wake Mixing

Most of the recent work on wake mixing behind wind turbines has centered around actuating the flow based on frequencies, f, scaled on the Strouhal number [3], $St=fD/U$, where D is the turbine diameter and U is the undisturbed upstream velocity usually at hub height. Note that the wake-mixing strategies considered herein target large-scale flow structures and are distinct from those that target smaller-scale tip vortex phenomena as in [4,5]. The techniques examined include the so-called pulse method, which applies an axisymmetric perturbation to the wake, and the helix method, which applies a helically winding perturbation to the wake. The blade pitch signal, ${\theta }_b$, that produces such perturbations can be represented as in Equation (1) derived from the multi-blade coordinate (MBC) transformation [6],

$${\theta }_b\left(t\right)={\theta }_0+\left[\begin{array}{ccc}1& \cos \left({\psi }_b\left(t\right)\right)& \sin \left({\psi }_b\left(t\right)\right)\end{array}\right]\left[\begin{array}{c}{\theta }_{axi}\left(t\right)\\ {\theta }_{tilt}\left(t\right)\\ {\theta }_{yaw}\left(t\right)\end{array}\right],$$

(1)

where subscript b denotes the blade number, ${\theta }_0$ is the nominal pitch command of the controller in degrees, ${\psi }_b$ is the blade azimuth in radians from the top-dead center, and where the quantities in the column vector are given by Equations (2)–(4)

$${\theta }_{axi}\left(t\right)=A_{axi}\sin \left({\omega }_{e}t\right)$$

(2)

$${\theta }_{tilt}\left(t\right)=A_{tilt}\sin \left({\omega }_{e}t\right)$$

(3)

$${\theta }_{yaw}\left(t\right)=A_{yaw}\cos \left({\omega }_{e}t\right),$$

(4)

where A is an amplitude of the pitch perturbation in degrees, ${\omega }_e=2\pi StU{D}^{-1}$ is the temporal angular frequency of the Strouhal-scaled excitation, and t is time [7]. The pulse method uses $A_{tilt}=A_{yaw}=0$, while the helix method uses $A_{axi}=0$ and $A_{tilt}=A_{yaw}$. The equations above produce a counter-clockwise-winding helix pattern when viewed in the same downstream direction as the flow.

These methods have been trialed both numerically [7,8,9,10,11,12,13] and experimentally [11], and the results show that improvements in the power production of arrays of two or three turbines in low-turbulence environments are on the order of 1–20% depending on the inflow conditions, actuation strength, actuation strategy, and turbine layouts. Further, the helix method has been shown in some cases to outperform the pulse method [7], and particularly it is the counter-clockwise-winding helix that performs most favorably. Another variation of the helix method is pure side-to-side or up-and-down forcing, which can be applied to the wake by retaining only the ${\theta }_{yaw}\left(t\right)$ and ${\theta }_{tilt}\left(t\right)$ terms in Equation (1). The early study by Frederik et al. [10] considered these cases, finding the up-and-down forcing produced better wake recovery than the side-to-side forcing. The fluid-dynamic understanding of the causes for the results above are not well known.

1.2. Possible Mechanisms Underlying Wake Mixing

Korb et al. [12] likely represent the most advanced work to date dissecting the mechanisms at play in the helix method. They pinpoint two different mechanisms by which the helix method affects the wake: (1) through deflection of the wake center, which reduces the overlap of the wake with potential downstream turbines and consequently increases the kinetic energy available to the downstream turbine, and (2) through increased meandering in the far wake, which has the same effect. Note that while wake meandering is sometimes used to refer to the movement of the wake as a passive tracer under the influence of large atmospheric flow structures, the periodic Strouhal-based oscillation of wakes has been observed irrespective of inflow dynamics [14], and it is this mechanism that is referred to as wake meandering in Korb et al. [12] and in this article, as well.

Although Korb et al. [12] suggest that the far wake meandering effect of the helix method on wake mixing is the less prominent of the two effects listed above, the present work shows that it may play a larger role than suggested. This assertion stems from the history of work on wake meandering both in wind energy and adjacent applications of fluid dynamics.

The first reported mention of a Strouhal-scaled unsteadiness behind a wind turbine was Medici and Alfredsson [15], who found $St\approx$ 0.1–0.3 similar to the $St\approx$ 0.2 observed behind bluff bodies due to periodic vortex shedding. Okulov et al. [16] and Iungo et al. [17] provided significantly more insight on this mechanism; Okulov et al. [16] found that for various wind turbine rotor regimes the $St$ is tied to large-scale, unstable, meandering structures in the far wake involving the precession of a helical vortex core, and Iungo et al. [17] similarly measured evidence that the most amplified frequency in the wake of a turbine model was that of a counter-winding helical structure. Drawing on studies of fluid dynamics behind bluff bodies, Okulov et al. [16] compared the helical structure observed behind the wind turbine rotor disk to the shed vorticity in the wake of a sphere where the vorticity is released from a point that moves along the surface at a shedding frequency that generates either a single or double helical structure [18]. Note that this helical phenomenon is present in high Reynolds number flows behind axisymmetric bodies of various types, including behind disks [19,20] and porous disks [20], geometries which are similar to the wind turbine rotor disk. A comparison of the antisymmetric helical structure produced by a nutating disk (i.e., a disk oscillating similarly to the motion of a swashplate) and by a wind turbine using the helix method is shown in Figure 1. Note that in the case of the disk, coherent helical structures naturally exist when the disk is at rest though their appearance is stochastic; applying the nutation motion “locks in” the structure to be stabilized in space and time [19].

The mathematical underpinnings to analyze such vortex shedding and subsequent flow structures have been well studied in the literature for free shear-flow instabilities related to wake [21,22] and jet flows [23,24,25]. In the classical linear stability approach, the time-varying perturbations, $\tilde{\chi }=(\tilde{u},\tilde{v},\tilde{w},\tilde{p})$, to the mean flow field are assumed to be initially small but can develop spatially or temporally depending on the wavelength and frequency of the excitation. These analyses generally assume a normal-mode representation for the perturbations with a temporal angular frequency $\omega$, spatial wavenumber $\alpha$, and azimuthal index n as a function of the cylindrical polar coordinates $(x,r,\varphi )$:

$$\tilde{\chi }(x,r,\varphi ,t)={\widehat{\chi }}_n\left(r\right)e^{i\left(\alpha x-\omega t+n\varphi +{\varphi }_{clock}\right)}.$$

(5)

Here, ${\varphi }_{clock}$ defines a clocking angle to adjust the orientation of the eigenfunction ${\widehat{\chi }}_n$, where ${\varphi }_{clock}=0$ corresponds to the vertical direction and ${\varphi }_{clock}=\pi /2$ corresponds to the horizontal direction. Using some assumptions regarding the parallel flow or inviscid nature of the problem, this expansion in the governing equations generally results in an eigenvalue equation that can be solved to find the most unstable disturbances in the flow.

Some characteristics that emerge from such spatial stability analyses, as well as from numerical and experimental studies on jet and wake free shear flows alike, are that the optimal $\omega$’s scale on $U/D$ (i.e., Strouhal scaling) and axisymmetric perturbations (i.e., $n=0$) often produce lower growth rates than perturbations of the first helical modes (i.e., $n=\pm 1$) [26]. For swirling flows, there is a preferential direction of the first helical mode to produce the largest growth rate of the instability structure [27]. The second helical modes also become more relevant for swirling jet flows [28]. Except for the findings related to the second helical modes (i.e., $n=\pm 2$), which have not been intentionally forced in wind turbine wakes, these observations corroborate the findings surrounding the pulse and helix methods [7,8,9,11,12] as well as those from Medici and Alfredsson [15] and Okulov et al. [16] and suggest that there may be a significant overlap of the physics of large-scale shear-flow instabilities between bluff-body wakes, jet flows, and wind turbine wakes, and are distinct from the physics of smaller-scale structures connected to the tip vortex instabilities.

1.3. Control of Large-Scale Shear-Flow Instabilities

The elegance of large-scale shear-flow structures from a flow-control perspective is their convectively unstable nature, i.e., they naturally grow in magnitude as the flow is convected. This suggests that a small actuation force near the wake origin might be leveraged to produce a large-scale mixing effect in the far wake, a characteristic that has been leveraged for control purposes in the literature from jet and bluff-body flows. For instance, Ho and Huerre [29] and Crow and Champagne [30] demonstrated that the excitation of large-scale coherent structures at the jet inlet can lead to large changes in the mean profile downstream, and the growth of the vortical structures is key to understanding this behavior. Furthermore, theory, simulation, and experiment have also suggested that the excitation of the first positive and negative helical modes (termed here the double-helix method) has a strong potential for increasing the mixing of the flow, such as in the case of bifurcating jets [31] or capillary jet instabilities [32].

It remains to be seen how the fluid-dynamic understanding of shear-flow instabilities and their control as derived from studies of canonical wake and jet flows can be applied to the wind turbine case beyond what has already been accomplished in the studies of the pulse and helix methods. Wind turbine wakes vary from canonical bluff-body wakes, for instance, because of differences in the geometry of the wake-producing element, the presence of non-uniform and unsteady inflow, and the presence of swirl in the wake.

The existing work in the wind energy and water power communities that has begun to study the control of shear-flow instabilities from a fundamental perspective include Mao and Sørensen [33], Gupta and Wan [34], and Li et al. [35]. These authors analyzed the stability of wind turbine wakes to identify the Strouhal range and azimuthal mode numbers most relevant to wind turbines. Mao and Sørensen [33] numerically studied the flow past a wind turbine at a reduced Reynolds number and with oscillating inflow perturbations and found the $n=1$ mode to be most energetic in the wake with a dominant $St$ of 0.16. Gupta and Wan [34] similarly used large-eddy simulation (LES) and harmonically oscillating inflow to show that the dominant frequency falls as the wake flow convects farther downstream, and this behavior was ascribed to the broadening of the velocity deficit, which affects the stability characteristics. Li et al. [35] studied a floating offshore turbine with perturbations imposed through the side-to-side motion of the wind turbine to simulate wave-induced turbine movement. They used a linear stability analysis and LES to demonstrate that wake meandering can be triggered with side-to-side motions in the $St$ range of 0.2–0.6. Mao and Sørensen [33], Gupta and Wan [34], and Li et al. [35] all observed the dominant frequency to decrease as the magnitude of the inflow perturbations increases, and Li et al. [35] noted that the amplification of the modes decreases as this magnitude increases, as well. Though not performing a stability analysis, Hodgson et al. [36] used LES to model the flow over a rotor with inflow perturbations that correspond to an $St$ range of 0.14–0.25. The observed increase in mean kinetic energy transport into the perturbed wake in Hodgson et al. [36] and the corresponding significant increases in the power output of virtual downstream turbines lends credence to the previous assertion that Strouhal-based wake excitation and the subsequent meandering play a prominent role in the success of the pulse and helix methods. Further, the velocity fluctuations at excited $St$ values in Hodgson et al. [36] and Mao and Sørensen [33] grow around an order of magnitude larger from the inflow to the far wake (i.e., the amplification factor is ∼10), which suggests the usefulness of shear-flow instabilities to effect large changes in wind turbine wakes at the price of potentially only a small actuation input.

The existing work in the previous paragraph stops short of analyzing the wake meandering behind a wind turbine in the case of the intentional actuation of the Strouhal-based instabilities using the turbine itself. Drawing on inspiration from the existing work in the wind turbine community and from adjacent fluid-dynamics communities for jet and bluff-body flows, we report for the first time in the literature the characteristics of the periodic wake meandering of the pulse- and helix-actuated methods in wind turbine wakes from a normal-mode perspective, doing so with both qualitative and quantitative methods, as well as relate for the first time this wake meandering growth to conventional quantities of interest. Further, we revisit the double-helix method for wind turbine wake control as inspired by the work on jet flows and as initially trialed for wind turbine flows in the early study by Frederik et al. [10].

The remainder of this article is organized as follows. Section 2 introduces the formulation used for the blade forcing, Section 3 introduces the LES setup, Section 4 offers the numerical results, Section 5 gives insights on the wind turbine wake meandering derived from the linear stability analyses, and Section 6 draws conclusions.

2. Formulation of Blade Forcing

The normal-mode representation for the wake flow perturbations as introduced above in Equation (5) can be readily adapted into the rotating frame and offers flexibility in specifying the forcing strategy. In this article, we choose to use this representation rather than the MBC transform of Equation (1), although both methods produce equivalent results as demonstrated in Appendix A. For each blade $b_j$, we compute the total phase angle, $P_{b,j}$, of the blade pitch excursion for a given mode index, j, according to Equation (6),

$$P_{b,j}\left(t\right)={\omega }_{e}t-n_j\left({\psi }_b\left(t\right)+{\varphi }_{clock}\right).$$

(6)

Next, the phase angle is converted into the complex pitch amplitude, ${\tilde{A}}_b\left(t\right)$, according to Equation (7),

$${\tilde{A}}_b\left(t\right)=A\sum _j{e}^{i{P}_{b,j}\left(t\right)},$$

(7)

where $i=\sqrt{-1}$ and the pitch amplitude of each mode to be superimposed is A, which is optionally left outside the summation to reflect the constant amplitude cases examined herein. Finally, the real pitch amplitude, ${\theta }_b\left(t\right)$, to be passed to the next step of the controller is calculated according to Equation (8),

$${\theta }_b\left(t\right)={\theta }_0+\mathbb{R}\left({\tilde{A}}_b\left(t\right)\right).$$

(8)

To help with the interpretation of the equation, we insert Equation (6) into Equation (7), insert that result into Equation (8), apply Euler’s formula, and extract the real-valued component. The result is Equation (9),

$${\theta }_b\left(t\right)={\theta }_0+A\sum _j\cos \left({\omega }_{e}t-n_j({\psi }_b\left(t\right)+{\varphi }_{clock})\right).$$

(9)

Equation (9) indicates that individual normal modes contribute harmonic content at distinct frequencies to ${\theta }_b\left(t\right)$.

Table 1. Table of different forcing schemes tested in this study. Note that the amplitude of pitch forcing in all cases was $A=0.5^{\circ }$.

Case	Azimuthal Modes, n	Clocking Angle, ${\mathit{\varphi }}_{\mathbf{clock}}$	Description
Baseline	N/A	N/A	no forcing
N0	0	${90}^{\circ }$	axisym. (i.e., pulse)
N1P	+1	${90}^{\circ }$	CW helical
N1M	−1	${90}^{\circ }$	CCW helical
N1P1M_CL90	+1, −1	${90}^{\circ }$	side to side
N1P1_CL00	+1, −1	$0^{\circ }$	up and down

provides a list of some different forcing schemes examined in this study. Simulations were run in which a single azimuthal mode, either $n=0,+1,-1$, was excited, as well as cases in which two opposing modes, the $n=+1$ and $n=-1$, were simultaneously excited. Figure 2 shows an example of the blade pitch signals for the cases in Table 1. For the single-mode cases of $n=+1$ and $n=-1$, the frequency of the blade pitching is near to the rotor shaft frequency since ${\psi }_b\left(t\right)=\mathsf{\Omega }t+{\psi }_{0,b}$ (where $\mathsf{\Omega }$ is the shaft frequency and ${\psi }_{0,b}$ is the azimuthal offset of blade b from blade 1) and $\mathsf{\Omega }>>{\omega }_e$, at least for cases where ${\omega }_e$ is based on a large-scale shear-flow instability $St$ value. For the double-mode $n=\pm 1$ cases, the two frequencies from the single-mode cases of $n=+1$ and $n=-1$ are superimposed in ${\theta }_b\left(t\right)$ according to Equation (9). This superposition creates the existence of a beat frequency, which oscillates at $2{\omega }_e$. At the peak of the beat, ${\theta }_b\left(t\right)$ becomes the sum of amplitudes from each mode, so the total amplitude of ${\theta }_b\left(t\right)$ for these cases is double that for the cases with only one forced mode, and the total pitch travel is also increased. For the single-mode case of $n=0$, the frequency of ${\theta }_b\left(t\right)$ is much lower than for the other cases because the second term under the cosine in Equation (9) becomes zero.

3. Computational Setup

The simulations in the current work are performed using the ExaWind/Nalu-Wind LES flow solver coupled with OpenFAST to model the aero-elastic behavior of the wind turbine. The Nalu-Wind solver [37] is based on an unstructured, node-centered finite volume approach to solve the incompressible Navier–Stokes equations with a low-Mach number approximation using an implicit BDF2 time-stepping algorithm. In all simulations, the subgrid-scale kinetic energy one-equation turbulence model was used for turbulence closure. Nalu-Wind was previously used to model the Cape Wind offshore ABL under various atmospheric conditions, as well as for turbine wake studies in onshore wind farms [38,39].

An actuator-line representation [40], coupled with the OpenFAST turbine simulation code, was used to model the effects of the wind turbine blades on the fluid. The control of the wind turbine and the desired pitch actuation was governed by the Reference Open-Source Controller (ROSCO) and capable of pitching each blade independently based on the azimuthal location and a predefined frequency.

The wind turbine model used to demonstrate the wake control theory is based on the publicly available IEA 3.4-130 reference model, which was scaled to match the general characteristics of the GE 2.8-127 turbine (see

Table 2. Characteristics of the OpenFAST wind turbine in the current study. These parameters were created by scaling an IEA 3.4-130 reference turbine to match the general characteristics of the GE 2.8-127 turbine.

Parameter	Value
Rotor diameter	127 m
Hub height	90 m
Rated power	2.8 MW
Rated wind speed	10.7 m/s
Rated rotor speed	12.8 rpm
Tip–speed ratio	8.0

). The inflow conditions for this study were chosen to correspond to the region 2 behavior of the turbine where the nominal blade pitch remains constant. Unless otherwise noted, we used a hub-height wind speed of 6.4 m/s with a shear exponent of $\alpha =0.169$ and neutral atmospheric stratification with no veer. For the ease of analysis and to isolate the behavior of the structures that appear in the turbine wakes, steady inflow conditions were used with no inflow turbulence.

The overall simulation domain was approximately 40.3 D × 15.1 D × 7.6 D in the streamwise, lateral, and vertical dimensions, respectively, with a total mesh size of 21.8 million elements. The grid resolution was 20 m in the freestream and was successively refined to be 1.25 m near the turbine rotor. At the upper and lower surfaces, a slip boundary condition was imposed, while there were periodic boundary conditions in the lateral boundaries. An outflow pressure boundary condition was used at the outlet surfaces, and the inlet boundary used a mass inflow boundary with the specified power-law velocity profiles. In all simulations, a time step of 0.015 s was used and typically computed using 1024–1152 cpu cores with 2.6 GHz Intel Sandy Bridge processors for at least 96 h of wall time.

The simulations used a power-law velocity as the initial condition and the runs included a transient initialization time of more than 9 min based on the duration required for the power output of the turbine to stabilize. The flow statistics were then accumulated at more than 2 Hz over a duration of 265 s, which corresponds to four periods at ${\omega }_e$ = 0.095 rad/s, or $St={\left(2\pi \right)}^{-1}{\omega }_{e}D/U_{hub}$=0.3. The applied $St$ was 0.3 unless otherwise noted, and the pitching amplitude per mode, A, was $0.5^{\circ }$.

4. Numerical Results and Analyses

Using the outputs of the LES and OpenFAST simulations discussed in Section 3, a series of analyses can be performed to explore the mechanisms of the wake modification and determine their effectiveness. As discussed in subsequent sections, these analyses focus on the following items:

• Using blade loading results to determine the actuation authority.
• Applying a modal POD analysis to calculate the energy growth of specific flow structures.
• Calculating momentum entrainment through the radial shear stress flux.
• Using a linear stability analysis to analyze the behavior of the excited instability modes.

4.1. Actuation Authority via Individual Pitch Control

Given the forcing schemes outlined in Table 1, it is of particular interest to determine whether the individual blade pitch actuation leads to the excitation of the correct azimuthal perturbations at the appropriate levels even in the presence of realities, such as the inflow shear and shaft tilt. Although the exact pitch amplitude to be employed depends on both the blade geometry and the ambient turbulent intensity in the atmospheric boundary layer, we can confirm that the blade pitch fluctuations lead to the appropriate loading response by examining the azimuthal and temporal variation in the axial blade loading.

Since the out-of-plane force, F, depends on the radial location r, azimuthal angle ${\psi }_b$, and time t (see Figure 3), we assume that it can be decomposed into the form in Equation (10),

$$F(r,{\psi }_b,t)=\sum {\widehat{F}}_{jk}\left(r\right)e^{i{\omega }_{k}t-i{n}_j{\psi }_b},$$

(10)

where k is a frequency index and j is again an azimuthal index. Note that $F(r,{\psi }_b,t)$ follows the blade and thus is not a full-field quantity that exists at every position in space and time on the rotor plane. Given a simple time series at every blade nodal location, special consideration therefore is needed to calculate ${\widehat{F}}_{jk}\left(r\right)$ as explained below.

In the limit of the constant rotor speed operation, we can define ${\psi }_b\left(t\right)=\mathsf{\Omega }t+{\psi }_{0,b}$ as before and combine this with an effective frequency, ${\omega }_{jk}^{\prime }={\omega }_k-n_j\mathsf{\Omega }$. For the case of blade 1, inserting the above relation into Equation (10) yields Equation (11)

$$F(r,{\psi }_b,t)=\sum {\widehat{F}}_{jk}\left(r\right)e^{i{\omega }_{jk}^{\prime }t}$$

(11)

so that ${\widehat{F}}_{jk}\left(r\right)$ can be calculated through a simple Fourier transform of the blade loading signal.

The results are shown in Figure 4 where a forcing strategy using $A=0.5^{\circ }$ leads to an axial blade loading variation of approximately 1–2% in all cases, with the largest loading variations seen near the tip section of the blade. This suggests that a reasonable level of initial perturbations can be excited at the rotor disk without incurring turbine damage from large blade pitch fluctuations, at least for cases with low inflow turbulence. In cases where both the $n=+1$ and the $n=-1$ modes are forced simultaneously, the axial blade load variation remains generally above 1% for each corresponding azimuthal mode, but the response at $n=+1$ is not identical to that at $n=-1$. This imbalance between the excited modes may be caused by the existing shaft tilt or interactions with the blade rotation and mean wind shear and may lead to differences in downstream wake behavior, as discussed in the sections below. In general, however, Figure 4 suggests that the appropriate modes are being perturbed by each respective case and further that no extraneous modes are being perturbed.

Note that the above analysis was conducted using a steady mean flow with no inflow turbulence in order to clearly demonstrate the effects of the actuation strategy. In the presence of atmospheric turbulence and unsteady inflow, the blade loading can become more complicated. Large-scale structures that are present in the atmospheric boundary layer can introduce additional loading for the $n=0$ or $n=\pm 1$ azimuthal modes at the actuation Strouhal number.

4.2. Tracking of Meandering Instabilities

4.2.1. Structure Visualization

The evolution of the perturbed large-scale structures and the differences between the modal forcing approaches are readily apparent in the visualizations of the computed turbine wakes. For instance, in Figure 5, the hub-height contours of the out-of-plane vorticity magnitude are displayed for all cases. The structures are analogous to the large-scale coherent structures observed in previous studies on instability wave excitations in jets [26,41]. For the forced cases, the development of vortical structures and the breakdown of the wake occur at earlier downstream locations compared to the baseline case. For the N0 case, organized vortical ring structures appear in the turbine wake, while for the N1P and N1M cases, the azimuthal forcing produces helical structures, which appear most coherent for the N1M case. When the turbine blades are forced at both the $n=\pm 1$ azimuthal modes for the N1P1M_CL90 case, a side-to-side flapping motion of the wake can be seen, and the vortical structures appear earlier than any of the other cases.

4.2.2. Modal Decomposition

A more quantitative perspective of the growth of large-scale flow-field structures is afforded by modal decomposition. Our approach follows directly from that of Citriniti and George [42]. The decomposition is applied to the streamwise component of the velocity on cross-stream planes. First, the velocity field is Fourier-decomposed in the stationary and homogeneous (i.e., temporal and azimuthal, respectively) dimensions, followed by a proper orthogonal decomposition (POD) in the non-homogeneous (i.e., radial) dimension. The POD is applied across the radial dimension from the axis of the turbine rotation to a radius of 1.4 times the rotor radius, and this allows for the flow dynamics near the wake edges to be captured even as the wake spreads. This approach assumes that the wake center remains on the axis of rotation. For the temporal decomposition, a single block is used with the record length corresponding to all 588 samples after the 9 min of transient initialization described previously. The eigenvalue problem at each streamwise position therefore produces a series of eigenvalues, ${\lambda }^1$, that indicate the relative turbulent energy per mode at that streamwise position.

Figure 6 shows the modal growth versus $x/D$. Approximately linear growth is observed until $x/D\approx 1.5$–2, after which nonlinear effects become prominent. In agreement with the previous literature on the related free shear flows, the baseline case shows markedly larger growth rates for the $n=0$, $+1$, and $-1$ modes with progressively decreasing rates for the higher modes. The ordering of these three fastest growing modes differs slightly from expectations from the jet flow literature [26] with the $n=0$ being most unstable in this case, followed closely by the $n=+1$ and $-1$ modes. For the cases with intentional forcing, the highest magnitude mode of any case is reached by the $\mathtt{N}\mathtt{1}\mathtt{M}$ case in Figure 6d. In an otherwise axisymmetric flow, the $\mathtt{N}\mathtt{1}\mathtt{P}$ and $\mathtt{N}\mathtt{1}\mathtt{M}$ cases would be expected to have the same growth trajectory of their respective forced modes, but the presence of swirl in our flow is believed to be responsible for the larger overall growth of the $n=-1$ mode [27]. This observation hints at a fluid-dynamic explanation for why the counter-clockwise helix forcing strategy for wind turbine wakes has been more successful than its clockwise counterpart as reviewed above. In all the forced cases, the $n=0$, $+1$, and $-1$ modes peak by $x/D=6$, though the local maxima also occur further downstream in several cases.

The streamwise development of the modal amplitudes also shows evidence of modal interactions and nonlinear energy exchange. While only the $n=0$ mode is explicitly excited in the $\mathtt{N}\mathtt{0}$ case (Figure 6b), the growth of the first helical $n=+1$ and $n=-1$ modes and second helical $n=+2$ mode is still present and can reach amplitudes comparable to the $n=0$ mode. The converse situation occurs for the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case (Figure 6e), where the axisymmetric mode grows despite only the $n=\pm 1$ modes being explicitly forced. This growth of higher harmonics and higher helical modes is an expected result of triadic interactions between instability modes and has been previously studied in mixing layers and jets [43]. For example, in the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case, the interactions of the $n=+1$ and $St=0.3$ mode with the $n=+1$ mean mode (i.e., $St=0.0$) can lead to the growth of the $n=+2$ mode at the fundamental frequency. While these interactions cannot be directly captured through a purely linear stability analysis (as described further below), more accurate predictions of mode competition can be achieved through approaches, such as the nonlinear parabolized equations.

4.3. Conventional Quantities of Interest

A consequence of the growth of the modes described above is an increase in the turbulent entrainment by the wake, especially at the wake boundaries. This entrainment is here quantified using the radial shear stress flux, $\overline{u_x}\overline{u_x^{\prime }{u}_r^{\prime }}$, where $\overline{u_x}$ is the mean streamwise velocity and where $u_x^{\prime }$ and $u_r^{\prime }$ are the fluctuating components of the velocity in the streamwise and radial directions, respectively. Lebron et al. [44] and Boudreau and Dumas [45] indicate that this radial turbulent transport is the dominant contributor to wake recovery for all locations downstream of the tip vortex breakdown.

Figure 7 shows cross-sections in the $yz$ plane of $-\overline{u_x}\overline{u_x^{\prime }{u}_r^{\prime }}$ for a distance 7D downstream from the rotor. At this distance, the stability modes and large-scale coherent structures arising from the forcing are more developed, and their impact on the wake behavior is more discernible. The effect of the wake-mixing strategies is to increase the transport of the mean flow kinetic energy back into the rotor-swept area as indicated by the more positive values of $-\overline{u_x}\overline{u_x^{\prime }{u}_r^{\prime }}$ for panels (b)–(f) as compared to panel (a). The cases with forcing of the $n=\pm 1$ modes demonstrate expected directional behavior with an apparent axis of roughly up-and-down and side-to-side fluxes appearing in the wakes in panels (e) and (f), respectively. The non-perpendicularity of these axes between panels (e) and (f) could be a result of the imbalance between the realized excitation levels of the $n=+1$ and $n=-1$ modes as discussed for Figure 4.

Quantitative tracking of the transport in the mean flow kinetic energy in Figure 7 can be achieved by averaging $-\overline{u_x}\overline{u_x^{\prime }{u}_r^{\prime }}$ along the circle projected from the blade tips. Figure 8 shows this result. For all the actuated cases, there is little turbulent entrainment in the near wake and stronger entrainment in the far wake. This observation corroborates the argument in Section 1 that the wake meandering phenomenon, which begins with small, unstable perturbations that grow into large-scale structures, is a relevant and maybe dominant mechanism in the recovery of wakes actuated with axisymmetric or helical modes. This averaged radial shear stress flux can be linked to additional quantities of interest, particularly the rotor-averaged velocity available downstream energy, as discussed below.

The most practical effect of the increases in the modal growth and turbulent entrainment due to intentional forcing may be the increases in the velocity recovery downstream of the rotor. Figure 9 demonstrates such increases using the metric of the rotor-averaged axial velocity, ${\langle {\overline{u}}_x\rangle }_{rotor}$, which is the mean velocity taken over the projected area of the rotor disk at different downstream locations. The $\mathtt{N}\mathtt{1}\mathtt{P}$ and $\mathtt{N}\mathtt{1}\mathtt{M}$ cases begin with relatively lower velocity recovery, but the $\mathtt{N}\mathtt{1}\mathtt{M}$ case outpaces the $\mathtt{N}\mathtt{1}\mathtt{P}$ case and eventually the $\mathtt{N}\mathtt{0}$ case by $x/D=6$. The linear superposition of the $n=+1$ and $n=-1$ in the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ and $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ cases produces an increase in far-wake recovery, especially for the former. The success of the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case compared to the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ case is in agreement with the results of Frederik et al. [10] and could be a consequence of the vertical flapping motion having greater efficacy at drawing high-momentum fluid from aloft down toward the rotor height. This success of the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case is especially notable considering that the actuation authority of this case is lower than the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ case in Figure 4, and more investigation is warranted to determine if the performance of the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case could be increased through improved actuation authority. Across all the cases, the increase in the rotor-averaged velocity relative to the baseline at a downstream distance of 7D, which might be considered a hypothetical position for siting a downstream turbine, is 20–32% of the hub-height inflow velocity depending on the forcing scheme. This increase in the rotor-averaged velocity directly translates to a higher possible energy capture for the downstream wind turbine or higher-density wind farm layouts, although additional studies are required across multiple wind conditions before the result can be applied in practice.

It should be noted that additional metrics for determining the impact of flow structures on the wake evolution are also available. For instance, the integrated momentum thickness and vorticity thickness have been shown to be well correlated to vortex pairing and coherent structure formation in jets and mixing layers [41,46,47,48]. While these measures are less directly connected to the available wind energy for downstream wind turbines, they may yield additional insights into the wake structures and are worth considering in future studies.

5. Insights from Linear Stability Theory

The behavior of the excited instability mode and its dependence on the excitation Strouhal number is an important consideration when optimizing the forcing for the maximum possible wake benefit. In the previous sections, the wake and turbine results for an excitation Strouhal number of $St=0.30$ were shown for various azimuthal forcing strategies. In the next section, we discuss how the choice of frequency modifies the instability mode behavior and link the results to linear stability theory.

A similar computational approach was adopted to study the effectiveness of the excitation frequency, where blade pitch fluctuations are applied to a representative GE 2.8-127 turbine model. For simplicity, a uniform flow of 6.4 m/s with no shear and no ground effect was considered, and only the individual $n=0$ and $n=-1$ azimuthal forcing modes were used. However, in these cases, the Strouhal number forcing varied from $St=0.225$ to $St=0.45$. The downstream energy content of the instability modes was then again calculated through Fourier decomposition, azimuthal decomposition, and POD, as described in Section 4.2.2.

The streamwise development of the modal energy eigenvalue, ${\lambda }^1$, for each frequency and azimuthal mode is shown in Figure 10a,b. In general, the higher frequency excitation leads to an earlier peak in the modal energy compared to the lower frequency modes. For Strouhal numbers $St$ = 0.375 or 0.45, the peak energy occurs near $x/D\approx 4.0$, while the lowest Strouhal number, $St$ = 0.225, peaks near $x/D=6.0$. In practice, this result implies that the actuation frequency can impact the downstream location where wake mixing occurs, i.e., for larger turbine spacing, a lower Strouhal number can be chosen to create the same wake-mixing effects.

These findings are consistent with the observations of Gupta and Wan [34] and Li et al. [35], who found that the frequency of the most amplified disturbances decreases as a function of the downstream distance. This phenomena can also be explained in greater depth through the use of a spatial linear stability analysis. Through such an analysis, the growth characteristics of a small initial disturbance can be determined from the temporal forcing frequency and a given mean flow wake profile. While the full details of the mathematical formulation are provided in Appendix B, the primary result is that a dispersion relation can be found that relates the spatial growth rate, −${\alpha }_i$, as a function of $St$. This can be integrated to provide the modal energy gain values G[43], which are similar to ${\lambda }^1$, according to Equation (12),

$$G\left(x\right)={\left(exp{\int }_{x_0}^x{\alpha }_i\left(x^{\prime }\right)d{x}^{\prime }\right)}^2.$$

(12)

The plots of the modal energy gain predicted by the linear stability theory are shown in Figure 10c,d. Similar qualitative trends as for the LES data in Figure 10a,b are seen in the near-wake region, where higher frequency modes initially grow faster for the axisymmetric and first helical modes compared to those for the lower Strouhal numbers. Due to the limitations of the linear stability theory, the saturation point for individual modes is not captured in Figure 10c,d, but the sensitivity of each frequency’s growth rate on the mean flow properties can be investigated.

In Figure 11, the non-dimensional spatial growth rate, $-{\alpha }_{i}R$, is plotted as a function of the downstream distance, $x/D$, for $St=0.22-0.45$ and for the $n=0$ and $n=\left|1\right|$ modes. For $x/D<5$, $-{\alpha }_{i}R$ remains fairly steady across this frequency range. In this region of the wake, the growth rates between the axisymmetric and first helical modes are also fairly similar, with slightly higher growth of the first helical modes at $St$ = 0.22–0.30. However, for $x/D>6$, all $-{\alpha }_{i}R$ rapidly decrease, with the fastest stabilization occurring for the highest frequency modes. This rapid change in behavior is brought about by the end of the potential core region of the wake profile and rapid thickening of the wake shear layers (Figure 12). Note that in this downstream region of the wake, the $n=\left|1\right|$ modes remain more unstable compared to the $n=0$ modes, especially at low frequencies.

In the comparisons of the theoretical and LES results, there are several effects that have not been accounted for in the current formulation of the linear stability theory. For instance, in the derivation of the Rayleigh equation in Appendix B, we assume that the flow remains locally parallel and the wake does not expand downstream. Furthermore, in the linearization process, we assume that the perturbations remain small relative to the mean flow and do not interact with other modes. The combined presence of both effects helps to limit the exponential growth of the instability modes as the energy is exchanged between finite amplitudes and the mean flow evolves with the wake downstream. In future work, nonlinear, non-parallel stability theories [43] can be used to capture these effects. Similarly, the effects of the wake swirl, wind shear, or veer can alter the growth rates of the stability modes and can be included in more sophisticated stability analyses. The effects of a turbulent inflow have also been currently neglected but are expected to have a strong influence on the mean flow wake profile shapes and hence the linear stability growth rates. This, along with the impact of turbulent loading, can be accounted for in future studies.

With further development, the results of such a linear stability model can be used to optimize the blade pitch forcing schemes given different inflow conditions and turbine operations. The best choice of the blade pitch amplitude, frequency, and azimuthal mode can be determined from a combination of the inflow wind conditions and baseline so that the wake recovery is maximized and the downstream turbines can minimize their performance losses.

6. Conclusions

This article demonstrates aspects of the growth and meandering of actuated, unstable normal modes in wind turbine wake flows. When intentionally excited, these modes produce an increase in turbulent entrainment and velocity recovery in the wind turbine wake, which can be of practical benefit to wind farms. Interpreting the wake perturbations in light of the mathematical framework developed for related free shear flows (i.e., jets and bluff-body wakes) provides insight on the success of certain methods of wake actuation. Specifically, the modal decomposition of LES data on an idealized wind turbine configuration showed that the counter-clockwise helix forcing strategy (i.e., the $\mathtt{N}\mathtt{1}\mathtt{M}$ case) that has proven successful in the wind turbine controls community had the largest peak modal energy gain of any case, corroborating findings about the most unstable mode in jet and bluff-body flows. Combining the clockwise and counter-clockwise helix forcing (i.e., the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ and $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ cases) in a double-helix approach produced higher recovery than the other strategies, with the up-and-down forcing pattern of the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case performing better than the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ case as was also found in an early study of wake-mixing techniques.

As implemented herein, however, the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ and $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{90}$ strategies also have higher total pitch travel and likely higher fatigue loading, which will be quantified in future work. An unexpected result of our analysis was that for the $\mathtt{N}\mathtt{1}\mathtt{P}\mathtt{1}\mathtt{M}\_\mathtt{CL}\mathtt{00}$ case, the axisymmetric $n=0$ mode grows significantly despite only the $n=\pm 1$ modes being explicitly forced. A further surprise was that the $n=0$ mode was the most unstable in the baseline (i.e., unactuated) wake, followed closely by the $n=+1$ and $-1$ modes. More understanding on the reason for the relative success of the $n=0$ and $\pm 1$ modes in different wind turbine wake environments should be pursued. A linear stability analysis offers insight on some of the above findings and may provide such understanding if more involved analyses are undertaken to include the effects of swirl, atmospheric stratification, and buoyancy, for instance.

An additional topic for future consideration is the impact of this wake control method on the wind turbine structural loading. Periodic actuation of the blade pitch can lead to different loading conditions on the upstream turbine depending on whether the $n=0$ or $n=\pm 1$ modes are used, but the excited coherent structures can also lead to wake-added turbulence impacting downstream turbines. These effects should be carefully assessed when accounting for the overall performance benefits of this wake control strategy and deserves devoted attention in future studies.