Effect of Atmospheric Stability on Meandering and Wake Characteristics in Wind Turbine Fluid Dynamics

Abstract

This study investigates the impact of atmospheric stability on wind turbine flow dynamics, focusing on wake deflection and meandering. Using the high-fidelity large-eddy simulation coupled with the Actuator Line model, we explore three stability conditions for the Vestas V80 turbine, both with and without yaw. The results indicate that wake meandering occurs predominantly along the deflected wake axis. Despite varying wake deficits and meandering behaviors, neutral and stable conditions exhibit similar wake deflection trajectories during yawed turbine operations. Spectral analysis of meandering reveals comparable cutoff and peak frequencies between neutral and stable cases, with a consistent Strouhal number ($St=0.16$). The unstable condition shows significant deviations, albeit with associated uncertainties. Overall, increased stability decreases both oscillation amplitude and frequency, highlighting the complex interplay between atmospheric stability and wind turbine wake dynamics.

1. Introduction

Offshore wind energy is emerging as the next large-scale energy production device, with better economic conditions, technological development, and favorable policies opening new markets and areas for production. According to the most recent estimates of the International Energy Agency (IEA), total offshore wind capacity is set to more than triple by 2026, reaching almost 120 GW of installed capacity, which would account for one fifth of the total capacity of installed wind energy [1].

When more wind farms with bigger turbines are being built offshore, the occurrence and importance of different atmospheric conditions are increasing. This makes understanding how this affects the flow dynamics of wind turbines more important in order to optimize energy production and the operation of wind farms.

As mentioned in a review paper by Porté-Agel et al. [2], it has been shown in numerous papers that the convective boundary layer displays stronger wake meandering and faster wake recovery compared to the neutral and stable atmospheric boundary layers [3,4,5,6,7,8,9,10]. Wake meandering is associated with one of the main impacts that stability has on the flows, namely changes in length and velocity scales of the atmospheric turbulence. Faster wake recovery is attributed to the increase in turbulence intensity, another consequence of decreasing stability, as found by Abkar and Porté-Agel [9]. They found that not only the magnitude but also the spatial distribution of the mean velocity deficit, turbulence intensity, and turbulent momentum fluxes were affected. In the present study, the Mann turbulence model is used to replicate different sized eddies in the different inflows and the corresponding spatial coherence.

There are two primary theories that attempt to explain the meandering of wind turbine wakes. One is that the wake is passively advected by large-scale turbulent motion originating from the inflow [11]. The other is an observation that wake meandering behind wind turbines exhibits similar behavior to that of vortex shedding behind bluff bodies in high Reynolds number flows. The non-dimensional Strouhal number represents the frequency associated with this vortex shedding, and its expected value for flow around a solid circular disk is around $0.12$, the same as for a turbine with a very high tip speed ratio. As the tip speed ratio decreases, $St$ increases. Medici et al. [12] found experimentally that for a variety of turbine configurations, $St$ was in the range of $0.12\to 0.2$. Trivellato et al. [13] found $St=0.16$ to be a recurring number both in their computational fluid dynamics (CFD) study of their wind turbine and in many other studies of seemingly unrelated flows presented in their literature review.

Wake steering shows potential in improving wind farm efficiency and reducing wake losses, both in wind tunnel studies [14] and field campaigns [15]. The research into wake steering under different atmospheric stabilities has not been as extensive as for the non-yawed conditions, with a few notable exceptions. Churchfield et al. [16] performed experimental and numerical measurements at the The Scaled Wind Farm Technology (SWiFT) facility with the aim of measuring wakes and wake deflection resulting from yaw misalignment under a variety of atmospheric conditions. Actuator line large-eddy simulation (AL-LES) simulations are initially run in order to better plan and predict subsequent experiments using light detection and ranging (LiDAR) measurements. These experiments indicated that wake deflection will be strongest under stable to neutrally stratified conditions, and the enhanced mixing of the unstably stratified conditions decreases the amount of expected deflection. Under stable and neutral conditions, a maximum wake deflection of about one third of a rotor diameter is expected 5 rotor diameters (5D) downstream, while this is reduced by roughly half under convective conditions.

In a study by Vollmer et al. [17], an actuator disc large-eddy simulation (AD-LES) representation of the 5 MW NREL turbine was simulated under different atmospheric conditions. This study suggested that it might not be reasonable to deflect the turbine wake through yaw in unstable conditions, as no correlation was found between the wake position and turbine yaw angle under convective conditions. In a later PhD thesis by Vollmer [18], wind farm control under different atmospheric conditions using LES was addressed. It was found that parameters such as atmospheric stability, wind veer, shear, and turbulence intensity are important parameters to predict the wake deflection of wind turbines, and that wake deflection can increase the energy yield of a two turbine array in a neutral and stable atmosphere, while the same could not be said about a convective atmosphere. Similar results have also been reported in field campaigns by Fleming et al. [19] and in LES simulations by Wei and Wan [20] and Wei et al. [21].

The Actuator Line model is particularly well-suited for simulating wind turbine wakes under varying atmospheric conditions due to its ability to accurately represent the aerodynamic forces on rotating blades without the computational overhead of fully resolving the blade geometry. This approach enables high-resolution simulations while using fewer grid cells and allowing for larger time steps, which is beneficial for studying large-scale wind farm scenarios with multiple turbines [22]. By applying the Actuator Line model in this study, we can closely examine the influence of atmospheric stability on wake behavior, particularly under yawed conditions, providing valuable insights for advancing wind farm control strategies.

In the research by Ning and Wan [23], large-eddy simulation (LES) studies were conducted using OpenFOAM v2206 to examine wind turbine wakes under neutral and unstable atmospheric boundary layer conditions. They presented a comprehensive analysis of wake meandering, including amplitude, spectrum, and probability distribution in both horizontal and vertical planes at various downstream locations. Additionally, they explored the impact of these wake dynamics on the structural loading stability of downstream turbines through correlation analysis and power spectrum evaluation. Rivera-Areba et al. [24] conducted a comparative study evaluating the engineering model for the DWM against LES in the context of horizontal and vertical steering of wind turbine wakes. Their analysis involved performing LES on the IEA 15 MW wind turbine under various yaw and tilt angles to assess the accuracy and performance of the DWM model.

The allure of capitalizing on the potential energy production gains of wake steering has prompted the construction of several analytical models. Analytical models for calculating the wake trajectory of yawed wind turbines have been proposed by Jiménez et al. [25], Bastankhah and Porté-Agel [26], Qian and Ishihara [27], and Shapiro et al. [28], in chronological order. Each model was based on previous work, attempting to improve the accuracy of wake deflection estimates. A mixture of theoretical derivation with different approaches, numerical simulations, and wind tunnel experiments were used for development and testing. One of the main distinctions between models is the assumed shape of the distribution of velocity deficit and skew angle, those being a top-hat and Gaussian distribution. The Jimenez model uses the top-hat, which is often used to explain why its predictions differ greatly from the others that use the Gaussian distribution.

Previous work done by the authors tested the effects of the yaw misalignment on two different wind turbines. RANS simulations were verified and validated and compared to the analytical wake deflection models described previously [29]. Selected figures and work from that study are included in Appendix C.

With the announced increase in energy production from wind farms, understanding the flow dynamics involved allows for more accurate modeling to help with design and optimization. Wake deflection is likely to be viewed as an attractive option within wind farm control. For this technique to mature, however, one of the questions that needs to be addressed further is how the wake behind yawed wind turbines behaves in non-neutrally stratified atmospheres, which are dominant offshore, and how well analytical models are able to predict this wake deflection.

This paper intends to investigate the wake dynamics of wind turbines with and without yaw under different atmospheric stabilities. This will be done through large-eddy simulations of the Vestas V80 wind turbine with inflow representing stable, neutral, and unstable atmospheres generated using the Mann turbulence generator superimposed onto the wind shear profile. A fast Fourier transform will be performed on the wake meandering, and the energy-containing frequencies and wake center position will be studied 8D downstream of the wind turbine. Finally, the wake deflection trajectories will be compared with the predictions made by analytical models.

The novelty of our research lies in its comprehensive approach to examining wind turbine wake dynamics by integrating multiple factors that have previously been addressed individually. While existing LES studies have explored the impact of atmospheric stability and wake deflection separately, this work uniquely combines these elements. Additionally, the study employs synthetic turbulence generation as the inflow condition, diverging from the conventional LES precursor approach. This choice is particularly significant as it enables the creation of a dataset tailored for direct comparison with existing engineering models, specifically the Dynamic Wake Meandering (DWM) models implemented in tools such as Fast.Farm, HAWC2, and DIWA. By doing so, the research not only enhances the understanding of wake behavior under varying atmospheric conditions and yaw but also provides a valuable benchmark for validating and improving current engineering models used in wind turbine design and analysis.

Since these numerical simulations are strictly aerodynamic representations of the wind turbines, the second big question that needs to be addressed for adoption of yaw steering strategies cannot be answered here, namely the structural loads on the wind turbine subject to different wake conditions. Different turbulence generators are not investigated either in the development of this model, and the inflow parameters are based on findings from Riverra-Arreba et al. [30].

2. Turbines

In this work, the Vestas V80 and Samsung S7.0-171 turbines are used. The V80 turbine is of interest because it is well described numerically and has relevant validation data available in the literature. The Samsung turbine is currently used for field research at Levenmouth [31], and it is therefore of academic interest to represent it numerically. Some key properties of the turbines are given in

Table 1. Turbine properties for both wind turbines.

	Vestas V80	Samsung S7.0-171
Rated power [MW]	2	7
Number of blades	3	3
Rotor diameter [m]	80	$171.2$
Hub diameter [m]	$3.3$	$4.2$
Blade length [m]	$38.35$	$83.5$
Rated wind speed [m/s]	15	$11.5$
Rated rotor speed [rpm]	$19.1$	$10.6$
Hub-height [m]	70	$110.6$

. The turbines will be implemented in openFOAM through an actuator line method where the aerodynamic forces of the blade are distributed along a line, which will be implemented in the Navier Stokes Equations (NSE) as a momentum sink term [32]. In this model, neither tower, hub, nor nacelle are included. Foti et al. [33] investigated the effect of this simplification and concluded that regardless of nacelle, the same results are achieved for velocity deficit in the far wake.

Creating Samsung S7.0-171 Turbine

A numerical model of the Vestas V80 turbine was already available for the authors, but the same was not true for the Samsung S7.0-171 turbine. Detailed aerodynamic data for the Samsung turbine is not publicly available, and it had to be constructed from turbine performance data from Serret et al. [34] using openFAST with a basis in the well-known 5 MW NREL turbine [35]. This was done due to the availability of field measurements of the turbine wake collected from the Levenmouth turbine through the Total Control project [31]. The changes made to the turbine are summarized in

Table 2. Alterations done when creating the Samsung S7.0-171 turbine.

	5 MW NREL Value	7 MW Samsung Value
Rotor diameter [m]	126	$171.2$
Hub diameter [m]	$3.0$	$4.2$
Chord increase factor	$1.0$	$1.35$
Pitch angle [°]	$0.0$	$-2.5$
Tower height [m]	$90.0$	$110.6$
Over hang	$-5.01910$	$-7.78$

.

The $c_P$-curve of the resulting turbine is presented in Figure 1 along with the same curve from Serret et al. [34]. A discrepancy can be seen between the two curves, where the maximum value of the power coefficients occurs at a tip speed ratio (TSR) of around 10 for the reference curve, while the curve produced by the authors gives a max $c_P$ at $TSR=7.27$. For this case, it was found that when using the $c_P$ curve produced by the authors in openFAST, the thrust and power of the turbine were in agreement with what is reported from literature, with the same not being true when relying on the $c_P$ curve form Serret et al. [34]. This was also found to be true when implementing the turbine in openFOAM later on, see Section 5.1.3. The $c_P$ curve produced from openFAST was therefore accepted.

3. Flow Analysis

Key methods used to post-process data obtained from the numerical simulations and to produce figures to analyze results are presented. The tool used for numerical wake tracking is introduced, as is the approach used to find the most consistent wake tracking method. A brief overview of analytical wake deflection models and their references, used in relation to yawed turbines, is given. The procedure for conducting spectral analysis of the meandering is presented.

3.1. Wake Center Tracking

How the location of the center of the wind turbine wake in a plane at a given downstream distance is calculated is highly relevant since the path of the wake for different inflow and operating conditions is of great interest. Different methods of finding the wake are described and implemented in Python by Quon [36] and illustrated in Figure 2 at 8D downstream of the Vestas turbine for a neutral atmosphere.

In addition to the four 1-D Gaussian methods, the contour-area and minimum-power approaches were also tested (labeled as ‘const area’ and ‘min power’, respectively, in Figure 2). The different methods were evaluated through visual inspection at downstream distances from 1D–8D for the stability cases of unstable, neutral, and stable including cases with and without yaw. One-dimensional Gaussian (ideal sigma), One-dimensional Gaussian (Bastankhah), and the contour-area approaches gave the most accurate and reliable results. The method eventually chosen was the contour area approach, due to its higher consistency. In this method, contours of constant velocity in the plane in question are calculated, and the contour that covers an area that most closely resembles the rotor area is assumed to represent the turbine wake. The center is given as the geometric center of the contour. This ensures that the estimated wake center is placed in the vicinity of this boundary, which reduces the likelihood of grave errors observed using the other methods.

3.2. Wake Deflection

For the yawed turbine simulations, the wake deflection will be compared to the analytical wake deflection models of Jiménez et al. [25], Bastankhah and Porté-Agel [26], Qian and Ishihara [27], and Shapiro et al. [28], using the wake expansion factor estimate described by Carbajo Fuertes et al. [37]. The equations giving the wake deflection estimates can be found in Appendix C.

3.3. Spectral Analysis

The wake center in the lateral (y-)direction at a fixed downstream position was plotted against time and further analyzed in the frequency, f, -domain by calculating the power spectral density (PSD). The location of interest is 8D, as it is a candidate for the placement of a new turbine in a wind farm.

In order to reduce noise from wrongly calculated wake centers, all points outside $3\sigma$ either side of the hub were removed, $\sigma$ being the standard deviation. The power spectral density was calculated using the Welch method with a Hanning window. The results were visualized in a log plot that was used to find the cutoff frequency $f_c$. This is the frequency at which the value of the PSD starts to taper off, shown for the neutral case without yaw in Figure 3. It represents the border between the frequencies containing the most energy in the signal, which are more relevant to the analysis.

Using a low-pass Butterworth filter with this $f_c$ and plotting the resulting PSD in a non-log plot, it is then possible to identify the single frequency with the highest power density $f_{peak}$, as done to investigate the frequency of meandering in Section 6.1.

Cutoff frequency is also calculated analytically using Equations (1) and (2) for comparison. The equations are similar, but one uses rotor diameter D, and the other wake diameter $D_w$. Finally, the Strouhal number $St$ is calculated using $f_{peak}$.

$$f_{c,r}={\displaystyle \frac{U_{hub}}{2D}}$$

(1)

$$f_{c,w}={\displaystyle \frac{U_{hub}}{2D_w}}$$

(2)

$$St={\displaystyle \frac{f_{peak}D}{U_{hub}}}$$

(3)

4. Inflow

The inflow of this study can be broken down into three separate parts, namely shear profile, Mann turbulence model, and turbulence intensity. A power law shear profile is used to determine the shape and profile of the wind shear at different atmospheric stabilities. The Mann spectral tensor is then used to model the spatial correlation due to turbulent eddies in the flow, with input parameters found in literature. The fluctuations generated by the Mann box are scaled using the specified turbulence intensity. This is all described in more detail in the following.

4.1. Shear Profile

The shear profile of the incoming wind velocity will affect the shape and magnitude of the wake. This could be particularly relevant for larger wind turbines since a larger span vertically means an increase in velocity difference between the top and bottom of the rotor plane. The shape of the shear profile will also be dependent on atmospheric stability. All of these factors necessitate the modeling of different shear profiles for the numerical simulations at different atmospheric stabilities. The average wind velocity profile is modeled using the power law in Equation (4).

$$U\left(z\right)=U_{hub}{\left({\displaystyle \frac{z}{z_{ref}}}\right)}^{\alpha }$$

(4)

Here $U_{hub}$ is the wind velocity at hub height, z is elevation above ground, $z_{ref}$ is the hub height, and $\alpha$ is an empirically derived exponent that includes both the effects of surface roughness and stability.

4.2. Mann Model

The Mann spectral tensor attempts to model one aspect of wind field turbulence at an affordable computational cost, namely spatial correlation that occurs due to turbulent eddies. It does this based on the three input parameters: the Kolmogorov constant multiplied by the turbulent kinetic energy viscous dissipation rate to the power of two thirds, ${\alpha }_k{ϵ}^{2/3}$, the length scale, $L_{Mann}$, and the non-dimensional shear distortion parameter related to the lifetime of the eddies, $\Gamma$. The resulting output is then a box of specified size containing the fluctuating velocity component at a specified number of points for the x, y, and z directions.

Taylor’s theory of frozen turbulence allows for an easy conversion between space- and time-dependent velocity. To quote Taylor [38], “… one may assume that the sequence of changes in u at a fixed point are simply due to the passage of an unchanging pattern of turbulent motion over the point.”. This means that the longitudinal step size $d{l}_x$ and total length $l_x$ of the Mann turbulence box can, together with average wind speed, be converted to a time step $dt$ and the total time T, respectively.

4.3. Turbulence Intensity

The generated fluctuations of the Mann box were also scaled using the specified turbulence intensity $TI$. It is done by taking the averaged standard deviation of the stream-wise wind velocity component of the four grid points closest to the turbine center using all time steps, here denoted ${\sigma }_1$. A scaling factor is then found such that Equation (5) is satisfied and subsequently applied to all grid points.

$${\sigma }_1=TI·U_{hub}$$

(5)

4.4. Turbulent Inflow

For LES simulations, the Mann turbulence generator is used with input parameters based on results found in the literature for offshore conditions at varying stability. Although the Mann model offers less accurate inflows for non-neutral atmospheric boundary layers, it provides a controlled and consistent method for generating turbulence that is practical for scenarios where computational resources are limited. It is also widely accepted and applied for modeling atmospheric boundary layers, particularly within wind energy research. However, due to its limitations, validating the generated inflow using either a precursor LES simulation or field measurements would be valuable to this study. The present study has not performed such a validation but has instead used values for ${\alpha }_k{ϵ}^{2/3}$, L, and $\Gamma$ found by Rivera-Arreba et al. [30], where the Mann model parameters were fit to wind fields simulated using LES. Three different wind speeds were simulated for the unstable, neutral, and stable conditions. De Maré and Mann [39] found the parameters to be largely independent of wind speed but not elevation. Values at hub height were therefore chosen and are presented in

Table 3. Mann inflow parameters.

	Vestas V80	Samsung S7.0-171
Unstable
${\alpha }_k{ϵ}^{2/3}$	0.0419	0.0429
$L_{Mann}$	80	86.6
$\Gamma$	4.4	3.5
Neutral
${\alpha }_k{ϵ}^{2/3}$	0.0257	0.0171
$L_{Mann}$	27.2	33.1
$\Gamma$	3.5	3.45
Stable
${\alpha }_k{ϵ}^{2/3}$	0.0129	0.0115
$L_{Mann}$	16.6	13.8
$\Gamma$	1.75	1.075

.

For the y- and z-direction, the length of the turbulence box exceeds the size of the mesh, and the cell lengths are equal to the maximum chord of the turbine blade. Total length and cell size in the x-direction correspond to the total time $T_{mann}$ and time step $d{t}_{mann}$ defined in the Mann model. They were chosen so that a total time of 3000 s could be run, 500 s to get past the statistically unsteady ramp-time, followed by 2500 s of simulation time to reach a converged solution. $dt$ is $0.25$ s. The effect of cell size and time step of the turbulence box on the solution was not explored in this study, but the values used are similar to other studies such as Rivera-Arreba et al. [30] and Nybø et al. [40].

The generated turbulence was scaled to give the desired $TI$, and the shear profile was superimposed on it. Again, results from the LES simulations in Rivera-Arreba et al. [30] were used.

Table 4. Inflow parameters from Rivera-Arreba et al. [30].

	Unstable	Neutral	Stable
$\alpha$ [-]	0.019	0.07	0.146
TI [%]	9.23	4.79	2.91
$U_{hub}$ [m/s]	7.5	7.5	7.5
$Ri$ [-]	<−0.569	$ϵ[-0.569,0.083]$	>0.083

gives their results for $\alpha$, $TI$, and $U_{hub}$.

Also given in Table 4 is the Richardson number, $Ri$, a parameter used to define stability. The given values were used by Rivera-Arreba et al. [30] to define inflow stability. Along with the Monin-Obukhov length, the Richardson number has been found to perform well in classifying atmospheric stability conditions; see Krogsæter and Reude [41]. It relates buoyancy-generated turbulence with turbulence produced by shear, as seen in Equation (6), where $\Theta$ is the potential temperature, T is the mean temperature, $\Delta u/\Delta z$ is the velocity gradient, and g the gravity constant.

$$Ri={\displaystyle \frac{g}{T}}{\displaystyle \frac{(\Delta \Theta /\Delta z)}{{(\Delta u/\Delta z)}^2}}$$

(6)

5. Numerical Simulations

CFD simulations are performed using OpenFOAM in order to analyze the flow dynamics behind wind turbines. The flow field is solved using the OffWindSolver, which is a solver under development by Balram Panjwani at SINTEF Industry [42]. Both RANS and LES simulations are used in this work. The RANS simulations use a RNG (Re-Normalization Group) k-$\epsilon$ turbulence model, based on the authors’ previous experience of this model’s suitability for solving wind turbine flows [29]. It models the effect of turbulence as an added viscosity, the effect of which is introduced into the NSEs. Separate transport equations for turbulent kinetic energy k and turbulent dissipation rate $\epsilon$ are solved to find the turbulent viscosity. The RNG k-$\epsilon$ model is an extension of the standard k-$\epsilon$ model, incorporating additional modifications that work to provide better accuracy in capturing the flow physics and reduce the dependency on user-defined constants. The LES simulations use the Smagorinsky turbulence model to close the NSE equations [43]. The Smagorinsky turbulence model is a method used in LES to approximate the effects of the unresolved small-scale turbulence on the simulated larger-scale flow. It is also an eddy viscosity model and therefore assumes the turbulent stresses are proportional to the local deformation of the flow. Their magnitude is determined using the Smagorinsky coefficient. Its value is usually based on empirical or theoretical considerations, and it affects the accuracy of the model.

5.1. RANS Simulations

RANS simulations were initially run in order to verify and validate the wind turbine models used for this study, in addition to being a building block towards the higher-fidelity LES simulations.

5.1.1. Mesh Description

Mesh parameters are given in

Table 5. Mesh parameters for RANS simulations.

	2 MW Vestas V80	7 MW Samsung
$L_x$, $L_y$, $L_z$ [m]	1400, 500, 250	2435, 1000, 600
$N_x$, $N_y$, $N_z$	190, 61, 55	241, 81, 71
$d{l}_{turbine}$ [m]	$6.0$	$9.0$

along with pictures of the mesh in Figure 4a,b for the Samsung turbine. $L_i$ is the domain length and $N_i$ the number of cells in the $i=x,y,z$ directions. $d{l}_{turbine}$ refers to the cell-size-lengths, which are equal in the x, y, and z directions, in the region of the mesh where the turbine is located. Near the ground, the cells are refined enough to capture and maintain the inflow shear profile, while at the top and back, cells are expanded. This reduces computational cost in the regions that are of less importance.

5.1.2. Verification

A total run-time of 700 and 1500 s for each simulation proved to give a converged solution with respect to time for the Vestas and Samsung turbines, respectively. A case with a reduced time step was also run to check the simulations sensitivity to a reduction in residuals. It showed a minimal effect on both wake, thrust, and power for both turbines. Residuals along with results for velocity deficit, turbine thrust, power, and time convergence can be found in Appendix A.

A mesh refinement study was performed to ensure that grid convergence for the RANS simulations was achieved. Three meshes of increasing refinement named coarse, orig, and fine were initially run for both turbines with minimum cell length $dx$ in the refined region. The cell length $dx$ was changed by a factor of $r=1.25$ between each mesh for the Vestas turbine and $r=1.33$ for the Samsung turbine. Based on these results, a fourth mesh, named medium, with a refinement level in between coarse and orig was proposed and verified for the V80 turbine, while the orig mesh was deemed adequate for the Samsung turbine. Figure 5a,b show the results of a GCI study performed using the medium and fine meshes, while output for power and thrust from the rotors is presented in

Table 6. Turbine thrust and power for RANS refinement study.

	Thrust [kN]	Power [MW]
Vestasmedium	146	$0.791$
Vestasfine	148	$0.809$
VestasDiff	$1.37$%	$2.28$%
Samsungmedium	790	$3.58$
Samsungfine	803	$3.74$
SamsungDiff	$1.57$%	$4.40$%

.

Shear in the inflow profile proved to have a pronounced effect on the velocity deficit in the wake, as can be seen in Figure 6. The shear profile was therefore included in subsequent iterations of the models. A no-slip boundary condition and wall functions were applied at the bottom wall, and, in order to capture the steep gradient, cells were refined in this lower region. A sufficiently low value for the non-dimensional wall distance y+ needed for an accurate near-wall solution was never achieved. However, a plot of the wind velocity profile for locations downstream of the inlet and upstream of the turbine shows good agreement with the input velocity profile, especially in the region where the turbine operates; see Figure 7. This analysis was done for the Vestas turbine because of the availability of inflow data from Keck et al. [7]. Changes based on the analysis were made to both turbine models.

5.1.3. Validation

RANS simulations in uniform inflow were run at different wind speeds to validate turbine thrust and power. The results were compared to performance data given by the manufacturer and found in literature and are given in Figure 8 and Figure 9. It can be seen in these figures that both thrust and power are modeled accurately. In Figure 9b, it can be seen that at a wind speed of 10 m/s, the thrust is overestimated. This is likely because the turbine would approach rated wind speed and go into a different operating region, which would mean altering the TSR. Tip speed ratio for the V80 turbine is $TSR=7.3$, and for the Samsung it is as described earlier in Section 2.

LES data from Keck et al. [7] of a Vestas V80 turbine was used to validate the wake of the RANS case. Keck et al. [7] used a hub-height wind speed of $U_{hub}=8$ m/s in an unstable atmosphere with turbulence intensity $TI=6.16\%$. A shear profile for the RANS simulation was generated with the power law using $\alpha =0.019$ as found by Rivera-Arreba et al. [30] for an unstable atmosphere; see Table 4. Another parameter with significant impact is the turbulent dissipation rate $\epsilon$ prescribed at the inlet. It is based on a turbulent length scale $T{u}_L$. Setting $T{u}_L=9.5$ gave the best agreement with validation data (see Figure 10a) and will be used in further simulations. The Samsung turbine was validated with LiDAR data from the TotalControl project [31], where measurements are done on the Levenmouth turbine operated by ORE Catapult. A rather large turbulent length scale of $85.6$ m, or $1R$ (1Radius), gave a solution most similar to that of the validation data, as shown in Figure 10b. It should be noted that the data used to validate this turbine is quite noisy, and there is an absence of information describing the inflow conditions when the measurements were made. With a basis in the information given from the TotalControl project, a wind speed of 8 m/s and a turbulence intensity of 10% were therefore assumed, while the shear profile was imposed in a similar manner to that of the Vestas turbine.

5.2. LES

A higher-fidelity simulation was run using LES for the 2 MW Vestas V80 turbine. It was verified and then validated using the same data as previously from Keck et al. [7] before being used to generate results. The Samsung turbine was not used for the LES simulations primarily due to lack of validation data, as explained in Section 5.3. For the results, six simulations were run, namely with and without a yaw angle of ${20}^{\circ }$ for each of the three stability conditions presented earlier in Section 4.4. The ${20}^{\circ }$ yaw angle was chosen based on previous research by the authors and is expected to produce a clear wake displacement with downstream flow structures, which has been observed in literature. This angle has also been among the yaw angles studied in the development of the analytical models by Bastankhah and Porté-Agel [26], Shapiro et al. [28], and Jiménez et al. [25].

5.2.1. Mesh Description

The new mesh has the same domain and cell size as the one used for RANS, only with added refinement regions around the turbine and wake. The reason being that, whilst RANS is designed for coarser meshes, LES needs a finer mesh in order to resolve eddies at smaller length scales. Running an LES simulation with the RANS solutions interpolated onto the finer mesh makes for a more accurate result. The mesh was also changed to be uniform in the x, y-, and z-directions. The refinement regions were created using the openFOAM utility snappyHexMesh, and the resulting cubic cells are described in

Table 7. Cell sizes for the various refined regions of the LES mesh.

	Turbine	Wake	Elsewhere
$dl$ [m]	$1.45$	$3.0$	$6.0$

. Pictures of the mesh are presented in Figure 11. For the simulations of yawed turbines, both wake refinement regions were moved 30 m in the negative y-direction, and the turbine refinement region was expanded 13 m in front of and behind the turbine. This ensures that both the turbine and the wake are still well captured by the finer cells. Due to the angle of the turbine, the wind velocity experienced by the rotor is decomposed by the cosine of the yaw angle. With the new wind speed assumed to be the component normal to the plane of the turbine, a new rotational speed was calculated keeping the tip speed ratio constant.

5.2.2. Verification

A simulation was run on a finer mesh with the number of cells increased by a factor of $r=1.2$ in the x, y, and z directions. The two solutions were then used to calculate the grid convergence index for the velocity deficit downstream of the turbine. Along with values for turbine output, the GCI study shows that a finer mesh is not needed; see Figure 12 and

Table 8. Turbine thrust and power for LES refinement study.

	Thrust [kN]	Power [MN]
V80medium	$151.2$	$0.728$
V80fine	$151.2$	$0.729$
Diff [%]	0	$0.1$

.

A total run time of 3000 s proved to give a converged solution with respect to wake, thrust, and power. A case with a halved time step of $dt=0.05$ was run to test the model’s sensitivity to this parameter. This also had the effect of lowering the residuals. Results show that $dt=0.1$ and the corresponding residuals are sufficiently low, which is shown in Appendix A.

Both the shear profile and turbulence intensity of the simulation were checked for positions upstream of the turbine. It is shown in Figure 13 that shear profiles for all positions and stabilities gave good agreement with the desired shear-profile shape. Upon inspection, turbulence intensity was found to decrease substantially from the inlet to the turbine. To correct for this, intensity at the inlet was increased.

Table 9. Average turbulence intensity in the inflow of the Vestas V80 turbine for LES simulations; all values are given in [%].

	Inlet	Turbine	Target
Unstable	$12.4$	$9.73$	$9.23$
Neutral	$7.75$	$4.64$	$4.79$
Stable	$5.1$	$2.81$	$2.91$

gives an overview of the achieved and target values for turbulence intensities.

In addition to the parameters presented above, a number of other simulations were run in order to examine the numerical models sensitivity and reliance on these parameters. This included comparing the Smagorinsky and a dynamic one equation eddy-viscosity turbulence models, testing for different values for the Smagorinsky coefficient, examining different damping functions within the Smagorinsky model and running a simulation with a larger domain. Of these, it was found that the turbulence model, the Smagorinsky coefficient, and the extended domain influenced the solution to a very small degree. The choice of damping function provided in openFOAM did impact the solution more, and among the Prandtl, van Driest, and cube-root volume damping functions, the simpler cube-root volume damping functions were shown to give results most representative of the validation case.

5.2.3. Validation

Simulated turbine thrust and power for all stabilities at 0 degrees yaw are compared with values from the turbine manufacturer given in

Table 10. Validation of turbine thrust and power for V80 LES simulations.

	Thrust [kN]		Power [MW]
	Simulated	Target	Simulated	Target
Manufacturer	$150.8$	151	$0.732$	$0.710$
Unstable	$132.1$	132	$0.607$	$0.600$
Neutral	$133.0$	132	$0.593$	$0.600$
Stable	$133.3$	132	$0.591$	$0.600$

. The wake deficit is then compared with Keck et al. [7], following the same procedure as in Section 5.1.3, and presented in Figure 14. Ideally, field measurement data or a higher fidelity model would be used to validate this LES model, thereby avoiding one LES model being used to validate another LES model. Time restrictions and data availability made this unachievable, but the validation results are an indication of a realistic turbine wake nonetheless.

5.3. Samsung Turbine

As previously stated in Section 2 and Section 5.1.3, the numerical Samsung turbine matches the performance expected from literature well. Even though the LiDAR measurements are noisy, the wake of the Samsung turbine was able to replicate the measured data to a reasonable degree with a turbulent length scale of one rotor radius for the RANS simulations (Figure 10b). For future implementations of the turbine, the tilt angle should be smaller. The value of this parameter followed from the 5 MW NREL turbine and was not altered. It can be seen in Figure 4b that the wake and wake center of the turbine have a downward trajectory, which is likely due to this high tilt angle. The wake center not going parallel to the horizon could also be a reason as to why this turbine needs such a high turbulent length scale to match the validation data.

Ultimately, however, the choice was made to use the V80 Vestas turbine in favor of the Samsung S7.0-171 turbine. This came down to the fact that for the Vestas turbine, the validation data were given for well-described inflow conditions, while the same could not be said for the Samsung turbine. Seeing that in many cases the wind speed was not well defined and that the turbulence intensity was not specified for any of the measurements, the authors could not validate the turbine with sufficient confidence to pursue high-fidelity LES simulations aiming at investigating the influence of atmospheric conditions on the flow.

However, the rotor performance of the turbine compares well with literature, and the RANS simulations give reasonable agreement with the TotalControl data. Other publications may therefore find the information useful, especially with access to validation data with a better description of the inflow.

6. Results and Discussion

6.1. Wake Deficit

Figure 15 shows the wake deficit for all atmospheric conditions for the Vestas turbine at different downstream locations. As expected, the deficit is larger for the stable atmospheric condition, followed by the neutral atmospheric condition, while it is significantly smaller for the convective atmosphere. At 6D and 8D, the wake has reached 85% and 90% of the free stream velocity, respectively, for the unstable atmospheric conditions. During the inspection of the wake meandering for this stability, it was also found that the wake tracking algorithm had trouble finding a well-defined wake center for all surfaces greater than 4D. This is thought to be due to the wake tracking algorithm struggling with finding a well-defined area of constant velocity resembling the rotor area beyond x = 4D. This could be due to the wake breaking into multiple velocity contours, and this points towards the contour area approach not being suitable for detecting the wake center in the far wake for unstable atmospheres.

6.2. Wake Deflection

The wake deflection is shown in Figure 16 by finding the wake center as described in Section 3.1 using the averaged velocity from the LES simulations. It is also plotted for the averaged position of the wake center at each downstream surface for the entire time series, with time steps sizes of 1 and 10 s. Filtering based on standard deviations, described in Section 3.3, is applied to the meandering signals such that the wake center measurements that are clearly unphysical are removed while keeping as much of the original measurements as possible. The deflection is then plotted against the analytical wake deflection models of Jiménez et al. [25], Bastankhah and Porté-Agel [26], Qian and Ishihara [27], and Shapiro et al. [28]. The equations used in these models are specified in the appendices.

Comparing the wake deflection for the neutral and stable atmospheres in Figure 16a,b, they appear to be very similar. The greatest difference between the simulated deflection using the mean velocity field of the numerical solutions is found to be 2 m at 2D downstream. For the analytical solutions, the biggest differences are in the far wake, and are found to be just under 5 m for Jimenez’s model, around 2 m for Shapiro’s model slightly less than 1 m for the two remaining models. These findings are in agreement with the findings of Churchfield et al. [16], and the maximum wake deflection of about one third of a rotor diameter at 5D found in that paper is in excellent agreement with the deflection of $23.5$ m or $0.29$ rotor diameters found in the current results.

The reduction of deflection by roughly half for the unstable case is not observed for the deflection shown in Figure 16c. It should be noted that due to the difficulties locating a wake center in the far wake of this turbine, the results are only shown until x = 4D. Further downstream at 8D, the wake deflection increases to roughly 0.37D in the lateral direction. It can be seen in Figure 15 that at 8D downstream, the wake is around 2 diameters wide. This means that the mean wake will still heavily interfere with a downstream turbine and that this turbine would be in partial wake conditions. If the meandering at 8D downstream is also considered from Figure 17, this indicates that this wake would oscillate with a magnitude of $0.25$ rotor diameters in both directions, which could induce additional unsteady loads on a downstream turbine. Given the difficulties in detecting the wake center in the far wake of unstable atmospheres with the wake center tracking algorithm used, this study is not able to analyze the wake deflection for this case. It is recommended for future work to also consider wake tracking algorithms aimed at detecting the wake center for this scenario.

Continuing with a comparison between the simulated and analytical wake trajectories, it can be seen that the initial wake displacement is greater for the numerical solution. The deflected wakes then follow a more gentle slope to the far wake, where a better agreement with the analytical solutions is found, and at 8D a good estimate is achieved for all models but Jimenez’s. It is interesting that the wakes of the more stable atmospheres appear to have a greater initial wake displacement than that of the unstable atmosphere. This could indicate that initial wake displacement decreases with decreasing atmospheric stability, but more research is necessary to say anything conclusive.

It was found in the project assignment [29] using a uniform flow RANS simulation without shear that the trajectories of the analytical solutions then followed the trajectories of the simulated wake displacement to an excellent degree, as can be seen in Appendix C. The differences between these simulations and the current ones are the turbulence model, turbulence intensity, and inflow profile. A subject for further work on these models could therefore be to investigate which of these parameters impact the wake deflection the most. Focusing now on the wake deflection plotted for different averaging times, it is seen that they all follow similar trajectories. The 1 and 10 s averages are found by using the average wake position over the wake center for each surface for the entire time series, which indicates that the meandering of the wake occurs around the deflected wake axis.

6.3. Wake Meandering

For the neutral and stable stabilities, meandering is analyzed 8 diameters downstream of the turbine, at x = 8D. As the unstable case had no clearly defined wake at this location (see Section 6.1), it was necessary to use x = 4D. The typical spacing between wind turbines in modern wind farms is currently between 6 and 10 times the turbine diameter [44]. In the present study, a streamwise distance of 8D is chosen as a reference case, representing a realistic scenario. The behavior of wake dynamics at this distance is of particular interest, as it may provide valuable insights into the flow dynamics of a region crucial to today’s common wind farm design practices.

The Welch method was used with a Hanning window and 1024 samples per segment with an overlap of $50\%$ for the window-averaging. The total duration of the time series is 2500 s, and the sampling frequency is $f_s$ = 1 s⁻¹. Using the method described in Section 3.3, results for cutoff frequency $f_c$ (see Figure 3) and peak frequency $f_{peak}$ were obtained and are presented in

Table 11. Results from spectral analysis; all frequencies $f^{*}$ are given in [s⁻¹].

	${\mathit{f}}_{\mathit{c},\mathit{r}}$	${\mathit{f}}_{\mathit{c},\mathit{w}}$	${\mathit{f}}_{\mathit{c}}$	${\mathit{f}}_{\mathit{peak}}$	$\mathit{St}$ [-]
Unstable	$0.0469$	$0.0276$	$0.0108$	$0.0069$	$0.0736$
Neutral	$0.0469$	$0.0234$	$0.02$	$0.0120$	$0.1632$
Stable	$0.0469$	$0.0276$	$0.0275$	$0.0153$	$0.1707$

. $f_{c,w}$ and $f_{c,t}$, the analytical versions of $f_c$ calculated with Equations (1) and (2), are also given in the table along with the Strouhal number $St$ calculated using Equation (3).

Figure 17 shows the low-pass Butterworth-filtered wake position for a fixed point over time and its corresponding PSD for all stabilities with and without yaw. The PSD of the inflow has not been included in the plots as its order of magnitude is over 1000 times smaller than that of the wake. See Nybø et al. [45] for more on the PSD of turbulent inflows. For the unstable and neutral cases, the meandering is very similar for the yawed and non-yawed turbines, while for the stable case, the yawed turbine appears to give a slightly larger meandering amplitude. Comparing stability conditions, amplitudes for unstable simulations appear to be larger than neutral, and neutral are again larger than stable, but perhaps only in non-yawed conditions. The frequency of oscillations for the meandering looks to decrease with increasing stability regardless of yaw, which is supported by results for peak frequency in Figure 17b,d,f.

Although the unstable case uses wake centers at a different downstream location (4D and not 8D), results for $f_c$, $f_{peak}$, and $St$ at 4D downstream in the neutral and stable cases gave similar results to 8D; see Appendix B. This indicates that results of this kind are somewhat independent of downstream position.

Results for stable and neutral inflow agree quite well with the analytical cutoff frequency calculated using the wake diameter, $f_{c,w}$. In regards to the Strouhal number findings for the stable and neutral cases, they are consistent with values found by Medici et al. [12] and Trivellate et al. [13] of $St=\langle 0.12\to 0.2\rangle$ and $St=0.16$, respectively.

$St$ for the unstable case is significantly lower. The result could indicate that for convective inflow, bluff body dynamics no longer apply to the meandering motion. Instead, it could be the result of the wake deficit being advected passively by the larger eddies in the inflow. A lower peak frequency is reminiscent of the lower frequencies, representative of the larger eddies, dominating the power spectral density of convective inflows. However, considering the limitations of the model and the difficulties faced when finding the wake center for this case, the large difference in $St$ warrants further inspection and verification before any conclusions can be drawn.

7. Conclusions

Understanding the impact of atmospheric conditions on wind turbine flow dynamics is crucial for optimizing energy production and operational efficiency. Offshore wind farms introduce challenges related to larger turbines and different operating conditions. The effect of atmospheric stability on wind turbine wake dynamics has been examined in this work, in particular wake deflection and meandering. High-fidelity numerical AL-LES simulations with Mann generated turbulent inflows were used to investigate three different stabilities for turbines with and without yaw.

An attempt was made to reproduce and use the Samsung S7.0-171 in the study. A successful creation of a numerical representation of the turbine has been achieved, which can be useful for later studies. However, due to a high degree of uncertainty in some parts of the validation data that are important for this work, sufficient accuracy of the AL-LES model could not be guaranteed, and the Vestas V80 turbine was used instead.

As anticipated, with increasing atmospheric stability and a consequent decrease in turbulence intensity, the wake deficit also increases. Despite these differences observed for wake deficit between the neutral and stable cases, they have very similar deflected wake trajectories when the turbines are in yaw. Both exhibit a greater initial wake deflection than what is expected from the analytical wake deflection models, but this initial sharp gradient flattens out, and at 8D, good agreement is found between the numerical and analytical results. At 5D, the wake center position was found to be 0.29D, which is in excellent agreement with Churchfield et al. [16], who reported a deflection of 0.33D for the same downstream distance. In the furthest wake position examined, at 8D, deflection increases marginally to 0.375D, exposing a new potential turbine at this location to partial wake conditions. The effect of this is made more substantial when considering that the oscillating motion from meandering has a magnitude equal to 0.25D. It was also found that the wake trajectories were very similar if the mean velocity was used to find wake centers or if the wake center position was found using the mean value of the meandering wake. This would suggest that the meandering behind yawed turbines occurs around the deflection axis.

Meandering analysis was conducted at 8D for neutral and stable stabilities, while the unstable case used 4D due to the absence of a clearly defined wake. The Welch method with specific parameters provided results for cutoff frequency, peak frequency, and Strouhal number. In the neutral atmosphere, meandering behavior was similar between yawed and non-yawed turbines, while slightly larger amplitudes were observed for yaw compared to non-yaw in the stable case. Neutral simulations exhibited larger amplitudes compared to stable, non-yawed simulations. Meandering frequency appeared higher in both stable cases, supported by the peak frequency results. Results at 4D in the neutral and stable cases were similar to those at 8D, indicating some independence from downstream position. The results for stable and neutral inflows aligned well with the analytical cutoff frequency using the wake diameter. Strouhal number findings were consistent with previous studies for stable and neutral cases, but significantly lower for the unstable case. This could suggest different dynamics; however, considering the model limitations and challenges in determining the wake center, further inspection and verification are needed to draw conclusive insights.