## 6 Results and Discussion
### 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. \(f_{c,w}\) 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 [PERSON] 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 [PERSON] et al. [12] and [PERSON] 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.  
\begin{table}
\begin{tabular}{l l l l l l} \hline  & \(f_{c,r}\) & \(f_{c,w}\) & \(f_{c}\) & \(f_{peak}\) & \(St\) [-] \\ \hline 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 \\ \hline \end{tabular}
\end{table}
Table 11: Results from spectral analysis; all frequencies \(f^{*}\) are given in [s\({}^{-1}\)].