**5. Discussion**

The previous section has shown the differences between the wakes computed from the AD and AS models for both uniform and turbulent inflow conditions.

For the uniform inflow cases, external perturbations are excluded and the wakes computed by both models develop on their own properties. The differences are easily identified already on the instantaneous and the time-averaged flow field: the instability near the wake boundary develops earlier for the AD model and results in a faster growth of turbulence near the wake boundary and quicker wake expansion and recovery in the near to intermediate turbine downwind locations. This phenomenon can be explained with the classical linear stability theory of shear flow [45]. According to this theory, the optimal spatial scale of perturbation to destabilize a shear flow is proportional to the transitional thickness so the sharp transition between the freestream flow and the wake of the AD case (as the tip loss effect is neglected) is more sensitive to small perturbations. This sharp transition across the wake boundary and the resultant stronger mixing and expansion in the near wake of the AD model are in accordance with previous wind tunnel experiments of Lignarolo et al. [8,46]. In the DMD analysis, we have compared the two models with the energy spectra and the dominant modes. For both models, a clear influence of the nacelle on the far wake is shown by the DMD modes, which is in agreement with previous numerical studies [21]. However, obvious disparities are also found that the spectrum of the AS case is concentrated in a lower frequency range, whereas that from the AD case has a distinct high frequency dominant mode corresponding to the shear layer instability. Moreover, the wake computed from the AS model has larger coherent structures and oscillates at a lower frequency than that from the AD model, which may also due to the differences of the shear layer near the wake boundary.

For turbulent inflow cases, the velocity deficit recovers faster than the uniform case due to the inflow turbulence, which is in agreement with previous studies [16,47]. In our test cases, the time-averaged streamwise velocity, TKE, and primary Reynold's stresses computed by the AD model reasonably agree with the AS model starting from *x* = 7*D*. This generally good agreement of the AD and the AS in turbulent inflow condition is in accordance with previous studies [2,17,48]. Furthermore, the DMD analysis shows inflow turbulence shifted the energy spectrum significantly to lower frequency ranges for both models. These findings are in accordance with previous studies where the wake is analyzed using Fourier Transform [6,49]. Compared with the Fourier Transform, the DMD analysis provides additional insightful information about the spatial scale of the coherent structures. These coherent structures reveal that for the turbulent inflow condition, the wakes are dominated by DMD modes of larger scale coherent structures related to the inflow eddies and some DMD modes are enhanced by the hub vortex. Moreover, a mode similar to the bluff-body vortex shedding at Strouhal number *St* = 0.17 is found to be dominant uniquely in the wake behind the AD model, which, on the other hand, happens at *St* = 0.23 for the AS model. The most dominant mode of the AS case appears at lower frequency of *St* = 0.08 and has larger scale coherent flow structures, which shall be related to the passive advection of the wake by the large scale inflow eddies. The origin of these large-scale motions of turbine wakes is often attributed to two different mechanisms, namely the bluff body shear layer instability [23] and the inflow large eddies [6], which convect turbine wake as passive scalars. Recent field measurements [50] and computational studies [49] suggested the co-existence of these two mechanisms. Furthermore, the hub vortex behind the nacelle is shown to have a significant impact on the start and enhancement of wake meandering [21,51]. A recent review on the meandering of turbine wakes can be found in [52]. In this work we observed a complex interaction between the turbine and the inflow eddies: (i) the modes at low frequencies are less affected by the turbine (Figure 7f); (ii) the DMD modes close to the bluff body vortex shedding frequency seem to be enhanced (Figure 7b,g); (iii) some DMD modes seem to be amplified by the hub vortex behind the nacelle (Figure 7c,h); (iv) the instability within the shear layer also seems to be a key factor for some modes (Figure 7d). Although the present work still can not provide a direct answer to the origin of the wake meandering, it suggests that the dynamic structures are different in wakes computed from the AD and the AS models. Due to this difference, the AD model should be used with more attention when the wake dynamics are of interest, e.g., to study the wake meandering, because the AD model can lead to different wake meandering patterns.
