4.1.2. Time-Averaged Flow Field

In Figure 3 we compare the time-averaged flow field computed from the AD and the AS simulations. As seen in Figure 3a, the mean velocity profiles of both models show an overall agreement in the far wake (*x* = 9*D*). However, immediately behind the wind turbine, the two velocity profiles differ significantly. The profile of AD is almost uniform (except for the region near the nacelle). In contrast, the profile of AS shows a clear radial variation, which is remarked by a weaker deficit behind the nacelle due to the root loss (*y* = 0) and a smoother transition on the wake boundary due to the tip loss (*y* = ±0.5*D*) at *x* = 1*D*. In the near wake, the thickness of the shear layer on the wake boundary is smaller for AD, but it grows faster than that in the AS simulation. At *x* = 5*D*, it is obvious that this transitional region is thicker for the AD than the AS. This faster growing of wake boundary thickness denotes a quicker recovery and expansion of the wake of the AD. By comparison, this transitional region has no remarkable development in the result of AS until *x* = 7*D* and expands faster from 7*D* to 9*D*. In Figure 3b,c, the turbulence kinetic energy (TKE; *k*) and the primary Reynolds stress (<*u*- *v*- >) both indicate that the AD model shows stronger turbulent effects on the wake boundary in the *x* < 5*D* region and is surpassed by the AS in the far wake (*x* > 7*D*). At *x* = 9*D* the velocity profile of the two models are in reasonable agreement, while the wake computed by the AS model contains more turbulence kinetic energy and larger Reynolds stress. This result confirms the observation from the instantaneous flow field in Figure 2.

**Figure 3.** Uniform inflow: horizontal profiles at hub height (*z* = *z*hub) of the time-averaged (**a**) streamwise velocity, (**b**) turbulence kinetic energy, and (**c**) the primary Reynolds stress <*u*- *v*- > at different downstream location.

#### 4.1.3. DMD Analysis

DMD is conducted to analyze the dynamic coherent structures in the wakes. The velocity field on the horizontal plane at the hub height (*z* = *z*hub) is analyzed. Figure 4 depicts the DMD amplitude spectra and the most dominants modes.

In Figure 4a,e, only the modes with Strouhal number less than three (*St* = *f D*/*U*hub < 3) are plotted although the entire spectra are much longer, because the most energetic modes are within this low frequency range. First, these amplitude spectra reveal a significant difference in the energy distribution over the frequencies of the AD and the AS models' wake. The wake from the AS simulations contains generally more energy in the low frequency range below *St* < 1.5 and shows a trend of concentration around *St* ≈ 0.7 (marked with *φ*<sup>1</sup> in Figure 4e). However, such a concentration trend does not appear in the spectrum in the AD's case, whose DMD modes are of almost the same amplitudes in 0 < *St* < 2 region, except for a distinct peak at *St* ≈ 1.8 (marked with *φ*<sup>1</sup> in Figure 4a). Secondly, when comparing the amplitudes between the two cases, it is found that modes of the AS have slightly larger amplitudes than that of the AD, especially in the low frequency region, showing the wake of AS contains more energetic low frequency oscillations. Thirdly, no energy concentration around the vortex shedding frequency of the bluff body (*St* ≈ 0.168) is observed in both spectra.

Figure 4b–d,f–h show the three most energetic modes of the AD and the AS cases, respectively. Overall, the spatial scale of the oscillation patterns enlarges as the Strouhal number decreases. However, the results from the AD and the AS models have very different dominant modes. Figure 4b shows the mode of the largest amplitude of the AD case. A spatial energy concentration on the wake boundary around 2*D* < *x* < 4*D* is found, which is in agreement with the instantaneous flow field (Figure 2c). Interestingly, no apparent source of disturbance can be traced in the upstream. It suggests that this mode should perhaps be related to the instability of the thin shear layer on the wake boundary that amplifies tiny disturbances in the flow field. It is noticed that this mode dominates for 2*D* < *x* < 4*D*,

but becomes weak in the far field. In contrast, the other two modes of the AD (Figure 4c,d) both stem from the nacelle and dominate the far wake.

**Figure 4.** Uniform inflow: Dynamic mode decomposition (DMD) analysis of the velocity field on the horizontal plane at the hub height (*z* = *z*hub). (**a**,**e**) the eigenvalue-weighted amplitudes of the DMD modes for the AD and the AS models; (**b**–**d**) the largest three DMD modes ordered by amplitude of AD; (**f**–**h**) the largest three DMD modes ordered by amplitude of AS. DMD modes are shown with the spanwise velocity contour.

As to DMD modes in the AS case (Figure 4f–h) all the three modes stem from the upstream nacelle and develop until the far wake. They differ from each other by the frequency and the wave length as shown by the spanwise velocity field. No AS mode similar to *φ*<sup>1</sup> of the AD is found after checking all the modes of the AS (including those not shown in the figure). This observation suggests that for the uniform inflow condition, the wake computed from the AS model is strongly affected by the vortex shedding behind the nacelle, while the AD model predicts a unique mode related to the shear layer instability near the wake boundary, which may not exist in real wind turbine wakes.
