3.1.1. Instantaneous wake characteristics

The influence of turbine side-to-side motion on the instantaneous wake is revealed in figure 4, where snapshots of the instantaneous wakes behind a fixed wind turbine and behind an oscillating one are compared. Figures 4(a) and 4(c) compare the contours of the instantaneous streamwise velocity deficit with the centreline and width of the wake extracted with a Gaussian fit defined as

(3.1)\begin{equation} {\rm \Delta} u(x,y) = u(x,y) - U_\infty = {\rm \Delta} u_c(x) \exp{\left(-\frac{(y-y_c(x))^2}{2 s^2(x)}\right)}, \end{equation}

where $u(x,y)$ is the streamwise velocity and $s(x)$ is the standard deviation of the Gaussian distribution to be fitted with the velocity profile. The wake centre position $y_c$ is defined with the peak of the Gaussian fit, and is plotted with red dash-dotted lines. The wake width (red dashed lines) is defined as $r_{1/2}(x) = \sqrt {2\ln 2} s(x)$, which gives the distance from $y_c(x)$ to the position where ${\rm \Delta} u = \frac {1}{2}{\rm \Delta} u_c$. This fitting technique based on self-similarity, which is often applied to extract the features of time-averaged wakes (Xie & Archer Reference Xie and Archer2015; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2016), has been demonstrated as being able to extract features of instantaneous wakes (Li & Yang Reference Li and Yang2021), especially when the inflow turbulence is subtracted. Other methods – e.g. calculating the instantaneous wake centre position $y_c$ as the weight centre of ${\rm \Delta} u$ (Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2017) or ${\rm e}^{-{\rm \Delta} u}$ (Muller et al. Reference Muller, Aubrun and Masson2015) – have been also employed when inflow turbulence presents.

Figure 4.

Instantaneous streamwise velocity deficit in the wake behind a fixed wind turbine (a), and a wind turbine subject to side-to-side simple harmonic motion with $St = 0.25$ and amplitude $A/D = 0.04$ (c). The wind turbines are illustrated with black solid lines. The red dash-dotted lines and the dashed lines indicate the centrelines and the wake width, respectively. They are obtained with the Gaussian fit of the velocity deficit profile, as shown in (b,d) for the wakes of the fixed wind turbine and the moving wind turbine, respectively.

As can be seen in figure 4(a), the centreline of the wake behind the fixed wind turbine is approximately straight in the near wake and fluctuates mildly in the far wake ($x>6D$). Moreover, the width of the wake behind the fixed turbine remains almost constant at all downstream locations, suggesting that the wake recovery is slow for the fixed turbine for the simulated case without inflow turbulence. In contrast, the oscillating wind turbine generates a wake with large-amplitude oscillations as shown in figure 4(c), revealing a significant impact of the side-to-side motion on turbine wakes. The meandering amplitude is found to be comparable to the turbine diameter in the far wake, while the amplitude of the side-to-side motion is only $A = 0.04D$. This rapid growth shows that the initial perturbation induced by the turbine motion is vastly amplified as one travels downstream. Besides the wake centreline oscillation, the velocity deficit in the far wake is also significantly affected, being weaker behind the oscillating turbine when compared with the fixed turbine as shown in figures 4(a) and 4(c). This difference is more evident in figures 4(b) and 4(d), where the transverse profiles of the velocity deficit at downstream location $x=8D$ are plotted. The blue curves represent the instantaneous velocity deficit, and the red dashed lines are their Gaussian fits defined with (3.1). It is clear that the centreline velocity deficit ${\rm \Delta} u_c$ is stronger and the wake width is narrower for the fixed wind turbine when compared with the oscillating wind turbine. As for the distribution of the velocity deficit, the wake behind the fixed turbine is symmetric and can be fitted well with the Gaussian curve. The wake of the oscillating wind turbine, on the other hand, shows obvious asymmetry. Despite the observed asymmetry, the essential features of the meandering wake – i.e. the centreline position, the velocity deficit and the width – are well captured by the Gaussian fitting. It is worth noting that these instantaneous wake profiles deform from one instant to another, especially in the meandering case. Further investigation on the time-averaged wake profile in the frame of references fixed in space and following the meandering wake centre will be provided in §§ 3.1.2 and 3.1.3, respectively.

The wakes downstream of the fixed and the oscillating wind turbines are further compared by examining the frequency spectra of the spanwise velocity fluctuations in figure 5, which are computed with instantaneous spanwise velocities ($v$) recorded along the time-averaged wake centreline ($y=0$) at four turbine downstream locations ($x = \{2D,4D,6D,8D\}$) using the discrete Fourier transform. As shown in figure 5(a), the velocity behind the fixed wind turbine contains only small-scale fluctuations, which are influenced by the hub vortex in the wake of the nacelle and are distributed over a wide range of frequencies. No obvious growth can be found for the fluctuation amplitude as one travels downstream. In contrast, the spectra behind the oscillating wind turbine consist of a distinct peak at the frequency of the imposed motion ($St=0.25$). Moreover, the fast Fourier transform (FFT) magnitude at $St=0.25$ grows significantly with downstream distances, revealing the onset of the instability of the wake, which significantly amplifies the disturbances initiated by the side-to-side motion. The FFT magnitude at other frequencies remains approximately the same except at three times the imposed frequency. It is also observed that the amplitude at the imposed frequency ceases to grow from $x=6D$ to $x=8D$, while the peak at a higher frequency ($St \approx 0.75$) starts to appear at $x=6D$, which indicates the saturation of the linear growth process and the energy transfer from low to high frequencies via nonlinear effects. According to the stability theory (Schmid, Henningson & Jankowski Reference Schmid, Henningson and Jankowski2002; Drazin & Reid Reference Drazin and Reid2004), nonlinear effects result in energy growth at the sum and difference frequencies of the base perturbation, so that the second harmonic ($St = 0.25 +0.25 = 0.5$) appears first, then it interacts with the first harmonic ($St=0.25$) to generate the third harmonic ($St=0.5+0.25=0.75$). The absence of the second harmonic on the wake centreline, which seems to contradict this theory at first glance, is related to this specific location, i.e. the wake centreline. To show this, we plot the coherent structures of the whole wake revealed by the spectral proper orthogonal decomposition (SPOD) (Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018) in figure 6. As can be seen in figure 6(a), an energy concentration does exist at the second harmonic ($St = 0.50$). Interestingly, its coherent structure, as shown in figure 6(c), is approximately anti-symmetric with very weak velocity fluctuations near the centreline, which explains why the FFT magnitude computed based on the centreline velocity fluctuations has no apparent component at $St = 0.50$ as shown in figure 5(b). In contrast, the first and third harmonics, appearing in the spectra of the centreline velocity fluctuation, both have symmetrical spatial modes (figure 6b,d)

Figure 5.

Fast Fourier transform (FFT) amplitude of the normalized spanwise velocity ($v/U_\infty$) in the wake behind the fixed wind turbine (a), and the wind turbine subject to side-to-side FOWT motion with $St = 0.25$, $A/D=0.04$ (b). Data are collected at four downstream locations ($x = \{2D,4D,6D,8D\}$) along $y=0$.

Figure 6.

Spectral proper orthogonal decomposition (SPOD) of the wake of an FOWT oscillating at $St = 0.25$, the normalized modal energy as a function of Strouhal number (a), the principal SPOD mode of the transverse velocity $v$ of the first harmonic $St = 0.25$ (b), the second harmonic $St = 0.50$ (c), and the third harmonic $St = 0.75$ (d).

Similar to the above analyses, the spectra of the wake centre displacement is computed and shown in figure 7. It is observed that the amplitudes of the wake centre oscillations behind the oscillating wind turbine are one to two orders of magnitude higher than those of the fixed wind turbine at the frequency of the imposed side-to-side motion. For other frequencies, the FFT magnitudes from the oscillating wind turbine are comparable to those of the fixed wind turbine, which does not have any distinct peak frequencies. Furthermore, it is observed that the FFT magnitude at the peak frequency grows monotonically as one travels downstream from the turbine and is comparable to the rotor diameter in the far wake ($x=6D, 8D$).

Figure 7.

FFT amplitude of the normalized wake centre displacement $y_c/D$ behind the fixed wind turbine (a) and the wind turbine subject to side-to-side FOWT motion at $St = 0.25$, $A/D=0.04$ (b). Data are collected at four downstream locations ($x = \{2D,4D,6D,8D\}$) along $y=0$.

Another interesting observation is that the spanwise distribution of instantaneous wake centre positions becomes very different for the oscillating turbine when compared with the fixed one. These instantaneous wake centres are sampled from instantaneous flow fields at evenly distributed time steps. In figure 8, the probability density functions (PDFs) of instantaneous wake positions at turbine downstream locations $x=6D, 7D, 8D$ are compared. It is found that the PDF for the wake centres of the fixed wind turbine approximately follows the normal distribution located in a very narrow band, which barely changes as one moves downstream, being consistent with the instantaneous wake shown in figure 4(a). In contrast, the PDF of instantaneous wake centres behind the oscillating wind turbine spreads much wider and expands as one travels downstream. More importantly, the PDF shows a distinct pattern, with two peaks near the two extremities and a trough near $y=0$, which is significantly different from that observed for the fixed wind turbine and that of wake meandering induced by inflow turbulence, whose PDFs are approximately Gaussian (Li & Yang Reference Li and Yang2021). In contrast, this PDF is close to that of a simple harmonic motion (red dashed line), with a trough around the equilibrium position and two peaks at extremities (Tennekes & Lumley Reference Tennekes and Lumley1972). This is expected as the turbine side-to-side motion is imposed with a harmonic function as well. More importantly, the differences between the PDF of the wake centres of an oscillating turbine and that of a simple harmonic motion should be noticed, namely that the two peaks near the two extremities move quickly towards the centre as one moves to further turbine downstream locations. From the practical point of view, this PDF implies that the wake of an oscillating wind turbine may partially shelter the downstream turbine more often than a fixed wind turbine.

Figure 8.

Probability density functions of wake centre positions $y_c$ at turbine downstream locations $x=6D, 7D, 8D$ for the fixed wind turbine (blue) and the oscillating wind turbine (orange) with $St = 0.25$, $A/D=0.04$. The PDF of the fixed wind turbine is multiplied by a factor of $1/5$ to plot them in the same figure. The red dashed lines show the PDF of an idealized harmonic motion with the amplitude defined by the extremity of $y_c$ of the oscillating wind turbine case.

3.1.2. Time-averaged wake in the meandering frame of reference

With the instantaneous wake centre positions tracked with the Gaussian fit method, wake characteristics on the meandering frame of reference (MFR) are studied, as in Trujillo et al. (Reference Trujillo, Bingöl, Larsen, Mann and Kühn2011). Figure 9 plots the velocity deficits ensemble-averaged on the MFR for both fixed and side-to-side oscillating cases. As can be seen, the ensemble-averaged velocity deficits are approximately symmetric with respect to the wake centreline for both cases. In the near wake (until $x=3D$ turbine downstream), it is observed that the velocity deficit profiles from the side-to-side oscillating case are almost the same as those from the fixed case, as shown in figure 9(c). Differences are observed at further downstream locations, where the wake behind the oscillating rotor recovers more rapidly and expands slightly wider, being consistent with the observation of the instantaneous velocity as shown in figure 4. For instance, at the $x = 8D$ turbine downstream, the wake centreline velocity deficits are ${\rm \Delta} u_c = -0.27 U_\infty$ and ${\rm \Delta} u_c = -0.50 U_\infty$ for cases with/without rotor oscillation, respectively. Such differences between the two cases increase with turbine downstream distance, indicating a stronger influence of wake meandering on wake recovery at far wake locations.

Figure 9.

Time-averaged wake deficit in the meandering frame of reference for the fixed turbine case (a) and the oscillating turbine case with $St = 0.25$ and $A/D = 0.04$ (b). The velocity deficit profiles are compared in (c). The dash-dot lines and the dashed lines in (a,b) illustrate the fitted wake centreline and the wake half-width, respectively.

The turbulence intensity defined in the meandering frame of reference is examined in figure 10, which is defined as

(3.2)\begin{equation} \text{TI} = \sqrt{\frac{2k}{3}}/U_\infty, \end{equation}

where $k$ is the turbulence kinetic energy $k = \frac {1}{2}(u'^2 + v'^2 + w'^2)$. It is noticed that $u^{\prime },v^{\prime },w^{\prime }$ are the root-mean-square of the fluctuations of the three velocity components defined with respect to the mean velocity averaged on the meandering frame of reference. As can be seen, the TI profiles have two peaks near the wake boundaries and gradually expand inwards and outwards for both cases. Such expansion is faster in the oscillating turbine case. At $x\ge 7D$, the two peaks of TI merge as one on the centreline for the oscillating turbine case, which is retained for the fixed case even at the $x=9D$ turbine downstream. The maximum TI is approximately the same for both cases, at approximately $6.8\,\%$ at $x=8D$ for both cases, while the distribution of TI is much wider for the oscillating turbine case.

Figure 10.

Time-averaged rotor added turbulence intensity in the meandering frame of reference for the fixed rotor case (a) and the moving rotor case with $St = 0.25$ and $A/D = 0.04$ (b). The turbulence intensity profiles from both cases are compared in (c).

3.1.3. Time-averaged wake in fixed frame of reference

The effects of turbine motion on the time-averaged wake are examined in this subsection in the ground fixed frame of reference (FFR). First, we show the contour of streamwise velocity deficit in figure 11. It is observed that the time-averaged velocity deficits of the fixed and oscillating turbine cases are close to each other in the near wake region ($x<3D$). At further downstream locations, the wake of the fixed wind turbine as shown in figure 11(a) shows only a mild expansion, and the shear layer grows gradually in both inward and outward directions. The maximum deficit along the wake centreline remains approximately ${\rm \Delta} U = U- U_\infty = -0.49 U_\infty$ at $x=8D$. By contrast, the wake behind the oscillating wind turbine as shown in figure 11(b) shows an obvious broader wake expansion starting from $x=4D$, as a result of large-amplitude meandering motion, which spreads the low speed region to a larger extent. A direct result of this broad wake expansion is the faster recovery of momentum deficit in the wake core. As shown in figure 11(c), the time-averaged velocity deficit is approximately ${\rm \Delta} U \approx -0.18U_\infty$ at $x=8D$, being significantly lower than that behind the fixed wind turbine (${\rm \Delta} U \approx -0.49U_\infty$). This deficit of the oscillating turbine is also weaker than ${\rm \Delta} u_c = -0.27U_\infty$ in the MFR, as shown in figure 12(a). The differences between the velocity deficits on the FFR and MFR are found to be much reduced by taking the meandering motion into account, as shown in figure 12(b), where ${\rm \Delta} U$ in the FFR shows a good agreement with the weighted averaged of ${\rm \Delta} U$ in the MFR using the wake centre PDF plotted in figure 8.

Figure 11.

Time-averaged streamwise velocity deficit in the fixed frame of reference in the wake behind a fixed wind turbine (a) and a wind turbine subject to side-to-side simple harmonic motion with $St = 0.25$ and amplitude $A/D = 0.04$ (b), and a comparison of velocity deficit profiles for both cases in (c).

Figure 12.

Comparison between the time-averaged velocity deficit profile in the fixed frame of reference (FFR) and the meandering frame of reference (MFR) at $x=8D$ for the case with $St=0.25$ and $A=0.04D$: (a) velocity deficits in both frames of references compared directly, (b) comparison of the velocity deficit in the MFR corrected with the PDF of wake centre locations with that in the FFR.

Except for the benefit of the faster wake recovery, the wake meandering also results in stronger unsteady loads for the downstream turbines, which is unfavourable for the structure of the downstream turbines. The obtained contours and profiles of the TI are shown in figure 13. As expected, the oscillating motion of the wind turbine has drastically intensified the turbulence intensity in the wake in terms of both magnitude and spatial extent. Specifically, it is observed that the TI behind the hub and the tip of the rotor is intensified in the near wake because of turbine oscillation. Starting from $x=2D$, the TI grows in the shear layer of the wake and extends both inward to the wake core and outward to the freestream as one travels downstream. At approximately $x=5D$, the shear layers from the tips meet at the wake centre for the oscillating turbine case. At further turbine downstream locations, the TI ceases to grow and slightly decreases, starting from $x=7D$. At $x=8D$, the TI in the oscillating turbine case is significantly higher than that in the fixed turbine case (maximum TI $\approx$ 13.4 %, 8.5 % for the two cases, respectively). It is noticed that this level of turbulence intensity is within the range of the wind turbine design standard ($\textrm {TI}_{ref} = 14\,\%$ for turbulence category B as per IEC 61400-1).

Figure 13.

Turbulence intensity (TI) in the ground fixed frame of reference of the wake behind a fixed wind turbine (a) and a wind turbine subject to the side-to-side harmonic motion ($St = 0.25$ and $A/D = 0.04$) (b). The transverse profiles of TI in both cases are compared in (c).

3.2.1. Local linear stability analysis

Previous studies on wind turbine wakes in the literature have demonstrated their convective instability (Iungo et al. Reference Iungo, Viola, Camarri, Porté-Agel and Gallaire2013; Sarmast et al. Reference Sarmast, Dadfar, Mikkelsen, Schlatter, Ivanell, Sørensen and Henningson2014; Mao & Sørensen Reference Mao and Sørensen2018; Gupta & Wan Reference Gupta and Wan2019), i.e. the wake behaves as an amplifier of upstream disturbances. For a prescribed base flow, the linear stability theory can predict the spatial amplification rate for an infinitely small disturbance at different excitation frequencies. In the following, a local LSA is performed to assess the response of the wake to small disturbances with different frequencies. The time-averaged flow behind a fixed wind turbine is simplified using locally axis-parallel and axisymmetric assumptions, i.e. the velocity is simplified as $(\bar U(r,x),0,0)$ in the streamwise, radial and azimuthal directions (where $\bar {U}$ is the temporally and azimuthally averaged streamwise velocity), and employed as the base flow for the LSA. The perturbation velocity components $(u_x,u_r,u_\theta )$ and the perturbation pressure $p$ are assumed to be the form $(u_x,u_r,u_\theta,p) = (\hat {u}_x,\hat {u}_r,\hat {u}_\theta,\hat {p}) (\exp ({\rm i}k x + {\rm i}n\theta - {\rm i}\omega t) )$, where $\hat {\cdot }$ denotes the shape function in the radial direction and $n$ is the azimutal wave number of the perturbation mode. For the cases considered in the present work, the azimuthal wave number is equal to one, i.e. $n=1$, as the side-to-side motion velocity ($u_y$) can be projected in the radial and azimuthal directions as $u_r = u_y \cos \theta$ and $u_\theta = u_y \sin \theta$. Spatial stability analyses are performed with angular frequency $\omega = 2{\rm \pi} f$ (real number) and wavenumber $k=k_r+{\rm i}k_i$ (complex number). Substituting the perturbation variables into the linearized Navier–Stokes equations yields

(3.3)\begin{gather} {\rm i} k \hat{u}_x + \frac{\hat{u}_r}{r} + \frac{{\rm d} \hat{u}_r}{{\rm d} r} + \frac{{\rm i} n \omega}{r}\,\hat{u}_\theta = 0, \end{gather}

(3.4)\begin{gather}{\rm i} ( k \bar{U} -\omega ) \hat{u}_x + \frac{{\rm d} \bar{U}}{{\rm d} r}\,\hat{u}_r ={-} {\rm i} k \hat{p}, \end{gather}

(3.5)\begin{gather}{\rm i} ( k \bar{U} - \omega ) \hat{u}_r ={-} \frac{{\rm d} \hat{p}}{{\rm d} r}, \end{gather}

(3.6)\begin{gather}{\rm i} ( k \bar{U} - \omega ) \hat{u}_\theta ={-} \frac{{\rm i} n}{r}\,\hat{p}. \end{gather}

The above equations are the same as those derived by Batchelor & Gill (Reference Batchelor and Gill1962) for axisymmetric jets/wakes, except that the viscous terms are excluded, because their contribution is less important for base flow velocity profiles with inflection points (Schmid et al. Reference Schmid, Henningson and Jankowski2002). The linearized Navier–Stokes equations are discretized with Chebyshev polynomials on $128$ Gauss–Lobatto nodes $\xi \in [-1,1 ]$ and stretched to $r \in [0, r_{max}]$ through an algebraic mapping (Schmid et al. Reference Schmid, Henningson and Jankowski2002) defined as $r = a ({(1+\xi )}/{(b-\xi )})$ with $a = {r_{max}}/{(r_{max}-2)}$ and $b=1+{2a}/{r_{max}}$. The size of the computational domain is chosen as $r_{max} = 15D$. The size of the domain and the number of discretization points are verified to be large enough to not affect the stability analyses. The far field boundary condition is that the perturbations vanish at $r = r_{max}$. On the wake centreline, $\hat {u}_x = 0$, $\hat {p}=0$ and $\hat {u}_r + i \hat {u}_{\theta } = 0$ are imposed following Batchelor & Gill (Reference Batchelor and Gill1962).

Figure 14 shows the results of local LSA conducted at different downstream locations. As can be seen in figure 14(a), the velocity deficit profiles in the region between the core of the wake and the freestream is characterized by a shear layer. The thickness of this shear layer gradually increases in the turbine downstream direction due to the momentum mixing between the wake and the freestream, and consequently makes the local stability properties vary with the downstream distance as shown in figure 14(b). In the near wake ($x<3D$), the steep velocity variation in the shear layer results in a broad range of unstable frequencies $St \in (0, 0.9)$ approximately, with the most unstable frequency at $St \approx 0.5$. At further turbine downstream locations, both the most unstable frequency and the range of the unstable frequencies decrease. This trend of decrease of the most unstable frequency via the downstream distance is in agreement with the work of Gupta & Wan (Reference Gupta and Wan2019), where LSA is applied to the time-averaged wake of a tidal turbine. Such downstream variations of the spatial growth rate indicate that perturbations that are the most unstable in the near wake become stable when propagating downstream and cease to grow subsequently. On the other hand, perturbations at lower frequencies may excel in the far wake since they can grow for a longer distance. To demonstrate this accumulation effect via downstream distances, figure 14(c) displays the contours of the integrated amplification factor defined as $G =\exp {(-\int _0^x k_i \,\textrm {d}\kern0.7pt x)}$. As expected, it is found that the frequency with the largest amplification factor $G$, shown with the red dashed line, also decreases in the streamwise direction until approximately $x=7D$. In the far wake, perturbations with evident amplifications fall in the frequency range $0.1< St<0.8$. The strongest amplification factors are predicted for $0.2< St<0.4$, explaining the large-amplitude wake meandering triggered by turbine side-to-side motion with $St=0.25$ as presented in § 3.1. Furthermore, this frequency range is rather low compared to the commonly encountered excitement frequencies generated by wind turbines, e.g. the rated rotor frequency and the blade passing frequency are approximately equal to $0.2\,\textrm {Hz}$ ($St \approx 2.2$) and $0.6\,\textrm {Hz}$ ($St \approx 6.6$), respectively, for the NREL 5 MW wind turbine (Jonkman et al. Reference Jonkman, Butterfield, Musial and Scott2009) as well as other wind turbines at utility scale. As a result, unstable modes of the wake are unlikely to be triggered by the rotation of blades alone, being consistent with the literature that the hub vortex at lower frequencies plays an important role in triggering the shear layer instability and the far wake meandering for the bottom-fixed turbine (Kang et al. Reference Kang, Yang and Sotiropoulos2014). Moreover, this range is also lower than the wave frequencies at operational sea states ($St \approx 1.0$) as shown in table 2, indicating that floating wind turbines moving at wave frequencies cannot trigger significant wake meandering. On the other hand, this low frequency range ($0.2< St<0.4$) overlaps partially with the natural frequencies of the roll or sway motions as shown in table 2. However, it is noted that the Strouhal numbers computed as $St = D/(TU_\infty )$ shown in table 2 are based on the diameter $D = 126$ m and the rated wind speed $U_\infty = 11.4$ m s$^{-1}$ of the NREL 5 MW wind turbine, thus may vary for other turbine designs and wind speeds.

Figure 14.

Local linear stability analysis results of the wake behind the fixed wind turbine: (a) the streamwise velocity deficit obtained from large-eddy simulations is employed as the base flow for the linear stability analysis; (b) contours of local spatial growth rate $-k_i$; and (c) the integrated amplification factor $G = \exp {(- \int _0^x k_i \,\textrm {d}\kern0.7pt x)}$. The red dashed lines in (b,c) indicate the most amplified frequency along the streamwise locations.

Although the above LSA can provide useful information on the instability properties of the wake, it is worth noting that the analyses are carried out based on several assumptions for the base flow and the disturbances, i.e. (i) the axis-parallel and axisymmetric assumption for the base flow, (ii) the small perturbation assumption for linearizing the Navier–Stokes equations, and (iii) the inviscid flow assumption. Often, these prerequisites are not fully satisfied in real applications, where the base flow is more complex and the disturbances are of finite amplitudes, such that nonlinear effects may manifest. To further investigate the wake characteristics and to verify the results from the LSA, results from LES cases with the frequencies and amplitudes presented in tables 1 and 2 will be analysed in the following subsections.

3.2.2. Instantaneous wake

Snapshots of the instantaneous streamwise velocity deficits are plotted for comparison in figure 15. As observed, the wake behaviour varies via the turbine oscillation frequency, being consistent with the LSA. For cases with the lowest and highest motion frequencies, i.e. for $St = 0.1$ and $St \ge 0.8$, no apparent wake meandering is observed, and the wake is almost straight. For the cases with the lowest motion frequency ($St = 0.1$), the spanwise motion of the wake is very mild and the wake is close to that behind a fixed turbine. Although increasing the amplitude of the side-to-side motion to $A/D=0.04$ can deform the wake slightly into a sinusoidal form, the amplitude of the wake centre displacement is still much smaller than the wake width. For cases with the largest frequencies ($St \ge 0.8$), the wake centreline is approximately straight, indicating that the overall meandering motion of the wake is negligible. On the other hand, small-scale fluctuations are observed on the wake boundary, which, however, do not show apparent amplifications as one travels downstream, and remain locally near the boundary without apparent interactions with the core of the wake. Moreover, it is observed that these small-scale fluctuations do not seem to grow in magnitude when increasing the amplitude of the side-to-side motion for which roughly the same instantaneous wakes are observed for cases at $St \ge 0.8$ with different amplitudes. In contrast, the cases with $0.20 \le St \le 0.60$ show a much stronger far wake meandering, which is consistent with the larger amplification factor predicted by the LSA. Specifically, the cases with $0.20 \le St \le 0.40$ exhibit apparent sinusoidal form deformations with a continuous growth of the wake meandering amplitude via the downstream distance until the far wake. Larger meandering amplitudes are also observed when the motion amplitude is increased. For cases with higher Strouhal numbers ($0.4 < St \le 0.6$), however, the sinusoidal deformation of the wake is limited only in a certain near-wake region, beyond which the sinusoidal shape wake structures can no longer be identified visually, and the effects of motion amplitude on the wake meandering are hard to tell directly from these instantaneous snapshots.

Figure 15.

Instantaneous streamwise velocity deficit behind a wind turbine subject to side-to-side simple harmonic motion for various frequencies and amplitudes. Figures in the same row have the same motion frequency, denoted by the Strouhal number given in the last column. Figures in the same column have the same motion amplitude.

The frequency characteristics of the oscillating wake motion are further examined with the spanwise velocity along the wake centreline as shown in figure 16. In this figure, the spanwise velocity $v$ is converted into the frequency domain with FFT to obtain the FFT amplitude $\phi _v$, which is then normalized by the magnitude of the turbine oscillating velocity $\phi _v^0 = 2{\rm \pi} f A$ to indicate the amplification effect of turbine wake. As can be seen, the peak frequencies of $\phi _v$ coincide with the turbine motion frequency for all cases, whereas the magnitudes of the peaks (indicating the amplification ratio of the spanwise velocity) vary with the motion frequency. The largest amplification is achieved at $x=8D$ for the case with motion frequency $St=0.3$. The peak at the three-times frequency is also observed for the case with motion frequency $St=0.3$ (figure 16b), but is not apparent for other motion frequencies (figure 16a,c,d). These observations are consistent with instantaneous flow fields shown in figure 15. For the case with low motion frequency, i.e. $St=0.2$, the meandering motion just starts even at $x/D=8$, and needs more downstream evolution for the nonlinear effect to become evident. For the cases with high motion frequency, i.e. $St=0.4$, 0.6, on the other hand, the sinusoidal shape pattern exists for only a short distance, and the wake transits to a chaotic pattern shortly, such that it is hard to distinguish the nonlinear interaction between frequencies.

Figure 16.

FFT amplitude of the spanwise velocity $\phi _v$ normalized by $\phi _v^0$ the turbine's oscillating velocity amplitude at $y=0$ for different turbine motion frequencies. The amplitude of motion is $A=0.01D$.

We compute the amplification factor of the spanwise velocity in the wake for different turbine oscillation frequencies and amplitudes, as shown in figure 17. The amplification factor is defined as $G^{LES} = \phi _v/\phi _v^0$, with $\phi _v$ the amplitude of the spanwise velocity at the turbine motion frequency. The amplification factors at every $0.5D$ for $x\in [0D, 10D]$ are computed. The obtained amplification factors are presented in figures 17(a), 17(b) and 17(c) for the cases with turbine motion amplitudes $A/D=0.01, 0.02, 0.04$, respectively. As can be seen, all three figures show a similar pattern. It is observed that the amplification of the spanwise velocity is the most intensive for turbine motions with $St \in [0.2,0.3]$ and gradually weakens for lower or higher frequencies. For the cases with the smallest turbine motion amplitude, i.e. $A = 0.01D$ in figure 17(a), the location and size of the region with high amplification factors are closer to those predicted by the local LSA (as shown in figure 14c) when compared with the cases with higher oscillation amplitudes. In terms of the magnitudes of the amplification factors, the LES predictions are observed as being, in general, lower than the LSA predictions, because the hypotheses employed in LSA – i.e. the infinite small perturbation, axis-parallel base flow and inviscid fluid – are not fully satisfied. Moreover, it is observed that the differences between LSA and LES predictions are enlarged with increasing turbine motion amplitudes as shown in figures 17(b) and 17(c), which suggests that the nonlinearity caused by the finite-size perturbations plays a role in reducing the growth rate, being consistent with nonlinear stability theory (Schmid et al. Reference Schmid, Henningson and Jankowski2002) and a previous work on tidal turbine wakes (Gupta & Wan Reference Gupta and Wan2019). Furthermore, figure 17 shows that the increase of turbine motion amplitudes slightly lowers the most amplified frequency and narrows the region with high amplification factors.

Figure 17.

Contours of the amplification factor $G^{LES}$ computed from the LES results for different turbine motion amplitudes: (a) $A/D =0.01$, (b) $A/D=0.02$, (c) $A/D = 0.04$. The amplification factor is defined as $G^{LES} = \phi _v/\phi _v^0$, where $\phi _v$ is the amplitude of spanwise velocity in the wake obtained with the discrete Fourier transform of the time series of the spanwise velocity obtained from LES along the time-averaged wake centreline, and $\phi _v^0 = 2{\rm \pi} f A$ is the velocity amplitude of the wind turbine's simple harmonic motion. The largest amplification factor is marked together with its Strouhal number.

3.2.3. Time-averaged wake

To investigate the influence of turbine motion on the wake recovery, we analyse the time-averaged velocity deficits computed from different cases. In figure 18, we show contours of the time-averaged streamwise velocity deficits. As can be seen, the recovery of the time-averaged wake for cases with $St = 0.1$ and $St \ge 0.8$ is slow, and the wake is close to that of a fixed wind turbine. In contrast, the velocity deficit contours are significantly altered for cases with motion frequencies in the range $0.20 \le St \le 0.60$. One interesting observation is that the time-averaged wake recovery is not always enhanced by increasing the turbine motion amplitude as one may expect. One example is the cases with $St = 0.6$ (figures 18i-1,i-2,i-3), where the wake recovers slower for $A/D = 0.04$ than for $A/D = 0.01$.

Figure 18.

Time-averaged streamwise velocity deficit at the hub-height plane. The wind turbine is subject to side-to-side simple harmonic motion defined with various frequencies and amplitudes. Figures in the same row have the same motion frequency, denoted by the Strouhal numbers given in the last column. Figures in the same column have the same motion amplitude.

The effect of turbine motion on wake recovery is further examined by the quantitative comparison shown in figure 19, where the velocity deficit averaged over one rotor diameter in the spanwise direction is displayed. In general, it is found that the velocity in the wake recovers faster for an oscillating turbine (lines in colour) than for a fixed turbine (the black line), except when the motion frequency is low ($St = 0.1$). It is also found that wakes with different turbine motion frequencies respond differently when increasing the motion amplitude. For cases with relative low motion frequencies ($St=0.2, 0.25, 0.3$), larger motion amplitudes enhance the wake recovery. The opposite is found for the far wakes in the cases with motion frequency $St \ge 0.5$, where wake recovery slows down when the motion amplitude is increased. For the cases with $St = 0.4$, wake recovery is almost the same for all three amplitudes, if focusing on $\widetilde {{\rm \Delta} U}$ at $x=10D$.

Figure 19.

Velocity deficit averaged in time and in space (over one rotor diameter) for cases with different turbine oscillation frequencies and motion amplitudes.

3.2.4. Wake turbulence

Figure 20 displays the contours of the turbulence intensity in the fixed frame of reference on the hub-height plane for cases with different motion amplitudes and frequencies. It is worth noting that the turbulence intensity compared in this subsection also considers the velocity fluctuations due to the large-amplitude wake meandering. A distinct separation of the two types of velocity fluctuations as done in figure 13 is not straightforward here, especially for cases with high motion frequencies where the meandering is hard to define. As can be seen, the cases with high turbulence intensity are those with large-amplitude meandering motions shown in figure 15, i.e. cases with turbine motion frequencies in the range $0.2 \le St \le 0.6$, indicating that the turbulence intensity defined on the fixed frame of reference includes significant contributions from the meandering motion for these cases.

Figure 20.

Turbulence intensity at the hub-height plane. The wind turbine is subject to side-to-side simple harmonic motion defined with various frequencies and amplitudes. Figures in the same row have the same motion frequency, denoted by the Strouhal numbers given in the last column. Figures in the same column have the same motion amplitude.

Figure 21 compares the turbulence intensity averaged in the spanwise direction over one rotor diameter on the hub-height plane. Except for the case with the lowest frequency ($St=0.10$) of turbine motion, all cases show significant increases of TI when compared with the fixed one (black line). For cases with the same Strouhal number in the frequency range $0.20 \le St \le 0.60$, it is found that increasing the amplitude of turbine motion increases the TI for a certain range of turbine downstream distances, which becomes shorter for large motion frequencies. At further downstream locations beyond this range, the TI values due to different motion amplitudes are similar to each other for the case with $St=0.25$, while the TI values are larger for smaller motion amplitude in cases with $0.3 \le St \le 0.6$. For the case with the highest frequency ($St=0.80$), the TI values from the cases with different amplitudes are approximately the same as each other, being higher than the fixed turbine case for $x/D<8$ and approximately the same as the fixed turbine at further downstream locations. For all cases, the TI first increases to a peak and then decreases. The value of the peak and its location are dependent on both the frequency and the amplitude of the turbine motion. When comparing cases with the same motion frequency, it is found that the peak location moves upstream with increasing motion amplitude for $0.2 \le St \le 0.6$. The maximum value of TI increases with turbine motion amplitude for cases with low frequencies ($0.2\le St \le 0.40$), while it decreases for $St =0.5, 0.6$.

Figure 21.

Turbulence intensity (averaged in space over one rotor width) for cases with different turbine oscillation frequencies and motion amplitudes.

3.3.1. Inflow turbulence

The effects of inflow turbulence on the onset of meandering induced by turbine side-to-side motion are examined in this subsection. As can be seen in figure 22, the wake of the fixed turbine is rather steady when the inflow turbulence is null or small, e.g. for cases with $\sigma _u/U \le 3.8\,\%$. At these low turbulence levels, the effect of side-to-side motion on triggering the meandering motion can be distinguished visually. For inflows with higher turbulence intensity ($\sigma _u/U \ge 4.7\,\%$), the meandering patterns between the fixed and oscillating turbines are similar to each other, as seen in figure 22. In figure 23, the standard deviations of instantaneous wake positions are compared for different inflow turbulence intensities. It is observed that the meandering amplitudes are in general increased because of the turbine side-to-side motion. However, such increases become smaller for higher inflow turbulence intensities, and are negligible for the case with $\sigma _u/U = 15.8\,\%$.

Figure 22.

Comparison of instantaneous velocity deficits behind a fixed turbine (in the first column) and an oscillating turbine (in the second column) for different inflow turbulence intensities $\sigma _u/U$. Inflows are synchronized for figures in the same row; turbine side-to-side motion synchronized in phase for figures in the second column. Fitted wake centrelines are plotted with red dashed lines.

Figure 23.

Standard deviation of instantaneous wake centre locations for cases with different inflow turbulence intensities $\sigma _u/U$.

The time-averaged velocity deficits for cases with different inflow turbulence intensities are examined in figures 24 and 25. For the fixed turbine cases, increasing the inflow TI monotonically accelerates wake recovery. In contrast, the wake recovery is slowed down when the inflow turbulence intensity is increased from 0 % to 1.5 % for the oscillating turbine case. As shown in figure 25, the wake recovery enhancement due to the side-to-side motion of the turbine is observed only for cases with $\sigma _u/U \le 3.8\,\%$, while it becomes negligible when $\sigma _u/U \ge 4.7\,\%$.

Figure 24.

Comparison of time-averaged velocity deficits in the fixed frame of reference behind a fixed turbine (in the first column) and an oscillating turbine (in the second column) for different inflow turbulence intensities $\sigma _u/U$.

Figure 25.

Velocity deficit in the wake behind a fixed and an oscillating wind turbine averaged in time and in space (over the width of the turbine) for cases with different inflow turbulence intensities $\sigma _u/U$.

The wake turbulence is analysed in figures 26 and 27. From both figures, it is found that the effect of side-to-side motion on the wake turbulence also diminishes with increasing inflow turbulence intensities. By carefully analysing figure 27(a), one can find that the pure side-to-side motion induces TI of approximately 4 % in the near wake (at $x=2D$). It is worth noting that this TI can be employed to indicate the energy of the perturbation induced by the turbine side-to-side motion. This energy level is found to be lower than that induced by the inflow turbulence for cases with $\sigma _u/U = 3.8\,\%$ (figure  27c), i.e. the inflow may violently destroy the harmonic perturbations induced by the turbine side-to-side motion, and hampers their development. This comparison could explain why the wake evolution is dominated by the inflow turbulence rather than the turbine side-to-side motion for cases with high inflow turbulence intensity.

Figure 26.

Comparison of the turbulence intensity $\sqrt {{{2k}/{3}}}/U_\infty$ in the fixed frame of reference behind a fixed turbine (in first column) and an oscillating turbine (in the second column) for inflow with different $\sigma _u/U$.

Figure 27.

Turbulence intensity in the wake behind a fixed and an oscillating wind turbine averaged in time and in space (over the width of the turbine) for inflows with different $\sigma _u/U$.

3.3.2. Wind shear effects

This subsection investigates the effects of vertical wind shear on the onset of wake meandering induced by turbine side-to-side motion. The vertical shear results in wind speed variation on the rotor's sweeping plane, and thus challenges the hypothesis of axisymmetric flow employed for local LSA. Results of LES are employed to verify how the wake meandering can be affected by including vertical wind shear. Figure 28 shows the instantaneous streamwise velocity deficit on the hub-height plane. As can be seen, similar wake meandering patterns are observed, although the axisymmetry is lost for cases with non-zero wind shear exponents. The quantitative comparison in figure 29 shows that the wake meandering amplitude, indicated by the standard deviation of the wake centre locations $\sigma _{y_c}$, decreases with increasing wind shear exponent $\alpha$, suggesting an attenuation effect of the wind shear on the wake meandering. It is worth noting that the shear-generated turbulence will also be involved in the streamwise evolution of the wake. We argue that the vertical shear is the major factor for the different trends shown in figure 29, as the turbulence intensities resulting from the vertical shear are fairly small and approximately the same for the considered non-zero shear exponents (see details in § 2.3).

Figure 28.

Instantaneous wake deficit behind the oscillating wind turbine for different wind shear exponents $\alpha = 0$ (a), $\alpha = 0.11$ (b), $\alpha = 0.14$ (c), $\alpha = 0.20$ (d).

Figure 29.

Standard deviation of the instantaneous wake centre location $\sigma _{y_c}$ for cases with different inflow wind shear exponents.