A Numerical Study of Wind-Turbine Wakes for Three Atmospheric Stability Conditions

Abstract

The effects of atmospheric stability on wind-turbine wakes are studied via large-eddy simulations. Three stability conditions are considered: stable, neutral, and unstable, with the same geostrophic wind speed aloft and the same Coriolis frequency. Both a single 5-MW turbine and a wind farm of five turbines are studied. The single-turbine wake is strongly correlated with stability, in terms of velocity deficit, turbulence kinetic energy (TKE) and temperature distribution. Because of the Coriolis effect, the wake shape deviates from a Gaussian distribution. For the wind-farm simulations, the separation of the core region and outer region is clear for the stable and neutral cases, but less distinct for the unstable case. The unstable case exhibits strong horizontal variations in wind speed. Local accelerations such as related to aisle jets are also observed, whose features depend on stability. The added TKE in the wind farm increases with stability. The highest power extraction and lowest power deficit are observed for the unstable case.

Appendices

Appendix

The version of WiTTS, including the Coriolis and stability effects, is validated by simulating the following two cases: the first is a buoyancy-driven convective planetary boundary layer used in Moeng and Sullivan (1994), where the turbulence generated by buoyancy is dominant. The second case is a moderately stable atmospheric boundary layer following the GABLS (GEWEX Atmospheric Boundary Layer Study) initiative (Beare et al. 2006), where the turbulence generated by wind shear is dominant. Both cases are simulated in the absence of wind turbines.

A Convective Planetary Boundary Layer

Following Moeng and Sullivan (1994), the domain size is $5~\text {km} \times 5~\text {km} \times 2~\text {km}$ in the x-, y- and z-directions, with the resolution of $N_{x} \times N_{y} \times N_{z}=96\times 96\times 96$, respectively. The total physical time of the simulation is $9000~\text {s}$, the prescribed surface heat flux $q_{3,{ wall}}=0.24$ K m s$^{-1}$, geostrophic velocity vector $(U_{g}, V_{g})=(10, 0)$ m s$^{-1}$ and Coriolis frequency $f_{c}=1\times 10^{-4}~\text {s}^{-1}$. The initial potential temperature is $300~\text {K}$ below the initial boundary-layer height of $937~\text {m}$ and increases by a total of 8 K across $6\triangle z$, and with a lapse rate of 3 K km$^{-1}$ above. More details of the problem can be found in Moeng and Sullivan (1994). Two values of ${ Pr}_{{ SGS}}$, i.e., 0.5 and 1, are tested here. Time- and horizontally-averaged simulation results from both WiTTS and the literature are compared in Fig. 14. In general, the current LES data match well with the results reported in the literature. Note that the momentum fluxes are normalized by the square of the Deardorff convective velocity $w_{*}$ (as defined in Moeng and Sullivan 1994) as

$$\begin{aligned} w_{*}=\Big (\frac{g}{\theta _{0}}q_{3,{ wall}}z_{i}\Big )^{1/3}, \end{aligned}$$

(10)

where $z_{i}$ is the boundary-layer height diagnosed using the “maximum-gradient method” (Sullivan et al. 1998) at each time-step. A relatively large deviation of the mean spanwise velocity component V is observed in the lower part of the boundary layer. The LES results from WiTTS are insensitive to the value of ${ Pr}_{{ SGS}}$ for the convective PBL, since the turbulence is intense. Even when the data in the literature are unavailable, such as in Fig. 14b, d, the profiles from both ${ Pr}_{{ SGS}}=0.5$ and 1 are well collapsed, although small deviations in $\varTheta$ are observed at about $z/z_{i}=1$, where the vertical gradient of $\varTheta$ is large. A slightly better match of the normalized momentum flux is observed for ${ Pr}_{{ SGS}}=1$ than that for ${ Pr}_{{ SGS}}=0.5$ (Fig. 14c). Therefore, for the unstable boundary layer, ${ Pr}_{{ SGS}}=1$ will be used. Hereafter, $\langle \ldots \rangle$ represents a horizontal average, overbar denotes the time average, and the time- and horizontally-averaged velocities and potential temperature are presented in upper case, and the tilde is omitted for the resolved properties for simplicity.

Fig. 14

Vertical profiles of time- and horizontally-averaged a wind speeds, b potential temperature $\varTheta$, c normalized total momentum flux (resolved $+$ SGS terms) and d total heat flux (resolved $+$ SGS terms) of the convective planetary boundary layer using two ${ Pr}_{{ SGS}}$ values compared with data reported in the literature (Sullivan et al. 1998). Note that the literature data were not available in (b) and (d)

Table 2 References for some models used in the GABLS cases

Fig. 15

Vertical profiles of time- and horizontally-averaged a wind speeds, b potential temperature $\varTheta$, c total momentum flux (resolved $+$ SGS terms) and d total heat flux (resolved $+$ SGS terms) of the stable boundary-layer simulations using three values of ${ Pr}_{{ SGS}}$ compared with the data from the GABLS database

B Stable Atmospheric Boundary Layer: The GABLS Case

Compared with the convective boundary layer, the simulation of a stable boundary layer is more challenging, since turbulence production is reduced and, therefore, the results are more sensitive to numerical errors. According to the GABLS description, the domain size is $400~\text {m} \times 400~\text {m} \times 400~\text {m}$ in the x-, y- and z-directions, with an initial potential temperature profile consisting of a mixed layer (with potential temperature $265~\text {K}$) up to $100~\text {m}$, with an overlying inversion of strength 0.01 K m$^{-1}$ above. A prescribed surface cooling of 0.25 K h$^{-1}$, a geostrophic velocity vector of $(U_{g}, V_{g})=(8, 0)$ m s$^{-1}$ and the Coriolis frequency of $f_{c}=1.39\times 10^{-4}~\text {s}^{-1}$ (corresponding to the latitude $73^{\circ }\text {N}$) are applied for 9 h, with the statistics produced from the last hour. A large amount of data are available from various participants of the GABLS project and can be downloaded from http://gabls.metoffice.com/, where some are used here to compare with the WiTTS results, as shown in Table 2. A good summary and intercomparison of those models can be found in Beare et al. (2006). For all simulations shown here, a resolution of $6.25~\text {m}$ in each direction is used monotonically. Three values of ${ Pr}_{{ SGS}}$, i.e., 0.375, 0.5 and 1, are considered in the current LES for comparison.

In Fig. 15, several time- and horizontally-averaged profiles, i.e., the mean wind speed (streamwise and spanwise components), mean potential temperature, momentum flux $\langle \overline{uw}\rangle$ and heat flux $\langle \overline{\theta w}\rangle$, are compared with results from the GABLS database. Unlike the convective boundary layer in Appendix A, a clear dependence of the results on the values of ${ Pr}_{{ SGS}}$ is observed. When ${ Pr}_{{ SGS}}=0.375$ and 0.5 are used, the results almost collapse, and fall within the threshold of the results of the GABLS database. Meanwhile, for ${ Pr}_{{ SGS}}=1$, large discrepancies are observed, especially for profiles of $\varTheta$ and the heat flux. Therefore, ${ Pr}_{{ SGS}}=0.5$ will be used for the stable case here.

The wind direction veers with height by the action of the Coriolis effect, and an obvious low-level jet is formed at about $200~\text {m}$, which is one of the distinctive characteristics of the stable boundary layer (Nieuwstadt 1984). The spanwise component V, however, has a maximum around the hub height (100 m) of most current wind turbines. The vertical profiles of the mean temperature reveal varying shapes, while a linear behaviour is observed in the momentum and heat fluxes, which have been successfully captured by the WiTTS model. Moreover, various models produced scattered results for both the mean and turbulence properties. Consistent with the observations reported in Beare et al. (2006), the main deviations occur at the top of the boundary layer, but less in the lower part where the wind turbines are located. Given the difficulty of simulating the stable boundary layer, which is very sensitive to the numerical configuration, the current results indicate a notable success of the WiTTS model.