**4.1. Wind-turbine parameterizations**

Figure 11(a) shows a typical three-blade horizontal axis upwind wind turbine, which usually exhibits much better power efficiency than other types of wind turbines [66]. The actuator line method [ALM, 34, 86, 91] combines a three-dimensional flow solver with a technique in which body forces are distributed radially along lines, which represent the blades of the wind turbine.

**Figure 11.** Schematic of the actuator line model: (a) three-dimensional view of a wind turbine; (b) a discretized blade; and (c) cross-section airfoil element showing velocities and force vectors.

Figure 11(b) shows one discretized element of a blade, and figure 11(c) shows a cross-sectional element at radius *r* defining the airfoil in the (*θ*, *x*) plane, where *x* is the axial direction. With the tangential and axial velocities of the incident flow denoted as *V<sup>θ</sup>* and *Va*, respectively, the local velocity relative to the rotating blade is given as **<sup>V</sup>***rel* = (*V<sup>θ</sup>* − <sup>Ω</sup>*r*, *Va*). The angle of attack is defined as *<sup>α</sup>* <sup>=</sup> *<sup>ϕ</sup>* <sup>−</sup> *<sup>γ</sup>*, where *<sup>ϕ</sup>* <sup>=</sup> *tan*−1(*Va*/(Ω*<sup>r</sup>* <sup>−</sup> *<sup>V</sup><sup>θ</sup>* )) is the angle between **V***rel* and the rotor plane, and *γ* is the local pitch angle. The turbine-induced force per radial unit length is given by the following equation

14 Computational and Numerical Simulations

**4. Wind energy applications**

for the development of the turbine wake.

**4.1. Wind-turbine parameterizations**

(a) (b)

cross-section airfoil element showing velocities and force vectors.

wind turbine.

The second high-Reynolds-number is based on measurements of nearly isotropic turbulence downstream of an active grid [41]. The initial Taylor micro-scale Reynolds number (*Reλ*) is approximately 720. The new model is implemented in LES of decaying isotropic turbulence with initial conditions that match the measured energy spectra at *x*1/*M* = 20. Figure 10(b) shows energy spectra at different times, for comparison, the results obtained using the standard dynamic nonlinear mixed model [41] were also included in the plot. The new

With the fast growing number of wind farms being installed worldwide, the interaction between ABL turbulence and wind-turbine wakes, and its effects on energy production and dynamic loading on downwind turbines, have become important issues in the wind energy and atmospheric sciences communities [92]. Optimizing the design of wind energy projects (placement of isolated wind turbines or layout of wind farms) requires the prediction of atmospheric turbulence and its interactions with wind turbines at a wide range of spatial and temporal scales. As a result, during the last decade, numerical modeling of wind-turbine wakes has become increasingly popular. Most of the previous studies of ABL flow through isolated wind turbines or wind farms have parameterized the turbulence using a RANS approach [3, 4, 30, 42]. Only recently there have been some efforts to apply LES to simulate wind-turbine wakes [16, 36, 37, 59, 78, 97]. In addition to the above-mentioned challenges in LES of the ABL, the accuracy of LES for wind energy applications hinges also on our ability to parameterize the forces induced by the turbines on the flow. These forces are responsible

Figure 11(a) shows a typical three-blade horizontal axis upwind wind turbine, which usually exhibits much better power efficiency than other types of wind turbines [66]. The actuator line method [ALM, 34, 86, 91] combines a three-dimensional flow solver with a technique in which body forces are distributed radially along lines, which represent the blades of the

**Figure 11.** Schematic of the actuator line model: (a) three-dimensional view of a wind turbine; (b) a discretized blade; and (c)

(c)

model clearly gives more accurate results at small scales that *<sup>k</sup>* > 6 m<sup>−</sup>1.

$$\mathbf{f} = \frac{\mathbf{dF}}{dr} = \frac{1}{2}\rho V\_{rel}^2 c \left(\mathbf{C}\_L \mathbf{e}\_L + \mathbf{C}\_D \mathbf{e}\_D\right) \, , \tag{14}$$

where *CL* = *CL*(*α*, *Re*) and *CD* = *CD*(*α*, *Re*) are the lift coefficient and the drag coefficient, respectively, *ρ* is the air density, *c* is the chord length, and **e***<sup>L</sup>* and **e***<sup>D</sup>* denote the unit vectors in the directions of the lift and the drag, respectively. The flow of interest is essentially inviscid, and viscous effects from the boundary layer of blade are introduced only as integrated quantities through the use of airfoil data. The airfoil data and subsequent loading are determined by computing local angles of attack from the movement of the blades and the local flow field. The model enables us to study in detail the dynamics of the wake and the tip vortices and their influence on the induced velocities in the rotor plane. The applied blade forces are distributed smoothly to avoid singular behavior and numerical instability. In practice, the aerodynamic blade forces are distributed along and away from the actuator lines in a three-dimensional Gaussian manner through the convolution of the computed local load, **f**, and a regularization kernel *ηǫ* as shown below

$$\mathbf{f}\_{\varepsilon} = \mathbf{f} \otimes \eta\_{\varepsilon} \, \qquad \eta\_{\varepsilon} = \frac{1}{\varepsilon^3 \pi^{3/2}} \exp\left(-\frac{d^2}{\varepsilon^2}\right) \, \, \, \tag{15}$$

where *d* is the distance between grid points and points at the actuator line, and *ǫ* is a parameter that serves to adjust the concentration of the regularized load. The value of this regularization parameter, *ǫ*, is typically on the order of 1 − 3 grid sizes and should be as small as possible so that the turbine-induced force is distributed over an area representing the chord distribution and does not affect the wake structure. However, small values will deliver significant numerical oscillations; thus, as a trade-off between numerical stability and accuracy [34, 90], it is set to be 1.5 times of the grid size in the rotor plane.

The main advantage of representing the blades by airfoil data is that many fewer grid points are needed to capture the influence of the blades than would be needed for simulating the actual geometry of the blades. Therefore, the ALM is well suited for wake studies since grid points can be concentrated in a larger part of the wake while keeping the computing costs at a reasonable level. Moreover, the ALM encompasses blade motions as well as their mixing mechanism, which is crucial for simulating more realistic wind-turbine wakes and achieving improved predictions with respect to the standard actuator disk model [7, 78, 97]. However, the ALM does not resolve detailed flows on blade surfaces and relies on tabulated two-dimensional airfoil data for providing lift and drag coefficients. This makes it dependent on both the quality of airfoil data and the method used to model the influence of dynamically changing angles of attack and stall.

### **4.2. Turbine model testing**

Validation of LES of wind turbine wakes is complicated by the difficulties associated with measuring turbulence in the field at the spatial and temporal resolution required for the validation of numerical models. Wind-tunnel experiments provide high-resolution spatial and temporal information characterization of the turbulent flow under controlled stationary inflow conditions and offer a valuable alternative to study the turbulent flow and vortical structures in wind-turbine wakes. Simulation results are compared with the high-resolution velocity measurements collected in the wake of a 3-blade miniature wind turbine placed in the Saint Anthony Falls Laboratory atmospheric boundary-layer wind tunnel. The miniature turbine adopted in the experimental study of Chamorro and Porté-Agel [19] consists of a 3-blade GWS/EP-6030x3 rotor attached to the small DC generator motor. In the simulations, the lift and drag coefficients of blade are determined based on a previous experimental study by Sunada et al. [89]. The computation domain has a height *Lz* = 0.46 m. The horizontal computational domain spans a distance *Lx* = 4.5 m = 30*D* in the streamwise direction and *Ly* = *Lz* in the spanwise direction, where *D* = 0.15 m denotes the turbine diameter. A wind turbine, which has a hub height *Hhub* = 0.125 m, is placed in the middle of the computational domain at 6*D* measuring from the upstream boundary. The resolution of the simulation is *Nx* × *Ny* × *Nz* = 192 × 64 × 64. The spatial distributions of key turbulence statistics, including time-averaged axial velocity, axial velocity variance and turbulent shear stress, are used to characterize wind-turbine wakes. Readers may find a detailed discussion of this wind-turbine wake in Porté-Agel et al. [78], Wu and Porté-Agel [97].

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good agreement with the measurements in the turbine wake. There is a clear evidence of the effect of the turbine extracting momentum from the incoming flow and producing a wake immediately downwind. As expected the velocity deficit is largest near the turbine and it becomes smaller as the wake expands and entrains surrounding air. Nonetheless, the effect

Large-eddy simulation of turbulent flows with applications to atmospheric boundary layer research

**Figure 13.** Axial velocity variance at different downwind distances. ◦: results from wind-tunnel measurements; solid line:

Figure 13 compares the measured and simulated axial velocity variances at selected locations. The turbulence profiles obtained using the ALM are in acceptable agreement with the wind-tunnel measurements. The results show a strong enhancement of the turbulence (compared with the relatively low turbulence levels in the incoming flow) at the top-tip level. The maximum turbulence is found at that level and at a normalized distance of approximately 3< *x*/*D* <5. It is important to point out that this is within the typical range of distances between adjacent wind turbines in wind farms and, therefore, it should

Further, due to the non-uniform (logarithmic) mean velocity profile of the incoming boundary-layer flow, it is found that a non-axisymmetric distribution of the mean velocity profile and, consequently, of the mean shear in the turbine wake. In particular, the strongest shear is found at the top-tip level. The turbulence distribution and the maximum enhancement of turbulence occurs at the top-tip level can be explained considering the non-axisymmetric distribution of velocity profiles and the fact that the mean shear and associated turbulence kinetic energy production are maximum at the top-tip height. It contrasts with the axisymmetry of the turbulence statistics reported by previous studies in the case of wakes of turbines placed in free-stream flows [22, 67, 91], and demonstrates the substantial influence of the incoming flow on the structure and dynamics of wind-turbine

be considered when calculating wind loads on the turbines.

results from simulations using the ALM.

wakes.

of the wake is still noticeable even in the far wake, at distances as large as *x*/*D*=20.

**Figure 12.** Time-averaged streamwise velocity at different downwind distances. ◦: results from wind-tunnel measurements; solid line: results from simulations using the ALM.

Figure 12 shows the measured and simulated time-averaged streamwise velocity profiles at selected downwind locations (*x*/*D*=2, 3, 5, 7, 10, 14, 20), together with the incoming (*x*/*D*=-1) flow velocity profile. LES with the ALM yields mean velocity profiles that are in good agreement with the measurements in the turbine wake. There is a clear evidence of the effect of the turbine extracting momentum from the incoming flow and producing a wake immediately downwind. As expected the velocity deficit is largest near the turbine and it becomes smaller as the wake expands and entrains surrounding air. Nonetheless, the effect of the wake is still noticeable even in the far wake, at distances as large as *x*/*D*=20.

16 Computational and Numerical Simulations

**4.2. Turbine model testing**

solid line: results from simulations using the ALM.

Validation of LES of wind turbine wakes is complicated by the difficulties associated with measuring turbulence in the field at the spatial and temporal resolution required for the validation of numerical models. Wind-tunnel experiments provide high-resolution spatial and temporal information characterization of the turbulent flow under controlled stationary inflow conditions and offer a valuable alternative to study the turbulent flow and vortical structures in wind-turbine wakes. Simulation results are compared with the high-resolution velocity measurements collected in the wake of a 3-blade miniature wind turbine placed in the Saint Anthony Falls Laboratory atmospheric boundary-layer wind tunnel. The miniature turbine adopted in the experimental study of Chamorro and Porté-Agel [19] consists of a 3-blade GWS/EP-6030x3 rotor attached to the small DC generator motor. In the simulations, the lift and drag coefficients of blade are determined based on a previous experimental study by Sunada et al. [89]. The computation domain has a height *Lz* = 0.46 m. The horizontal computational domain spans a distance *Lx* = 4.5 m = 30*D* in the streamwise direction and *Ly* = *Lz* in the spanwise direction, where *D* = 0.15 m denotes the turbine diameter. A wind turbine, which has a hub height *Hhub* = 0.125 m, is placed in the middle of the computational domain at 6*D* measuring from the upstream boundary. The resolution of the simulation is *Nx* × *Ny* × *Nz* = 192 × 64 × 64. The spatial distributions of key turbulence statistics, including time-averaged axial velocity, axial velocity variance and turbulent shear stress, are used to characterize wind-turbine wakes. Readers may find a detailed discussion

of this wind-turbine wake in Porté-Agel et al. [78], Wu and Porté-Agel [97].

**Figure 12.** Time-averaged streamwise velocity at different downwind distances. ◦: results from wind-tunnel measurements;

Figure 12 shows the measured and simulated time-averaged streamwise velocity profiles at selected downwind locations (*x*/*D*=2, 3, 5, 7, 10, 14, 20), together with the incoming (*x*/*D*=-1) flow velocity profile. LES with the ALM yields mean velocity profiles that are in

**Figure 13.** Axial velocity variance at different downwind distances. ◦: results from wind-tunnel measurements; solid line: results from simulations using the ALM.

Figure 13 compares the measured and simulated axial velocity variances at selected locations. The turbulence profiles obtained using the ALM are in acceptable agreement with the wind-tunnel measurements. The results show a strong enhancement of the turbulence (compared with the relatively low turbulence levels in the incoming flow) at the top-tip level. The maximum turbulence is found at that level and at a normalized distance of approximately 3< *x*/*D* <5. It is important to point out that this is within the typical range of distances between adjacent wind turbines in wind farms and, therefore, it should be considered when calculating wind loads on the turbines.

Further, due to the non-uniform (logarithmic) mean velocity profile of the incoming boundary-layer flow, it is found that a non-axisymmetric distribution of the mean velocity profile and, consequently, of the mean shear in the turbine wake. In particular, the strongest shear is found at the top-tip level. The turbulence distribution and the maximum enhancement of turbulence occurs at the top-tip level can be explained considering the non-axisymmetric distribution of velocity profiles and the fact that the mean shear and associated turbulence kinetic energy production are maximum at the top-tip height. It contrasts with the axisymmetry of the turbulence statistics reported by previous studies in the case of wakes of turbines placed in free-stream flows [22, 67, 91], and demonstrates the substantial influence of the incoming flow on the structure and dynamics of wind-turbine wakes.

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Global Energy and Water Cycle Experiment Atmospheric Boundary Layer Study (GABLS) initiative [11]. It represents a typical quasi-equilibrium moderately SBL, similar to those commonly observed over polar regions and equilibrium nighttime conditions over land in

Large-eddy simulation of turbulent flows with applications to atmospheric boundary layer research

To study the effect of a wind farm on the GABLS case [59], a V112-3.0MW wind turbine is "immersed" (using the ALM) in the GABLS domain such that the wind-turbine center is located at *xc* = 80 m, *yc* = 280 m, and *zc* = 119 m (hub height). This wind turbine has a rotor diameter of *D* = 112 m, and rotates at 8 RPM, corresponding to a tip speed ratio of approximately 7 for an optimal performance at a free-stream wind speed of approximately 6 m/s. Three blades consist of Risø-P airfoil. Like in the original GABLS case, the vertical height of the computational domain is *Lz* = 400 m. The domain size in the y-direction is fixed to be *Ly* = 5*D* = 560 m, and two x-direction dimensions corresponding two typical wind-turbine spacings that are studied: (i) *Lx* = 8*D* = 896 m (the corresponding LES is abbreviated as the 8D case); (ii) *Lx* = 5*D* = 560 m (the corresponding LES is abbreviated as the 5D case). Periodic boundary conditions are applied horizontally to simulate an infinitely large wind farm. It should be noted that the baseline case (without turbines) attains a quasi-steady state in 8 - 9h [10, 11]. Therefore, in order to examine the wind-turbine effects relative to the baseline case, the wind turbine is only introduced in the last hour of simulation. Figure 15 shows the time evolutions of the boundary height, the surface momentum flux and the surface buoyancy flux. Data are saved each 15 seconds. When wind turbines are installed, there exists a significant increase of the boundary-layer height. Specifically, over the last 15 min, the 8D case yields a SBL height of approximately 225 m (increased ≈ 28%), and the 5D case yields a SBL height of approximately 250 m (increased ≈ 43%). The current simulation results support the tendency that smaller wind-turbine spacing yields larger boundary-layer increases. Further, the magnitudes of the surface momentum flux and the surface buoyancy flux decrease with time. Specifically, over the last 15 min, the 8D case yields a momentum-flux magnitude of approximately 0.05 m2/s2 (reduced ≈ 30%), corresponding to a friction velocity of 0.23 m/s; the 5D case yields a momentum-flux magnitude of approximately 0.043 m2/s2 (reduced ≈ 40%), corresponding to a friction velocity of 0.21 m/s. The 8D case yields a buoyancy-flux magnitude of approximately −3.8 × <sup>10</sup>−<sup>4</sup> <sup>m</sup>2/s3 (reduced ≈ 15%), corresponding to a heat flux of −13.5 W/m2; the 5D case yields a buoyancy-flux magnitude of approximately −3.2 × <sup>10</sup>−<sup>4</sup> <sup>m</sup>2/s3 (reduced ≈ 28%), corresponding to a heat flux of −11.4 W/m2. It is interesting to note that it takes some time before wind turbines are able to affect surface fluxes. This delay in the change of the fluxes is likely associated to the time it takes for the multiple wakes to expand horizontally and affect the entire surface area. Overall, the reduced surface heat flux phenomenon obtained from the current research is consistent with the results from low-resolution wind-farm simulations performed by [7]. The reduced surface momentum and heat flux magnitudes indicate a reduction in the level of turbulent mixing and transport near the surface. Regarding the overall thermal-energy budget, this reduced heat flux is consistent with the increase of air temperature in the

Figure 16(a) shows the formation (initial stages) of blade-induced three-dimensional helicoidal tip vortices, detected using |*ω*|-definition (0.3 times of the maximum vorticity, 35) in the 5D case. Due to the strong shear and non-uniformity of the incoming boundary-layer flow, helicoidal vortices are stretched as they travel faster at the top tip level compared with

mid-latitudes. The GABLS case is used here as a baseline case (no-turbine case).

boundary layer as shown later in figure 17(b).

**Figure 14.** Kinematic shear stress at different downwind distances. ◦: results from wind-tunnel measurements; solid line: results from simulations using the ALM.

Figure 14 compares the measured and simulated total shear stress (summation of the resolved part and the SGS part) at selected locations. Again results obtained using the ALM are in good agreement with the wind-tunnel measurements. It is clear that the turbine introduces stresses that are locally much larger in magnitude than the stresses in the incoming flow. In the near-wake region, a region above the turbine hub bears large negative stress and a lower region bears large positive stress. As the wake grows with downwind distance, the relative change becomes smaller. Also, similar to the other turbulence statistics, the change in kinematic stress with respect to the incoming flow is not negligible until the far-wake at a distance of x/D=20.
