A large-eddy simulation framework for wind energy applications

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

A large-eddy simulation framework for wind energy applications Fernando Port´e-Agela, b , Hao Lub , Yu-Ting Wua, b a School

of Architecture, Civil and Environmental Engineering, ´ Ecole Polytechnique F´ed´erale de Lausanne, Switzerland b Saint Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota - Twin Cities, USA ABSTRACT: Accurate prediction of atmospheric boundary layer (ABL) flow and its interactions with wind turbines is of great importance for optimizing the design (turbine siting) of wind energy projects. Large-eddy simulation (LES) can potentially provide the kind of high-resolution spatial and temporal information needed to maximize wind energy production and minimize fatigue loads in wind farms. However, the accuracy of LES in simulations of ABL flow with wind turbines hinges on our ability to parameterize subgrid-scale (SGS) turbulent fluxes as well as turbine-induced forces. This paper focuses on recent efforts to develop and validate a LES framework for wind energy applications. Both SGS stress and SGS heat flux are parameterized using tuning-free scaledependent Lagrangian dynamic models (Stoll and Port´e-Agel, 2006a). These models optimize the local value of the model coefficients based on the dynamics of the resolved scales. The turbineinduced lift and drag forces are parameterized using two types of models: an actuator disk model (ADM) that distributes the force loading on the rotor disk, and an actuator line model (ALM) that distributes the forces on lines that follow the position of the blades. Simulation results are compared to wind-tunnel measurements collected with hot-wire and cold-wire anemometry in the wake of a miniature 3-blade wind turbine at the St. Anthony Falls Laboratory ABL wind tunnel. In general, the characteristics of the turbine wakes simulated with the proposed LES framework are in good agreement with the measurements. Finally, results are presented from simulations performed with the new LES framework to study the effects of wind farms on local and regional-scale turbulent fluxes of momentum and heat. 1

INTRODUCTION

With the fast growing number of wind farms being installed worldwide, the interaction between atmospheric boundary layer (ABL) turbulence and wind turbines, and the interference effects among wind turbines, have become important issues in both the wind energy and the atmospheric science communities (e.g., Petersen et al., 1998; Vermeer et al., 2003; Baidya Roy et al., 2004). Accurate prediction of ABL flow and its interactions with wind turbines is of great importance for optimizing the design (turbine siting) of wind energy projects. In particular, it can be used to maximize wind energy production and minimize fatigue loads in the evaluation of wind-farm layouts. Additionally, numerical simulations can provide valuable quantitative insight into the potential impacts of wind farms on local meteorology. These are associated with the significant role of wind turbines in slowing down the wind and enhancing vertical mixing of momentum, heat, moisture and other scalars. The turbulence parameterization constitutes the most critical part of turbulent flow simulations. It was realized early that direct numerical simulations (DNSs) are not possible for most engineering

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

and environmental turbulent flows, such as the ABL. Thus, the Reynolds-averaged Navier-Stokes (RANS) approach was adopted in most previous studies of ABL flow through single wind turbines or wind farms (e.g., Xu and Sankar, 2000; Alinot and Masson, 2002; Sørensen et al., 2002; G´omezElvira et al., 2005; Tongchitpakdee et al., 2005; Sezer-Uzol and Long, 2006; Kasmi and Masson, 2008). However, as repeatedly reported in a variety of contexts (e.g., AGARD, 1998; Pope, 2000; Sagaut, 2006), RANS computes the mean flow and models the effect of the unsteady turbulent velocity fluctuations according to a variety of physical approximations. Consequently, RANS is too dependent on the characteristics of particular flows to be used as a method of general applicability. Large-eddy simulation (LES) was developed as an intermediate approach between DNS and RANS, the general idea being that the large, non-universal scales of the flow are computed explicitly, while the small scales are modeled. Large-eddy simulation can potentially provide the kind of high-resolution spatial and temporal information needed to maximize wind energy production and minimize fatigue loads in wind farms. The accuracy of LES in simulations of ABL flow with wind turbines hinges on our ability to parameterize subgrid-scale (SGS) turbulent fluxes as well as turbine-induced forces. Only recently there have been some efforts to apply LES to simulate wind-turbine wakes (Jimenez et al., 2007, 2008; Calaf et al., 2010). In this study, we introduce (section 2) a new LES framework for wind energy applications. The SGS turbulent fluxes of momentum and heat are modeled using the Lagrangian scale-dependent dynamic models, which optimize the local values of the SGS model coefficients and account for their scale dependence in a dynamic manner (using information of the resolved field and, thus, not requiring any tuning of parameters). They have been shown to represent the resolved flow statistics (e.g., mean velocity and energy spectra) near the surface better than the traditional Smagorinsky model or the standard dynamic models (Stoll and Port´e-Agel, 2006a, 2008). The turbine-induced forces are parameterized with two types of models: an actuator disk model (ADM) that distributes the force loading on the rotor disk, and an actuator line model (ALM) that distributes the forces on lines that follow the position of the blades. In order to test the performance of the new LES framework (with scale-dependent dynamic models and different turbine parameterizations), simulation results are compared with high-resolution wind-tunnel measurements collected in the wake of a miniature wind turbine in a boundary layer flow (section 3). Section 4 presents the application of the new LES framework to a case study of a large wind farm in a stably stratified ABL. 2

LES FRAMEWORK

2.1 LES governing equations The LES numerical code used in this study is a modified version of the code used for previous studies of ABL flows (e.g., Albertson and Parlange, 1999a; Port´e-Agel et al., 2000; Port´e-Agel, 2004; Stoll and Port´e-Agel, 2006a,b; Basu and Port´e-Agel, 2006; Wan et al., 2007; Lu and Port´eAgel, 2010). The code solves the filtered continuity equation, the filtered momentum conservation equations written in rotational form, and the filtered heat transport equation ∂e ui =0, (1) ∂xi D E   e e ∗ θ − θ uj ∂τi j ∂e ui ∂e ui ∂e 1 ∂ pe fi + uej − =− − + δi3 g + fc εi j3 uej − + Fi , (2) ∂t ∂x j ∂xi ρ ∂xi ∂x j θ0 ρ

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

∂q j ∂e θ ∂e θ + uej =− , ∂t ∂x j ∂x j

(3)

e uei is the resolved velocity in the i-direction where the tilde represents a spatial filtering at scale ∆, (with i = 1, 2, 3 corresponding to the streamwise (x), spanwise (y) and vertical (z) directions), e θ is the resolved potential temperature, θ0 is the reference temperature, the angle brackets represent a horizontal average, g is the gravitational acceleration, fc is the Coriolis parameter, δi j is the Kronecker delta, εi jk is the alternating unit tensor, pe∗ = pe + 12 ρe u j uej is the modified pressure, pe is the filtered pressure, ρ is the air density, fi is an immersed force (per unit volume) for modeling the effect of wind turbines on the flow, and Fi is a forcing term (e.g., geostrophic wind or imposed mean pressure gradient). τi j and q j are the SGS stress and the SGS heat flux, which are defined as ei uej , τi j = ug iu j − u

ej e q j = uf θ. jθ − u

(4)

Molecular dissipation and diffusion have been neglected since the Reynolds number is very high. In equation (2), the rotational form is used to ensure conservation of kinetic energy. Buoyancy effects are accounted for by using the Boussinesq approximation. 2.2 Lagrangian scale-dependent dynamic models A common parameterization approach in LES consists of computing the deviatoric part of the SGS stress with an eddy-viscosity model 1 τi j − τkk δi j = −2νsgs Sei j , 3

(5)

and the SGS heat flux with an eddy-diffusivity model qj = −

νsgs ∂e θ , Prsgs ∂x j

(6)

where Sei j = (∂e ui /∂x j + ∂e u j /∂xi )/2 is the resolved strain rate tensor, νsgs is the SGS eddy viscosity and Prsgs is the SGS Prandtl number. A popular way of modeling the eddy viscosity is to use the mixing length approximation (Smagorinsky, 1963). As a result, the eddy viscosity is modeled as  1/2 2 2 e e e e e νt = Cs ∆ |S|, where |S| = 2Si j Si j is the strain rate, and Cs is the Smagorinsky coefficient. When applied to calculate the SGS heat flux, the resulting eddy-diffusion model requires the speci−1 . fication of the lumped coefficient Cs2 Prsgs One of the main challenges in the implementation of the eddy-viscosity/diffusivity models is the specification of the model coefficients. The value of the model coefficients Cs and Prsgs (and there−1 ) is well established for isotropic turbulence. In that case, if a cut-off filter is used in the fore Cs2 Prsgs e is equal to the grid size, then Cs ≈ 0.17 and Prsgs ≈ 0.4 (Lilly, inertial subrange and the filter scale ∆ 1967; Mason and Derbyshire, 1990). However, anisotropy of the flow, particularly the presence of a strong mean shear near the surface in high-Reynolds-number ABLs, makes the optimum value of those coefficients depart from their isotropic counterparts (e.g., Kleissl et al., 2003, 2004; Bou-Zeid

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

et al., 2008). A common practice is to specify the coefficients in an ad hoc fashion. For Cs , this typically involves the use of dampening functions and stability corrections based on model performance and field data (e.g., Mason and Thomson, 1992; Mason, 1994; Kleissl et al., 2003, 2004). Similarly, a stability dependence is usually prescribed for the SGS Prandtl number, for example, Prsgs = 0.44 under free convection and increasing to 1 for very stable stratification (e.g., Mason and Brown, 1999). An alternative was introduced by Germano et al. (1991) in the form of the dynamic Smagorinsky e model, in which Cs is calculated based on the resolved flow field using a test filter ( f ) at scale ∆ = α∆ (with typically α = 2). This procedure was also applied to the eddy diffusivity model by Moin et al. (1991) to calculate the lumped coefficient. While the dynamic model removes the need for prescribed stability and shear dependence, it assumes that the model coefficients are scale invariant. Simulations (e.g., Port´e-Agel et al., 2000) and experiments (e.g., Bou-Zeid et al., 2008) have shown that the scale invariance assumption breaks down near the surface, where the filter and/or test filter scales fall outside of the inertial subrange of the turbulence. The assumption of scale invariance in the standard dynamic model was relaxed by Port´e-Agel et al. (2000) and Port´e-Agel (2004) through the development of scale-dependent dynamic models for the SGS stress and flux, respectively. In this study, we employ the recently developed scaledependent Lagrangian dynamic models (Stoll and Port´e-Agel, 2006a, 2008) to compute the local optimized value of the model coefficients Cs and Prsgs . Based on the Germano identity (Germano et al., 1991; Lilly, 1992), the procedure of minimizing the error associated with the use of the model equations (5) and (6) results in the following equations

 L M hKi Xi iL i j i j −1 e e =

L , Cs2 Prsgs (∆) = , (7) Cs2 (∆) hXi Xi iL Mi j Mi j L where h·iL denotes Lagrangian averaging (Meneveau et al., 1996; Stoll and  Port´e-Agel, 2006a),    e e ∂ θ ∂ θ 2 2 2 2 e |S| e |S| e Sei j , Ki = uee e e Sei j − α β|S| e − α βθ |S| θ − ueie θ, and Xi = ∆ . Li j = uei uej − uei uej , Mi j = 2∆ ∂xi

∂xi

2 e 2 e Here, β = Cs2 (α∆)/C s (∆) is the ratio of Cs at the test filter scale and the filter scale, and βθ = −1 (α∆)/C 2 −1 e 2 −1 e Cs2 Prsgs s Prsgs (∆) is the ratio of Cs Prsgs at the test filter scale and the filter scale. Instead of assuming β = 1 and βθ = 1 as scale-invariant models do, the scale-dependent Lagrangian dynamic models leave the two parameters as unknowns that will be calculated dynamically. The implemenb = α2 ∆. e In addition, the models assume a tation requires to use a second test filter ( fb) at scale ∆ power-law dependence of the coefficients with scale, and thus −1 (α∆) −1 (α2 ∆) e e e e Cs2 Prsgs Cs2 Prsgs C2 (α2 ∆) C2 (α∆) = s , βθ = = . (8) β= s −1 e −1 e e e (∆) (α∆) Cs2 (∆) Cs2 (α∆) Cs2 Prsgs Cs2 Prsgs It is important to note that this assumption is much weaker than the assumption of scale invariance (β = 1 and βθ = 1) in scale-invariant models, such as dynamic Smagorinsky model, and has been shown to be more realistic in recent a-priori field studies (e.g., Bou-Zeid et al., 2008). Again minimizing the error associated with the use of the model equations (5) and (6), another set of equations for Cs and Prsgs are E D 0 0 Li j Mi j hK 0 X 0 i −1 e e =D EL , Cs2 (∆) Cs2 Prsgs (∆) = i0 i0 L , (9) Xi Xi L M0 M0 ij

ij

L

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010









b c be e2 |[ e2 |[ eb e Sei j − α2 β2 |b e ∂eθ − α2 β2 |b e ∂eθ ej − b uej , Mi0 j = 2∆ Sei j , Ki0 = uee θ − uei θ, and Xi0 = ∆ ueib S| S| where Li0 j = ued S| iu θ S| ∂xi . ∂xi

More details can be found in Port´e-Agel et al. (2000), Port´e-Agel (2004), and Stoll and Port´e-Agel (2006a, 2008). Tests of different averaging procedures (over horizontal planes, local and Lagrangian) simulations of a stable boundary layer with the scale-dependent dynamic model have shown that Lagrangian averaging produces the best combination of self consistent model coefficients, first- and second-order flow statistics and small sensitivity to grid resolution (Stoll and Port´e-Agel, 2008). Based on the local dynamics of the resolved scales, these tuning-free models compute Cs and Prsgs dynamically as the flow evolves in both space and time, hence are better suited for ABL flows associated with inhomogeneities.

(a)

(b) (CS2 )

Figure 1: (a) Smagorinsky coefficient obtained with the scale-dependent Lagrangian dynamic model for the SGS stress tensor; (b) Scale dependence parameter (β) obtained with the scale dependent Lagrangian dynamic model for the SGS stress tensor. Results are averaged over time and spanwise direction. −1 The scale-dependent Lagrangian dynamic models have also been used to compute CS and CS2 Prsgs in simulations of neutral boundary layers over simple topography patterns (using terrain following coordinates) consisting of sinusoidal hills (Wan et al., 2007) and single hills (Wan and Port´e-Agel, 2010). As illustrated in Figure 1(a), the model for the SGS stress tensor is able to dynamically (without tuning) adjust the value of the Smagorinsky coefficient and scale-dependence coefficient to have smaller values near the hill crest, where the flow is more anisotropic. As a result, it yields results that are more accurate and less dependent on resolution than the standard Smagorinsky and scale-invariant dynamic models (see Fig. 2 for the prediction of the velocity profile above the hill crest).

2.3 Surface boundary conditions The surface boundary conditions require calculation of the instantaneous local surface shear stress and surface heat flux at each surface grid point. This is accomplished through the application of Monin-Obukhov similarity theory (Businger et al., 1971) between the surface and the local velocity and potential temperature fields. Although this theory was developed for mean quantities, it is

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

(a)

(b)

Figure 2: Effect of grid resolution on the simulated non-dimensional velocity profile over the wave crest obtained from (a) the Lagrangian dynamic model and (b) the scale-dependent Lagrangian dynamic model for the SGS stress tensor. The wind-tunnel data of Gong et al. (1996) (symbols) are also shown.

common practice (e.g., Stoll and Port´e-Agel, 2006b) in LES of atmospheric flows to use it for instantaneous (filtered) fields as follows 2  uei uer κ (i = 1, 2) , (10) τi3,s = − ln (z/z0 ) − ΨM uer where τi3,s is the instantaneous local wall stress, z0 is the roughness q length, κ is the von K´arm´an constant, ΨM is the stability correction for momentum, and uer = ue21 + ue22 is the local instantaneous (filtered) horizontal velocity at the height z = ∆z /2 with ∆z corresponding to the vertical grid spacing. The surface heat flux is calculated in a similar manner from the instantaneous (filtered) local potential temperature at z = ∆z /2   ue∗ κ θs − e θ , (11) q3,s = ln (z/z0 ) − ΨH q where θs is the local surface temperature, ue∗ = 4 τ213,s + τ223,s is the friction velocity, and ΨH is the temperature stability correction. The surface-layer stability corrections are calculated at every point based on a locally defined Obukhov length L=−

ue3∗e θ . κgq3,s

(12)

Following the recommendations of previous stable ABL studies (e.g., Beare et al., 2006), we adopt κ = 0.4, ΨM = −4.8 Lz and ΨH = −7.8 Lz in this study. The upper boundary condition is a stress/flux free condition.

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

2.4 Wind-turbine parameterizations The actuator disk model is the simplest approach to represent the wind-turbine forces in numerical models of flow through propellers and turbines. The approach was introduced by Froude (1889) as a continuation of the work of Rankine (1865) on momentum theory of propellers. The RankineFroude actuator disk model assumes the load is distributed uniformly on the circular disk. Since the resulting force acts only in the axial direction (i.e., no rotational component is considered), here we refer to this method as ‘actuator disk model without rotation’ (ADM-NR). The axial thrust force Fx u20 ACT , where ue0 is the acting on the actuator disk in the streamwise direction is modeled as Fx = 21 ρe unperturbed resolved velocity of the axial incident flow in the center of the rotor disk, A is the frontal area of the cells within the rotor region and CT is the thrust coefficient. This model provides only a one-dimensional approximation of the turbine-induced thrust force; however, due to its simplicity and capability to deliver reasonable results with coarse grids, this model is still widely used in the context of both RANS (e.g., G´omez-Elvira et al., 2005) and, more recently, also LES (Jimenez et al., 2007, 2008; Calaf et al., 2010; Wu and Port´e-Agel, 2010).

(c) (a)

(b)

Figure 3: Schematic of the blade element momentum approach: (a) three-dimensional view of a wind turbine; (b) a discretized blade; and (c) cross-section airfoil element showing velocities and force vectors.

A major advancement in the wind-turbine modeling was the introduction of the blade element momentum (BEM) theory (Glauert, 1963). Since the flow is assumed to be inviscid, the blade element approach does not require resolving the boundary-layer flow around the turbine blade surface, which greatly reduces the computational cost requirements. In Figure 3(c), a cross-section element at radius r defines the airfoil in the (θ, x) plane, where x is the streamwise direction. Denoting the tangential and axial velocity in the inertial frame of reference as Vθ and Vx , respectively, the local velocity relative to the rotating blade is given as Vrel = (Ωr −Vθ ,Vx ). The angle of attack is defined as α = ϕ − γ, where ϕ = tan−1 (Vx /(Ωr −Vθ )) is the angle between Vrel and the rotor plane, and γ is the local pitch angle. The actuator-disk surface is defined as the area swept by the blades, where the turbine-induced lift and drag forces are parameterized and integrated over the spatial and temporal resolution used in LES. Here we refer to this method as ‘actuator disk model with rotation’ (ADM-

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

R). Unlike the ADM-NR, the ADM-R considers the effect of the turbine-induced flow rotation as well as the non-uniform force distribution. To further determine the forces acting on the rotor disk, we consider an annular area of differential size dA = 2πrdr. The resulting force is given by f2D =

dF 1 2 Bc = ρV (CL eL +CD eD ) , dA 2 rel 2πr

(13)

where B is the number of blades, CL = CL (α, Re) and CD = CD (α, Re) are the lift coefficient and the drag coefficient, respectively, c is the chord length, and eL and eD denote the unit vectors in the directions of the lift and the drag, respectively. More details on the ADM-NR are given by Wu and Port´e-Agel (2010). In the generalized actuator disk and related models, the turbine-induced forces (lift, drag and thrust) are parameterized and integrated over the temporal and spatial resolutions, which limits the ability to capture the helicoidal vortices (Vermeer et al., 2003). To overcome this limitation, a threedimensional actuator line model (ALM) has been developed by Sørensen and Shen (2002). The ALM calculates the turbine-induced lift and drag forces and distributes them along lines representing the blades. As a result, it has the strength to capture important physics of the wind-turbine wake such as tip vortices and coherent periodic helicoidal vortices in the near-wake region (Sørensen and Shen, 2002; Troldborg et al., 2007; Ivanell et al., 2009). The resulting turbine-induced force is calculated as f2D =

dF 1 2 = ρVrel c(CL eL +CD eD ) . dr 2

(14)

In all above-mentioned wind-turbine models, the induced 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 by taking the convolution of the computed local load, f, and a regularization kernel ηε as shown below  2 d 1 ε (15) f = f2D ⊗ηε , ηε = 3 3/2 exp − 2 , ε ε π where d is the distance between grid points and points at the actuator line, and ε is a parameter that adjusts the distribution of the regularized load. The main advantage of representing the blades by airfoil data is that much fewer grid points are needed to capture the influence of the blades compared to what would be needed for simulating the actual geometry of the blades. Therefore, the ADM-R and the ALM are 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. The effects of the nacelle and the turbine tower on the turbulent flow are modeled as drag forces fnacelle and ftower , respectively, by using a formulation similar to the ADM-NR and a drag coefficient based on the specific geometry. More details on this approach are given by Wu and Port´e-Agel (2010). As a result, the total immersed force fi associated with wind-turbine effects in the filtered momentum equation (Eq. 2) is given by fi = (fε + fnacelle + ftower ) · ei , where ei is the unit vector in the i-th direction.

(16)

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

3

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. Despite the limitations in the Reynolds numbers that can be attained, windtunnel experiments can still reproduce important characteristics of wind-turbine wakes such as the velocity deficit, the local enhancement of turbulence intensity, and the formation of helicoidal tip vortices. In addition, they can provide high-resolution spatial and temporal characterization of the turbulent wake flow under controlled stationary inflow conditions, which is of great value for LES validation. 3.1 Simulation details In this study, 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 Port´e-Agel (2010) consists of a 3-blade GWS/EP-6030x3 rotor attached to the small DC generator motor. In the simulations, the blade section is viewed as a flat plate, whose lift and drag coefficients are determined based on a previous experimental study by Sunada et al. (1997). As shown in Figure 4, the domain has a height Lz =0.46m. The horizontal computational domain spans a distance Lx =28.8d in the streamwise direction and Ly =4.8d in the spanwise direction, where d=0.15m denotes the turbine diameter. A wind turbine, which has a hub height Hhub =0.125m, is placed in the middle of the computational domain at a distance of 6d from the upstream boundary. Ly = 0.72 (m)

Lx = 28.8d = 4.32 (m)

Buffer zone

R = 0.075 (m)

Lz = 0.46 (m)

Lz = 0.46 (m)

Vx

R = 0.075 (m)

Hhub = 0.125 (m)

0.36 (m) (~2.4d)

0.36 (m) (~2.4d)

Hhub = 0.125 (m) 2d

4d

22.8d

Figure 4: Schematic of the simulation domain. Front view (left) and side view (right).

3.2 Comparison with wind-tunnel measurements The spatial distribution of two key turbulence statistics is used to characterize wind-turbine wakes: the time-averaged streamwise velocity u and the streamwise turbulence intensity σu /uhub . The overbar represents a temporal average. The experimental data were collected at x/d=-1, 1, 2, ..., 9, 10, 12, ..., 18, 20 in the middle of the domain as well as y/d=-0.7, -0.6, ..., 0,6, 0.7 in a cross section downstream x/d=5 of the turbine. Figure 5 displays contours of the time-averaged streamwise velocity obtained from the wind-tunnel experiment and simulations with ADM-NR, ADM-R and ALM on a vertical plane perpendicular to the turbine. Furthermore, to facilitate the quantitative comparison

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

of the results, Figure 6 shows vertical profiles of the measured and simulated time-averaged streamwise velocity at selected downwind locations (x/d=2, 3, 5, 7, 10, 14, 20), together with the incoming flow velocity profile. There is clear evidence of the effect of the turbine extracting momentum from the incoming flow and producing a wake (region of reduced velocity) immediately downwind. As expected the velocity deficit (reduction with respect to the incoming flow) 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. Further, due to the non-uniform (logarithmic) mean velocity profile of the incoming boundary-layer flow, we find a non-axisymmetric distribution of the mean velocity profile and, consequently, of the mean shear in the turbine wake. In particular, as also reported by Chamorro and Port´e-Agel (2009, 2010), the strongest shear (and associated turbulence kinetic energy production) is found at the level of the top tip. This result contrasts with the axisymmetry of the turbulence statistics reported by previous studies in the case of wakes of turbines placed in free-stream flows (Crespo and Hern´andez, 1996; Medici and Alfredsson, 2006; Troldborg et al., 2007), and demonstrates the substantial influence of the incoming flow on the structure and dynamics of wind-turbine wakes.

Figure 5: Contours of the time-averaged streamwise velocity u (ms−1 ) in the middle vertical plane perpendicular to the turbine: (a) wind-tunnel measurements, (b) ADM-NR, (c) ADM-R, (d) ALM.

The Fifth International Symposium on Computational Wind Engineering (CWE2010) Chapel Hill, North Carolina, USA May 23-27, 2010

2

z/d

x/d =−1

x/d = 2

x/d = 3

x/d = 5

1

0

1.2

1.6

2

2.4

2.8

1.2

1.6

2

2.4

2.8

1.2

1.6

2

2.4

2.8

1.2

1.6

2

2.4

2.8

2

2.4

2.8

Wind speed (ms−1) 2

z/d

x/d = 7

x/d = 10

x/d = 14

x/d = 20

1

0

1.2

1.6

2

2.4

2.8

1.2

1.6

2

2.4

2.8

1.2

1.6

2

2.4

2.8

1.2

1.6

Wind speed (ms−1)

Figure 6: Comparison of vertical profiles of the time-averaged streamwise velocity u (ms−1 ): wind-tunnel measurements (◦), ADM-NR (dashed line), ADM-R (solid line) and ALM (dotted line).

As shown in Figures 5 and 6, LES with the ALM or the ADM-R yields mean velocity profiles that are in good agreement with the measurements everywhere in the turbine wake (near wake as well as far wake). The ADM-NR is able to capture the velocity distribution in the far-wake region (x/d>5), but it clearly overpredicts the velocity in the center of the wake in the near-wake region (x/d