Experimental data were collected through a series of wind tunnel experiments conducted by in the Portland State University wind tunnel depicted in Fig. . The tunnel test section measures 5 m long with a cross section of 1.2×0.8 m. Inflow to the test section was conditioned with vertical strakes to produce a logarithmic boundary layer following and . Chains measuring 0.005 m were placed across the tunnel at fixed intervals of 0.1 m to maintain the boundary layer profile throughout the test section. A single scaled model turbine was positioned in the center of the tunnel 3.5 m downstream of the strakes. The model turbine measured 0.084 m in height with a rotor diameter of 0.08 m and was manufactured via 3D printing using a 3D Systems ProJet MJP 3600 printer. A Faulhaber 1016SR motor measuring 0.01 m in diameter was used as a DC generator. Turbine operation was controlled by means of a variable resistance potentiometer tuned such that the tip-speed ratio measured λ≈4, CT≈0.65, and CP≈0.15 for a hub-height inflow velocity of 6.5 m s -1. The Reynolds number based on model turbine diameter was 3.3×104 for the chosen inflow velocity, and the measured turbulence intensity was 11 %. A reference case was created by orienting the model turbine normal to the inflow. Four yaw and tilt angles of ±10 and ±20∘ were considered by rotating the model turbine about its base. As a consequence, tilt misalignment varied nacelle elevation between 0.0045 m lower or higher and 0.026 m downstream or upstream, respectively. Stereoscopic particle image velocimetry (SPIV) measurements were recorded in the wake of a single model wind turbine at downstream distances of 2, 3, 5, and 7 rotor diameters. Inflow conditions were captured by removing the model turbine and recording free stream behavior. Neutrally buoyant aerosolized diethylhexyl sebacate seeding particles were maintained at constant density throughout the experiment. Measurement planes were oriented perpendicular to the mean flow and captured using two 4 M pixel CCD cameras in conjunction with a Litron Nano double-pulsed Nd:YAG (532 nm, 1200 mJ, 4 ns duration) laser; 1500 image pairs spanning 0.24×0.18 m were recorded at a rate of 4 Hz. Images were processed in LaVision DaVis 8.4 with decreasing multipass kernels of 48×48 and 24×24 px at 50 % overlap for a spatial resolution of 0.001 m. Further details pertaining to the experimental setup including inflow profile measurements and turbine characterization are available in .

LES data were obtained from a series of SOWFA-6 simulations with the IEA-15 MW reference turbine and a 1.5 MW turbine shown in Figs.  and , respectively. The IEA-15 MW reference turbine rotor diameter measures 240 m and the hub height 150 m. The 1.5 MW turbine rotor diameter was 77 m and the hub height 80 m. Turbine behavior was simulated with the actuator disk model through OpenFAST coupling . Yaw and tilt misalignments of ±10 and ±20∘ were imposed by rotating the turbine nacelle. The total domain size measured (7100; 3000; 1000) m for the 15 MW cases and (3000; 3000; 1000) m for the 1.5 MW cases with a grid resolution of 10 m. Two levels of mesh refinement were added for the 1.5 MW cases to increase grid resolution in the turbine vicinity, so the 15 and 1.5 MW cases shared similar grid resolution across the rotor. The first refinement zone spanned (2310; 390; 300) m and was located at (690; 1320; 0) m. The secondary refinement measured (2160; 310; 250) m and was located at (770; 1350; 0) m. Grid resolution was increased to 5 m in the first zone and again to 2.5 m in the second. A single turbine was located at (1000; 1500; H) m for all SOWFA-6 simulations. A neutral atmospheric boundary layer inflow was generated with a 20 000 s precursor simulation on each base domain with a hub-height inflow velocity of 8 m s-1 and a surface roughness of 0.15 m. Hub-height turbulence intensities were computed from turbulent kinetic energy and averaged from x/D=-0.25 to x/D=-1.25. Mean turbulence intensity for the 15 MW cases was 6.2±0.3 %, and mean turbulence intensity for the 1.5 MW cases was 8.4±0.2 %. Inflow to the domain was driven by a pressure gradient which was adjusted at each time step to maintain the desired hub-height velocity . The measured thrust coefficients for each turbine configuration at the specified condition are presented in Table . In each simulation, 500 s was allotted for startup transients followed by 3000 s of data collection.

Eddy viscosity values are obtained at each downstream location from the slope of a least squares linear regression between S13|w and u′w′‾|w. Wake flow is isolated downstream of the turbine following Eqs. () and (). Both S13|B and u′w′‾|B are obtained from the reference plane at X for experimental data, then as streamwise averages spanning x=-1.25 D through x=-0.25 D for simulation data. In the experimental setup, ∂W/∂x=0 upstream of the turbine by design and is not included in S13|B. Furthermore, ∂W/∂x is found to be at least 2 orders of magnitude less than ∂U/∂z across all data sets. However, its contribution to S13|w is included for completeness. Example planes of the streamwise-vertical component of the velocity gradient tensor and its corresponding component of the Reynolds stress tensor are shown in Fig. . Slope fit error is computed with the standard regression error and presented as shaded 95 % confidence bounds on νT,w in Fig. . Since ground interactions introduce substantial strain near the surface, data below a height of z=H-D/1.75 are discarded to ensure eddy viscosities reflect wake rather than surface phenomena. Additionally, only data contained within (y±1.15 D, z±H-D/1.75) are considered at each streamwise location, so the measurement area is consistent across all data sets. Neglecting LES data outside this area does not influence the quality of the linear regression as quantities outside the wake are near zero from Eq. ().

Eddy viscosity evolves downstream of the turbine as the wake flow recovers, highlighted in Fig. . Here, νT,w* is normalized relative to the maximum value for each case to facilitate consistent comparisons across cases. Immediately downstream of the turbine rotor, x/D≲1, the flow is dominated by the momentum deficit and pressure gradients from turbine operation. Shear stresses are small in this region and develop downstream as energy is converted from mean flow gradients into turbulence. A linear relationship is present between S13,w and u′w′‾|w with negligible average fit errors below 1 % of νT,w*. S13|w peaks just behind the rotor and begins to decay as the initial momentum deficit recovers. Eddy viscosity displays a convex increase in response to the decreasing rate of strain coupled with increasing turbulent stresses.

From 1≲x/D≲3, the momentum deficit recovers and strain transfers to turbulence with maximum shear stresses present near x/D≈3. Linear regression error is low here as well with average fit errors near 2 % of νT,w*. S13|w diminishes from continued wake recovery while Reynolds stresses continue to develop. However, after u′w′‾|w peaks near x/D=3, wake eddy viscosity is driven by the rate of momentum recovery relative to turbulent dissipation.

Beyond, the remaining velocity deficit is recovered and turbulence follows the energy cascade towards small scales. Since S13|w and u′w′‾|w are decreasing, a shift from concave to convex curvature is visible in Fig. . As the wake dissipates, the quality of the linear fit decreases as shown by increasing regression error magnitudes in Fig.  up to 10 % of νT,w*. The rate of strain is small, S13,w≈1×10-4, but remains as long as turbulent fluctuations are present in the wake. Eddy viscosity reaches its maximum here as the mechanisms for continued growth have deteriorated and νT,w is now driven by exclusively Reynolds stress decay. At large downstream distances, x/D≳20, the wake flow has dissipated and both S13|w and u′w′‾|w are near zero. As the wake has returned to the background flow, performing a linear regression on wake flow components produces erroneous values. This is not the case for the background flow which is treated separately.

Based on these observations, we propose the streamwise behavior of eddy viscosity in the wake may be modeled as a Rayleigh function: 11νT,w(x)=Axσ2e-x2/2σ2, where A parameterizes amplitude and σ is the scale parameter. A is determined by performing a scale analysis on the eddy viscosity hypothesis in which each component of Eq. () is written in terms of their respective units. By selecting a velocity scale Us, a length scale ls, and noting ∂W/∂x≪1, we can write 12Us2∼2νT,wUsls. Rearranging to isolate eddy viscosity yields 13lsUs2∼νT,w. The wake velocity scale is selected as Us∼UB1-CT following , where CT is the turbine thrust coefficient and UB is the mean inflow velocity at hub height. The length scale is selected as ls∼R, where R is the rotor radius. Note that radius is selected rather than diameter as both the rate of strain tensor and shear stresses are symmetric about the wake center. Additionally, the chosen velocity scale is derived from 1 D momentum theory to estimate the mean velocity in the far wake. Substituting the velocity and length scales into Eq. () yields an expression for the eddy viscosity magnitude: 14νT,w∼RUB1-CT2, where CT is the turbine thrust coefficient, UB is the inflow velocity at hub height, and R is the rotor radius. Radius is selected rather than diameter as both the rate of strain tensor and shear stresses are symmetric about the wake center. The scale parameter σ is obtained by substituting A and fitting Eq. () to the eddy viscosity curves for aligned turbines. The parameter σ represents a characteristic distance of where the wakes achieve a fully developed far-wake state. Substituting for A and σ into Eq. () produces 15νT,w(x)=A0.01+xσ2e-x2/2σ2, where A=RUB1-CT/2 and σ=5.5. A constant offset of 0.01 is added to prevent νT,w=0 immediately behind the rotor and far downstream as this would imply wake diffusion is absent. This scaling nondimensionalizes eddy viscosity from each data set and demonstrates agreement across the range of turbine sizes in Fig. .

The impact of nacelle misalignment on eddy viscosity is presented in Fig. . Despite the presence of the counter-rotating vortex pair, the streamwise evolution of eddy viscosity is consistent with earlier trends. However, introducing wake deflection reduces the peak viscosity magnitude as the momentum deficit from a misaligned turbine is lower than that of a turbine operating in nominal conditions. Additionally, the formation of the counter-rotating vortex pair downstream of a misaligned turbine serves to deform and deflect the wake accelerating its recovery and lowering νT(x). This is particularly the case for 20∘ tilt with the 15 MW reference turbine since the large deflection angle and turbine aspect ratio combine to drive the wake into the ground where it experiences rapid dissipation.

In the proposed formulation, nacelle misalignment is accounted for by varying the thrust coefficient. Yet this parameter alone is insufficient to capture the dynamics of the curled wake as evidenced by the departure of the eddy viscosity curves for yawed and tilted turbines from the unmodified case. While the normal velocity imparted by the counter-rotating vortex pair increases ∂w/∂x, this term remains at least 2 orders of magnitude below ∂u/∂z and as such has a negligible effect on eddy viscosity. The primary drivers are variations in Reynolds shear stress and surface interactions with the ground resulting from wake deflection. Under yaw misalignment, the wake experiences asymmetric growth as momentum is entrained into the wake center by the counter-rotating vortex pair . Asymmetric expansion forces the lower portion of the wake close to the ground where it experiences additional strain leading to heightened Reynolds shear stresses. Ground interactions are inherent to tilt misalignment as this approach directs the wake either up into the boundary layer where it is advected by the mean flow or into the ground . The extent of additional strain from surface interactions depends on the wake expansion rate, nacelle deflection angle, height of the bottom tip above the ground, and local surface characteristics. Describing this interaction is beyond the scope of the present work, although such a study is well warranted.

The proposed model Eq. () is incorporated within the FLORIS curled wake formulation to assess the effectiveness of including a high-fidelity eddy viscosity representation in wake modeling applications. For the experimental comparison, the boundary layer height in FLORIS was reduced to match the wind tunnel layer height described by . Streamwise flow fields were computed for each turbine type, misalignment angle, and inflow velocity. Contours of streamwise velocity in y–z planes of each case are presented at x/d=3 in Fig.  (yaw) and Fig.  (tilt). Quantitative comparisons are computed as error between flow field estimates and the corresponding experimental or simulation data with 16ϵ(x)=100%⋅ℓ2Ux-Ux,Fℓ2Ux, where ℓ2 is the Euclidean norm, Ux is the measured or simulated streamwise flow field at a given downstream location x, and Ux,F is the flow field estimate at x. Error using the proposed model is presented for each case in Fig. .

Maximum error occurs immediately downstream of the turbine as FLORIS is not designed to represent near-wake phenomena. Non-zero error is expected from comparing a wake modeling utility with experimental or LES data. Additionally, because the curled wake model uses a mirror condition at the ground to satisfy no-slip, estimated flow fields have a steeper boundary layer profile than the reference data. Aligned turbine cases produce the greatest errors since the proposed model overpredicts near-wake diffusion. Similarly, low misalignment angles are similar to typical turbine operation and the proposed model underestimates the velocity deficit for these cases as well. Overall, flow field error decreases with distance behind the turbine and reaches a minimum by x/D=5. However, beyond x/D=15 error increases since the proposed model limits turbulent stress formation and thus wake diffusion. Error increases the fastest under positive tilt deflection since wakes in this scenario are deflected into the ground.

Comparisons between the proposed model and existing formulation are considered to evaluate the impact of increasing eddy viscosity fidelity in the curled wake model. Contours of streamwise velocity in y–z planes are presented in Figs. –. Flow field error is computed for the current formulation following Eq. () and shown in Fig. . Relative error between flow field estimates is presented in Fig.  and calculated with 17Δϵ(x)=100%⋅ℓ2Ux-Ux,νT=C-ℓ2Ux-Ux,νT=f(x)ℓ2Ux-Ux,νT=C, where ℓ2 is the Euclidean norm, Ux is the measured or simulated streamwise flow field at a given downstream location x, Ux,νT=C is the flow field estimate computed with the existing formulation, and Ux,νT=f(x) is the flow field estimate computed with the proposed model.

Error peaks immediately downstream of the turbine with a secondary peak visible near x/D=2 for the 1.5 MW cases. As noted previously, such errors are expected as FLORIS is not designed to model near-wake phenomena. The presence of a secondary peak for the 1.5 MW turbine implies the assigned eddy viscosity value and thus near-wake diffusion is too high. There is not a clear trend between nacelle misalignment and error with the current approach. Because the existing formulation maintains a constant eddy viscosity, error is relative to how well the prescribed value represents wake diffusion for a given turbine. For the 15 MW cases, the aligned turbine produces the greatest error, while for the 1.5 MW cases, positive tilt deflection results in greater error. Flow field error decreases with distance behind the turbine for all cases except positive tilt deflection, which results in increased wake diffusion by directing the wake into the ground.

Qualitatively, the proposed model captures the wake shape and velocity deficit as shown in Figs. –. However, both formulations overpredict the far-wake velocity deficit for the 15 MW turbine with a lingering momentum deficit visible in Fig. . Despite its large velocity deficit, the 15 MW turbine has low ground clearance such that the wake experiences shear from the ground and dissipates within a relatively short distance. Consequently, the total error for the 15 MW cases is large compared to the relative error between eddy viscosity formulations. Further downstream, the impact of the curled wake mirror condition is apparent with a region of low velocity extending up from the ground into the wake flow. Since the boundary layer profile is assigned by the curled wake model, this region is present in all flow field estimates.

The proposed model outperforms the existing formulation for the majority of cases. Although relative error is inconsistent in the near wake, our model reduces net error for x/D≤15. Because the proposed model limits Reynolds stress production, and thus wake diffusion, it is less accurate far downstream of the turbine due to overpredicting the wake velocity deficit. While nacelle misalignment is not accounted for in either formulation outside of the thrust coefficient, the proposed model produces less error for negative tilt misalignment at all downstream locations. Under these circumstances, the wake is deflected into the boundary layer and advected downstream rather than deformed. As such, it maintains a coherent structure and lingering velocity deficit which the proposed model reproduces. The improvement in flow field prediction for these cases is attributed to better representation of Reynolds stress formation and is not assumed to convey additional fidelity in modeling the impact of the counter-rotating vortex pair on wake recovery. The notable exception is for positive tilt angles where the wake is directed into the ground. As discussed previously, the proposed model overpredicts the wake velocity deficit leading to comparatively large errors.