The numerical grid is a Cartesian domain with a refined region around the horizontal center at which the turbines are placed. The grid topology and boundary conditions are the same as specified in ; however, the domain dimensions are case-specific. For the single-wind-turbine case, a domain with dimensions 1024D×1005D×25D is employed for the streamwise, lateral, and vertical directions, respectively. The inner refined domain is employed to resolve the wind turbine wake(s). It has dimensions 24D×5D×3D, is placed at 4≤x/D≤20 and 2.5≤y/D≤2,5, and contains a mesh with uniform horizontal spacing of D/8. Vertically, the cells grow with height up to D/8 in the rotor area and D/6 at the end of the refined domain using a first cell height of D/200. The cells are distributed as 288×128×64, leading to a total number of 2.4 million cells. The cell spacing of D/8 is sufficient to resolve the wind farm flow under neutral conditions . A finer grid spacing may be necessary for stable-inflow cases depending on the user's quantity of interest. For the present study, we use the same grid spacing of D/8 for all stability cases for simplicity. Our goal is to evaluate the RANS-LUT surrogate model against RANS-AD wind farm simulations using the same grid spacing. If one would like to apply a finer spacing, then the refined single-wake database would also improve the RANS-LUT model. A grid refinement study of RANS-AD single-wake simulations is performed in Appendix . The numerical domains of the wind turbine row and square wind farm are larger in the horizontal extent, and the corresponding grid dimensions are listed in Table .

The inflow conditions represent an atmospheric surface layer including atmospheric stability following MOST . We employ a two-equation k–ε turbulence model that is in balance with MOST , which is important for isolating the turbine and wind farm flow effects. The turbulence model has the following form: 2νT=CμfP(σ/σ̃,CR,CB)k2ε,3DkDt=∂∂xjν+νTσk∂k∂xj+P+B-ε+Sk,4DεDt=∂∂xjν+νTσε∂ε∂xj+(Cε,1P-Cε,2ε+Cε,3B)εk, with xj={x,y,z} as the Cartesian coordinates in streamwise, lateral, and vertical directions, respectively. The turbulence model solves equations for the turbulent kinetic energy, k, and its dissipation, ε, from which the eddy viscosity, νT, is calculated. In addition, the Boussinesq hypothesis is applied to compute the Reynolds stresses. The source terms P and B represent the production of turbulence due to shear and buoyancy, respectively; the latter is negative for stable conditions. The effect of the non-neutral conditions is mainly modeled by the turbulent buoyancy source term since a momentum buoyancy source and a temperature equation are not employed. The MOST profiles are in balance with the turbulence model using an additional source term Sk and a height-dependent Cε,3 . They are derived by substituting the MOST similarity functions of the normalized shear and potential temperature gradient into the k-ε turbulence model; the full expressions of Sk and Cε,3 are provided in . The original turbulence model used a buoyancy source that was a function of the stability and local shear. However, this can lead to a non-physical wake recovery for unstable conditions, where an increase in turbulent buoyancy production could lead to less wake recovery. In the present work, we use a constant turbulent buoyancy source term, B=u*3/(κL), which solves the wake recovery problem for unstable conditions . Here, u* is the friction velocity, and κ=0.4 is the von Kármán constant. However, we find that using a constant buoyancy source term for stable conditions can lead to numerical instabilities. For stable conditions, the constant buoyancy source is negative and reduces k. When the k equation is solved for, the other source terms can vary, while the buoyancy source remains constant. This can lead to negative k values. The numerical problem is solved by multiplying the constant buoyancy source term by the ratio of local eddy viscosity, νT, to inflow eddy viscosity, νT,MOST, for stable conditions only: 5B=u*3κLνTνT,MOST=u*3κLνTu*κzϕm(ζ), with ϕm as the MOST function of normalized wind shear. The idea of using νT/νT,MOST in B is based on the fact that both the original buoyancy production and the turbulent shear production also scale with νT. The modification does not change a single wake under stable conditions significantly. The eddy viscosity is limited with a near-wake-length-scale limiter, fP , extended to non-neutral conditions and later revised for unstable conditions . The fP function depends on the local shear, σ=k/ε(∂Uj/∂xj)2, normalized by the inflow shear, σ̃=1/Cμϕm/ϕε, and two model constants, CR=4.5 and CB=5.0. The other constants in the turbulence model are set as {Cμ,σk,σε,Cε,1,Cε,2}={0.03,1.0,1.3,1.21,1.92}. Finally, the molecular viscosity, ν, is set as 1.78×10-5m2s-1 but has no influence on the solution since νT≫ν .

The non-dimensional inflow is defined by the TI based on the turbulent kinetic energy, Iref=2/3k/Uref, and the stability parameter, ζref, with Uref as the freestream wind speed. Iref and ζref are used to set the roughness length, z0, and the friction velocity, u*: 6u*Uref=f(Iref,ζref)=Iref3/2Cμ14Φε(ζref)-14Φm(ζref)14,z0zref=f(Iref,ζref)=exp⁡κUref/u*+Ψm(ζref)-1-1, with Φm, Φε, and Ψm as MOST functions defined in . Even though a turbulence model is employed that is analytically in balance with MOST, numerical deviations can occur. Therefore, a 1D precursor is used to simulate each MOST profile using the same vertical grid as used for the 3D successor simulations. The friction velocity from Eq. () is rescaled to get the desired wind speed at the reference height, Uref, using Reynolds number similarity. These scaling factors, computed as Uref/U1D precursor, are 0.9901, 0.9959, and 0.9962, for the stable, neutral, and unstable cases, respectively. The scaling factors indicate the numerical error in the streamwise velocity at the reference height. The scaled precursor inflow profiles are compared with the MOST analytic solutions in Fig. . While the precursor profiles of wind speed and eddy viscosity compare well to the analytic solutions, the precursor profile of TI has a small but visible deviation from MOST. This stems from numerical errors in the k profile leading to TI values at hub height equal to 0.0493, 0.0592, and 0.0995, for the stable, neutral, and unstable cases, respectively.

The velocity deficit normalized by the inflow velocity and wake-added TI are independent of the Reynolds number for fixed values of the inflow TI and stability conditions . Hence, the inflow wind speed, Uref, and turbine size are not relevant parameters. The prior holds if the operational parameters as CT, CP, and λ are set correctly. The latter applies for a constant ratio of turbine hub height to rotor diameter, zH/D, which also defines the ground clearance. The Reynolds number similarity reduces the number of independent parameters to a total of eight: two inflow parameters, Iref and ζref, and six turbine-related parameters, CT, CP, λ, zH/D, rotor tilt angle θtilt, and yaw misalignment angle θyaw. Here, we use a reference height equal to the turbine hub height, zref=zH. However, it is not necessary to consider all possible combinations of the aerodynamic coefficients and tip speed ratio since they are related. One could use the general wind turbine model of Eq. () to replace CP(U) and λ(U) with CP(CT)=CP,r(CT/CT,r)(3/3.2) and λ(CT)=λr(CT/CT,r)(1/3.2) and simulate a range of CT values up to CT,r. The number of dimensions is still eight, but the variation in CT,r, CP,r, and λr between existing utility-scale turbines is relatively small . For the more simple AD model using a normalized axi-symmetric fixed thrust force distribution , without tangential forces, and zero tilt and yaw misalignment angles, the number of turbine parameters reduces to two, namely, CT and zH/D. For such a simplified setup, it is practical to create a general single-wake RANS database that could be used for any wind turbine model. However, in this work we employ a more realistic AD model including tangential forces and analytic force distributions, based on a generic version of the NREL-5 MW reference turbine, as discussed in Sect. .

The RANS-AD single-wake database of the generic NREL-5 MW reference turbine is made by a parametric study of Iref, ζref, and CT. We use Iref={0.05,0.06,0.08,0.1,0.15,0.2,0.3}, ζref={-0.5,0,0.5}, and CT=0 to CT=0.8 with an interval of 0.1. The corresponding CP and λ values are obtained from their respective relationship with the wind speed and using CT(U). Simulating stable conditions for a high TI inflow leads to unrealistic roughness length values obtained from Eq. (). This is overcome by replacing the stable single-wake cases for Iref≥0.1 in the database with the neutral single-wake cases, which is justified by the fact that a high Iref often correlates with neutral conditions.

For every fixed TI and stability, the variation in CT is simulated consecutively by first running a zero thrust coefficient simulation until convergence, followed by adjusting the aerodynamic coefficients from low to high for the other CT cases while maintaining the inflow wind speed at hub height at 1.0 ms-1. The results are not affected due to Reynolds number similarity, but the total number of required iterations is an order of magnitude less compared to running all simulations separately. In total, about 500 CPU hours are used to simulate 153 single-wake cases. Note that the stable single-wake cases for Iref≥0.1 are not simulated, leading to 189-36=153 cases.

The single-turbine deficit and wake-added TI model by is built by normalizing the streamwise velocity deficit and added TI fields with the background flow, such that for a single RANS simulation 7ΔŨ(x′)=U(x′)-Ub(z′)Ub(z′),ΔĨ(x′)=2/3k(x′)-2/3kb(z′)Ub(z′), with normalized deficit ΔŨ and added TI ΔĨ, computed from streamwise velocity U and turbulent kinetic energy k. The background flow is denoted by ⋅b and corresponds to a simulation with CT=0. All variables are given in the turbine hub coordinate system x′=(x-xH)/D, with turbine hub position xH in the RANS frame of reference and x=(x,y,z).

As illustrated in Fig. a and b, the RANS-LUT wake model employed here is built from a RANS-AD single-wake database DL, which is a discrete collection of numerical solutions sampled across a range of thrust coefficients, inflow turbulence intensities, and stability parameters. To reconstruct the continuous velocity deficit and turbulence intensity fields from these discrete samples, a multi-dimensional interpolation operator I is employed. The final single-wake model takes the functional form 8ΔŨ,ΔĨ=I(x′,CT,Iref,ζref|DL), where I maps the discrete entries in DL to a continuous domain. Within PyWake this is done using the xarray package for multi-dimensional data handling. However, PyWake's native 1D grid interpolator is employed by collapsing all dimensions into a 1D array, which is faster than the multi-dimensional xarray interpolator. Furthermore, we use linear interpolation, due to its robustness and speed. To optimize memory usage, DL retains only the vertical CFD layers spanning the rotor-swept area – defined by the range between the minimum and maximum tip heights across all turbines within a farm. For wind farms with heterogeneous turbine types, DL is expanded to include specific entries for each turbine model. Figure c and d show how the single-wake deficit database is split into up- and downstream regions to build PyWake blockage and wake deficit models. This results in some effects related to blockage, as for example the speed-up in wind speed around the turbine wake, becoming part of the wake deficit model. As depicted in Fig. f, this split is not done for the wake-added TI model.

To move from the non-dimensional quantities provided by the single-wake model in Eq. () to a physical representation of the flow field, they must be scaled by some reference inflow values, which are referred to as effective wind speed, U^, and TI, I^, in PyWake. As turbines inside a farm respond to the local waked inflow, we compute quantities at each rotor location. In general the effective wind speed at the ith turbine in PyWake in the wind farm coordinate system is given by 9 10〈U^jΔŨj〉Ai=∑nwnU^jΔŨ(xi,n-xH,j,CT(U^j),I^j,ζref), where wake deficits ΔŨW from upstream turbines W and blockage deficits ΔŨB from downstream turbines B are superimposed (self-induction is excluded) and rotor-averaged 〈⋅〉Ai separately and deducted from the wind farm background flow, which is evaluated at the rotor center Ub(xH,i). This signifies that background flow variations over the rotor area (shear, veer) do not impact the effective wind speed and consequently neither turbine power nor thrust. Note that this differs from the way the RANS-AD reacts to the flow, which responds to the local variation over the AD instead, i.e Ub(xi)-U^jΔŨj(xi,n-xH,j). Rotor averaging, executed preceding superposition, is performed by weighing the scaled deficits U^ΔŨ over the rotor area Ai with weights wn assigned to discrete sampling points, representing a numerical integration across the rotor disk where ∑wn=1. Here, the normalized deficits are scaled by the turbine local wind speed; however, in PyWake it is generally possible to switch to freestream scaling, meaning that U^j=Ub(xH,j) in the above. Furthermore, superposition and rotor-averaging methods are allowed to differ between blockage and wake deficits in PyWake, and, whilst not used here, it is possible to introduce multi-dimensional thrust curves. Generally the background flow in PyWake is allowed to vary freely in space, yet here we ensure it follows the RANS inflow conditions that only include vertical shear following MOST: 11Ub(z)=Urefln⁡(z/z0)-Ψm(ζ)ln⁡(zref/z0)-Ψm(ζref), with Ψm as the integrated normalized shear from MOST, as defined in . The effective TI is only computed using upstream turbines: 12 where ⊕ denotes the superposition of background with added TI. As the single-wake database underlying the RANS-LUT model is generated through a parametric variation in inflow TI, our approach inherently assumes that turbines respond identically to ambient inflow and wake-added TI.

Due to the inclusion of up- and downstream effects, an iterative approach needs to be employed to obtain the effective wind speed and TI at all turbines. As shown in Fig. m, here we use PyWake's All2AllIterative solver. Once the operating conditions of each turbine have been determined, they are fixed, and the final wind farm flow field at a point xi is determined by aggregating contributions from all turbines N–omitting all other dimensions except x for ΔŨ,ΔĨ: 1314

As deficits and power production scale with the effective wind speed, the accuracy of our RANS-LUT model predictions strongly depends on our choice of superposition models. Whilst Eq. () shows an explicit dependency on the deficit superposition models, their is also an implicit connection to the TI superposition, as the normalized deficit ΔU is a function of the effective TI I^. While found that linear superposition of velocity deficits and wake-added TI yielded the most accurate results, identified the superposition of the maximum added TI – I^i=Ib(xH,i)+max⁡j〈ΔĨj〉 – as optimal; however they also utilized a different definition of wake-added TI, namely ΔĨ=2/3(k-kb)/Ub. As we follow the one by in Eq. (), we similarly linearly superimpose wake-added TI contributions as depicted in Fig. l such that 15I^i=Ib(xH,i)+∑j∈W〈ΔĨj〉Ai. Since blockage deficits are limited, they are also linearly superimposed – a valid approach as demonstrated by .

As mentioned above, similarly superimposed wake deficits linearly and found them to agree with RANS-AD simulations in terms of the wind farm velocity and TI fields. However, as shown in detail in Appendix , we find that this is due to velocity superposition errors canceling those from using rotor center – 〈ΔŨj〉=ΔŨ(xH,j) – instead of rotor-averaged values. As reported by , linear superposition overestimates wake deficits in deep arrays, and they consequently devised a momentum-deficit-conserving superposition method instead, which effectively acts as a weighted sum, such that the combined wind farm deficit is given by 16ΔU=∑iuciUcΔui, where lower- and uppercase letters refer to the single-turbine and combined wind farm velocities, respectively. The weights are given by the ratio between single-turbine uc and wind farm Uc wake convection velocities, which are determined by performing an integral over the plane perpendicular to the mean flow, A⟂: 17Uc=∫A⟂(Ub-ΔU)ΔU∫A⟂ΔU. For a single turbine just replace with lowercase letters. Equations () and () are coupled, thus requiring an iterative approach. Hence, to employ the weighted sum, cross-plane integrals need to be performed for single-turbine wakes and the combined wind farm flow at every iteration until convergence. If the cross-plane integrals are performed numerically, the computational cost would be prohibitive. PyWake circumvents this by employing analytical solutions based on assuming Gaussian-shaped wake deficits, defined here as 18Δu=δexp⁡-r22σ2, with centerline deficit δ, cross-wind distance r, and wake expansion σ. already showed that for a single Gaussian wake there is an analytical solution to Eq. (), as it has a finite integral. Here, we show that there is also an analytical solution to the wind farm flow; substituting Eq. () into Eq. () and assuming that Ub does not vary over A⟂, the update of Uc(n+1) can be written entirely in terms of single-wake parameters that can be precomputed: 19Uc(n+1)=Ub-1Uc(n)∫A⟂∑iuciΔui2∫A⟂∑iuciΔui=Ub-1Uc(n)×∑iuciδiπσi2+2∑i∑j<iuciucjδiδjIij∑iuciδi2πσi2Iij=2πσi2σj2σi2+σj2exp⁡-dij22(σi2+σj2)dij=(yH,i-yH,j)2+(zH,i-zH,j)2, where Iij represents the interaction between two wakes, meaning that if the combined deficit downstream of the ith turbine is to be computed, the interaction of its single wake with all preceding wakes (i<j) needs to be determined. As this nested sum can become expensive, small deficits less than 1 % of Ub are skipped and added linearly (uci=Uc(n)). As the superposition model assumes that the pressure has fully recovered (∂p/∂x≈0), it does not hold in the near-wake, causing the algorithm to become unstable, as locally the individual wake convection velocity might exceed that of the combined wakes. Referring to Eq. (), if uc>Uc(n), Uc→0, as the numerator grows faster than the denominator, and Uc continues to drop. Since uc>Uc(n) implies that individual wakes are convected faster than the background flow and that momentum deficit would be created during superposition (see Eq. ), we enforce uc≤Uc(n). This means that the weights – uc/Uc(n) – in the superposition method do not exceed 1. The iterations are initialized by setting Uc=max⁡i(uci), and stopped when max⁡|(Uc(n+1)-Uc(n))/Uc(n)|<10-3, or five iterations have been completed.

Due to its accuracy, speed, and numerical stability, we adopt the analytical approach in our RANS-LUT model by fitting Gaussian profiles to the RANS-AD database. The fitting is performed over the downstream plane at hub height, z′=0 and 2≤x′≤100, with an additional fit at x′=0 where the magnitude is limited to the axial induction from 1D momentum theory; see Fig. e. Subsequently, the normalized wake expansion, σ/D, and centerline deficit, δ̃, are provided by additional LUT-based surrogate models: 20σ/D,δ̃=Lx′,CT,Iref,ζref|GL, where δ from the RANS-AD simulations is normalized by the hub height wind speed. It should be noted that the assumption of the Gaussian velocity deficit is only applied to determine the superposition weights that are employed to superpose the velocity deficit shapes of the RANS-LUT model. The performance of the weighted superposition method and effect of enforcing the uc≤Uc(n) limit is further discussed in Sect. .

Finally, in this work we use a rotor-averaging model based on a Gaussian quadrature method with eight points, which is sufficient to obtain a rotor-average wind speed with an error of 0.65 % .