A simple steady-state inflow model of the neutral and stable atmospheric boundary layer applied to wind turbine wake simulations

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P.; Kelly, M.; Baungaard, M.; Dicholkar, A. Published in: Wind Energy Science Link to article, DOI: 10.5194/wes-9-1985-2024 Publication date: 2024 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): van der Laan, M. P., Kelly, M., Baungaard, M., & Dicholkar, A. (2024). A simple steady-state inflow model of the neutral and stable atmospheric boundary layer applied to wind turbine wake simulations. Wind Energy Science, 9, 1985-2000. https://doi.org/10.5194/wes-9-1985-2024 Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024 © Author(s) 2024. This work is distributed under the Creative Commons Attribution 4.0 License. A simple steady-state inflow model of the neutral and stable atmospheric boundary layer applied to wind turbine wake simulations Maarten Paul van der Laan, Mark Kelly, Mads Baungaard, Antariksh Dicholkar, and Emily Louise Hodgson DTU Wind and Energy Systems, Technical University of Denmark, Risø Campus, Frederiksborgvej 399, 4000 Roskilde, Denmark Correspondence: Maarten Paul van der Laan ([email protected]) Received: 9 March 2024 – Discussion started: 19 March 2024 Revised: 8 July 2024 – Accepted: 5 September 2024 – Published: 24 October 2024 Abstract. Wind turbines are increasing in size and operate more frequently above the atmospheric surface layer, which requires improved inflow models for numerical simulations of turbine interaction. In this work, a steadystate Reynolds-averaged Navier–Stokes (RANS) model of the neutral and stable atmospheric boundary layer (ABL) is introduced. The model incorporates buoyancy in the turbulence closure equations using a prescribed Brunt–Väisälä frequency, does not require a global turbulence length-scale limiter, and is only dependent on two non-dimensional numbers. Assuming a constant temperature gradient over the entire ABL, although a strong assumption, leads to a simple and well-behaved inflow model. RANS wake simulations are performed for shallow and tall ABLs, and the results show good agreement with large-eddy simulations in terms of velocity deficit from a single wind turbine. However, the proposed RANS model underpredicts the magnitude of the velocity deficit of a wind turbine row for the shallow ABL case. In addition, RANS ABL models can suffer from numerical problems when they are applied as a shallow-ABL inflow model to large wind farms due to the low-eddy-viscosity layer above the ABL. The proposed RANS model inherits this issue, and further research is required to solve it. 1 Introduction Wind turbine and farm interaction can lead to energy losses and increased turbine loads, mainly due to wakes from upstream turbines and farms but also because of blockage effects (Porté-Agel et al., 2020). The magnitude of these effects is strongly influenced by the atmospheric conditions such as ambient turbulence intensity (Nilsson et al., 2015), buoyancy, and boundary layer depth (Hansen et al., 2012). Traditionally, models for simulating wake losses assume simple atmospheric conditions that only represent the first 10 % of the atmospheric boundary layer (ABL), known as the atmospheric surface layer (ASL). Examples are wind speed profiles based on a power law or on the Monin–Obukhov similarity theory (Monin and Obukhov, 1954). However, wind turbines are increasing in size and operate more frequently above the ASL, especially for shallow ABLs. Hence, there is a need for improved inflow models that can capture the effect of the ABL on the wind farm flow. Wind farm flow models based on computational fluid dynamics (CFD) can be employed to simulate wake losses (Porté-Agel et al., 2020). High-fidelity turbulenceresolving and transient CFD methods as a large-eddy simulation (LES) is a popular method in academia because it can simulate the complex interaction between the ABL and a wind farm; however, it is too expensive to simulate all wind direction and wind speed flow cases that are necessary to calculate wake losses in terms of annual energy production (AEP). For the latter, the industry employs engineering wake models because of their computational speed. However, such models require calibration and are often not general enough to perform well for a wide range of atmospheric conditions and wind farm layouts due to the need to assume a single wake shape and wake superposition method. Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V. 1986 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs Reynolds-averaged Navier–Stokes (RANS) is a mediumfidelity steady-state CFD method that is several orders of magnitude faster than LES and does not require the engineering wake model assumptions. An idealized RANS setup of a large wind farm (16 by 16 turbines with 8Dinterspacing) can simulate AEP wake losses in roughly a day using 624 CPUs (van der Laan et al., 2022). However, RANS requires a turbulence model, which is not trivial, but reasonable results in terms of velocity and power deficits can be achieved (Politis et al., 2012; van der Laan et al., 2015c, b; Baungaard et al., 2022b). In addition, atmospheric inflow modeling in RANS is challenging because the inflow needs to be a solution of the RANS model, and numerical convergence is not guaranteed when ABL models beyond the neutral ASL are employed as inflow models to wind farm simulations (van der Laan et al., 2023b). Transient ABL models can be employed for inflow to complex terrain and wind farm simulations via unsteady RANS (URANS; Koblitz et al., 2015; Castro et al., 2015). Such model setups typically include buoyancy terms in the momentum and turbulence transport equations that are linked to an active-temperature equation. However, URANS has significant disadvantages, starting with the need to solve in time and the inflow developing downstream; the former requires computational time an order of magnitude larger than RANS, and the latter results in nontrivial complications in model setup to obtain the desired inflow at a given wind farm location. Instead of using URANS, some authors use a RANS setup by including an inflow that is not a steady-state solution of the employed RANS model, by including, e.g., an active-temperature equation (Bleeg et al., 2015; Quick et al., 2024). In that case the same issue of nonstationary and horizontally inhomogeneous inflow is encountered, which introduces the distance between the upwind domain edge and wind farm as a parameter upon which the results depend. Steady-state ABL inflow models generally rely on the global length-scale limiter of Apsley and Castro (1997), where a maximum turbulence length scale is chosen that indirectly determines the ABL height. Neutral or stable atmospheric conditions can be represented by setting relatively large or small values of the maximum turbulence length scale to obtain tall or shallow ABLs, respectively. However, unstable conditions, i.e., convective ABLs (CBLs), cannot be modeled without additional model components; it is not trivial to obtain realistic results in the surface layer without getting nonphysical ABL heights (van der Laan et al., 2020). One can argue that CBLs are inherently unsteady, which challenges steady-state model prescription; therefore, the present article focuses on neutral and stable atmospheric conditions. When an inflow model based on the global turbulence length-scale limiter of Apsley and Castro (1997) is applied to a 3D RANS simulation (Koblitz et al., 2015; Arroyo et al., 2014; van der Laan et al., 2015a; Avila et al., 2017; Ivanell et al., 2018; Freitas et al., 2024), as for a wind farm or complex terrain, then all turbulence length scales will also be limited, which can result in nonphysical solutions. In previous work (van der Laan et al., 2023b), an alternative ABL inflow model was proposed, where the global turbulence length-scale limiter of Apsley and Castro (1997) was replaced by a turbulent buoyant-destruction term, using a prescribed potential temperature profile to represent conventionally neutral ABLs. This model works well for tall ABLs but can have problems for shallow ABLs (as shown later in Appendix A). In the present work, a new ABL inflow model is proposed that does not require a global length-scale limiter and that is further suited to model stable and shallow ABLs. The model employs a turbulent buoyancy source that depends on a prescribed Brunt–Väisälä frequency by assuming a constant temperature gradient over the entire ABL. While this is a strong assumption, the resulting model is simple and well-behaved. In addition, the proposed model can simulate the effect of neutral and stable atmospheric conditions on a wind turbine wake in RANS. The two existing and the proposed RANS inflow models are discussed in detail in Sect. 2. The three RANS inflow models are applied to single-wake simulations following a methodology described in Sect. 3, and the results are compared with the results of LES in Sect. 4, for both a shallow and a tall ABL. The shallow ABL case is also applied to a wind turbine row. 2 RANS inflow models of the ABL RANS inflow models of the ABL are based on a numerical solution of the 1D momentum equations for streamwise and lateral velocity components, Uand V, respectively, including a prescribed pressure gradient in the form of a geostrophic wind speed, G=qU2 G+V2 G, and Coriolis forces. Here, UG and VGare the streamwise and lateral components of the geostrophic wind vector. The momentum equations only depend on a single Cartesian coordinate, namely, the vertical coordinate, z: fc(V−VG)+d dzνT dU dz=0, −fc(U−UG)+d dzνT dV dz=0,(1) with fcas the Coriolis parameter. In addition, we have employed the Boussinesq hypothesis with νTas the eddy viscosity for which a turbulence model is required. In the present work, we use the k–ε–fPeddy viscosity model (van der Laan et al., 2015c) that employs a transport equation for both the turbulent kinetic energy, k, and its dissipation, ε: Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs 1987 νT=CµfP k2 ε, d dzνT σk dk dz+P−ε+B+Sk,amb =0, d dzνT σε dε dz+C∗ ε,1P−Cε,2ε+Cε,3Bε k+Sε,amb =0,(2) with fPas a scalar function that acts as a local turbulence length-scale limiter in regions with high-velocity gradients to assure realizable Reynolds stresses, which is mainly applicable to a wind turbine (near) wake. However, fPis not of importance to an inflow model but is applied to be consistent with a 3D RANS simulation of a wind turbine wake. Furthermore, Pand Bare the turbulent production due to shear and buoyancy, respectively: P=νT"dU dz2 +dV dz2#, B=g θ0 θ0w0=−νT σθ g θ0 d2 dz,(3) with g=9.81 m s−2as the magnitude of the gravitational acceleration vector and 2as the mean potential temperature, with θ0as the hydrostatic background temperature (here we use the value at the wall boundary), and a simple flux–gradient relationship for the heat flux, θ0w0= −(νT/σθ)d2/dz, is employed. Note that in order to obtain a steady-state solution of the ABL, one cannot employ an active-temperature equation in combination with a nonlinear temperature profile, as such a setup would effectively become an unsteady RANS method due to a forever-growing ABL height. Sk,amb and Sε,amb are additional source terms used to maintain a small ambient value of turbulence for numerical robustness, such that k=kamb and ε=εamb in the absence of any velocity gradients are applicable to the flow above the ABL (van der Laan et al., 2015a): Sk,amb =εamb, Sε,amb =Cε,2 ε2 amb kamb , kamb =3 2G2I2 amb, εamb =C3/4 µ k3/2 amb `amb ,(4) with `amb and Iamb as the ambient turbulence length scale and turbulence intensity (based on k) above the ABL, respectively, and Camb as a model constant (van der Laan et al., 2020). The values of Iamb and Camb are set small enough to not influence the inflow model solution. Furthermore, the definition of `amb differs with the chosen inflow model and is discussed in Sect. 2.1–2.3. In addition, the following turbulence model constants are used: (Cµ,Cε,1,Cε,2,σk,σε,σθ)= (0.03,1.21,1.92,1.0,1.3,0.74), and turbulence model parameters C∗ ε,1and Cε,3are also discussed in Sect. 2.1–2.3. 2.1 RANS–`max: inflow model using the turbulence length-scale limiter of Apsley and Castro (1997) The global turbulence length-scale limiter of Apsley and Castro (1997) can be employed to model a neutral and a stable inflow model without the need for a turbulent buoyancy source term (B=0). The limiter represents a variable C∗ ε,1in the transport equations of ε: C∗ ε,1=Cε,1+Cε,2−Cε,1` `max ,(5) where `≡C3/4 µk3/2/ε is a model-based turbulence length scale. When `exceeds `max then the source terms in the εequation cancel, and this prevents the turbulence length scale from growing larger than the maximum set value, `max. The height of the ABL can be set implicitly using `max. For `max →0 and `max →∞, the analytic ABL solutions of Ekman (1905) (constant νT) and Ellison (1956) (linear νTwith z) are obtained, respectively, which bounds the numerical RANS model, as shown in van der Laan et al. (2020). When the global turbulence length-scale limiter of Apsley and Castro (1997) is applied as an inflow model to a 3D RANS simulation, then all turbulence length scales are limited, and this can lead to a nonphysical recovery of a wake generated by, for example, a wind turbine, a wind farm, or a hill (Koblitz et al., 2015; van der Laan et al., 2015a; Avila et al., 2017). An ad hoc solution has been proposed in previous work (van der Laan et al., 2015a) by switching off the turbulence lengthscale limiter in the wake region using the fPfunction as a wake identifier: C∗ ε,1=f1Cε,1+Cε,2−Cε,1` `max , f1=1 2tanh50fP−0.9+1.(6) Here, f1is a blending function that switches between the global (`max) and local (fP) turbulence length-scale limiters. The impact of this solution on a single wake is further investigated in Sect. 4. The ambient values of kand εare set by Eq. (4), where the ambient turbulence length scale is defined as `amb =Camb`max,(7) with Camb =10−6and Iamb =10−6(van der Laan et al., 2020). We label the inflow model as the RANS–`max model. 2.2 RANS–Θ: prescribed temperature inflow model The RANS–`max can lead to nonphysical wake recovery when it is applied as an inflow model to the wind farm, especially for shallow ABLs. To overcome this issue, an alternative RANS inflow model has recently been developed (van der Laan et al., 2023b), here labeled as the RANS–2model, where the global length-scale limiter of https://doi.org/10.5194/wes-9-1985-2024 Wind Energ. Sci., 9, 1985–2000, 2024 1988 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs Apsley and Castro (1997) has been replaced (C∗ ε,1=Cε,1) by the use of a nonzero turbulence buoyancy from Eq. (3) and an analytic prescribed temperature profile that includes a constant temperature in the surface layer and a constant inversion: d2 dz=1 21+tanhz/zi−1 zT/zid2 dzc ,(8) where ziis the inversion height, d2/dz|cis the inversion strength, and zTcharacterizes the distance over which the temperature gradient changes from 0 to d2/dz|c(we take zT/zi=0.2). The temperature profile can be obtained upon integration, and its final form is described in van der Laan et al. (2023b). The temperature profile remains constant when the model is applied as inflow to a 3D RANS simulation, since 2(z) from Eq. (8) is prescribed instead of solving a temperature equation. Note that the original RANS–2model was employed with a slightly different implementation of the buoyancy compared to Eq. (3), namely, B=−(νT/σθ)(g/2)d2/dz. However, 2≃θ0, since zid2/dz|cθ0for the values of ziand d2/dz|cencountered in the ABL (permitting us to also use the wall temperature for θ0). The ambient turbulence length scale above the ABL is defined as `amb =Cambzi.(9) In addition, we use Iamb =10−5and Camb =10−7. Finally, the Cε,3constant is defined as Cε,3=1+Cε,1−Cε,2,(10) following Sogachev et al. (2012) for `max →∞. The RANS–2model is suited to model a conventionally neutral ABL (CNBL). However, if one selects an inconsistent combination of ziand d2/dz|cthen an unphysical inflow profile (with effectively two ABL heights) can result. This problem is further illustrated in Appendix A for a (too- )shallow ABL. 2.3 RANS–N: a new inflow model based on a constant Brunt–Väisälä frequency We propose to write the buoyant destruction of turbulent kinetic energy from Eq. (3) as B=−νT σθ g θ0 d2 dz=−νT σθ N2;(11) the Brunt–Väisälä frequency is described by N≡sg θ0 d2 dz.(12) The Brunt–Väisälä frequency is a measure of stable stratification, normally applied to the inversion layer of the ABL or “free atmosphere” above. The problems with the RANS–`max and RANS–2models outlined above can be overcome by prescribing a constant gradient of temperature throughout the entire ABL in Eq. (11), giving a constant N→NABL in Eq. (12). The turbulence model constant σθ(turbulent Prandtl number) is set to 1 for simplicity, as it could be absorbed into NABL. The RANS–2model can also be written in the form of Eq. (11) but with a vertically varying temperature gradient and N(z), where as z→ziin the upper ABL d2/dz→(d2/dz)|cand N→Nc. The simple form of Bwith a constant Nalso implies that the heat flux profile is same as the eddy viscosity profile times a constant, θ0w0=−N2(θ0/[gσθ])νT. A constant temperature gradient was also assumed by Chougule et al. (2017) to simulate atmospheric boundary turbulence with a spectral tensor model including the effects of buoyancy. Using a constant Nor constant temperature gradient for the entire ABL is not always realistic, but this model choice results in a simple RANS ABL inflow model, which we label the RANS–Nmodel, that can yield reasonable results of the ABL; this is further discussed in Sect. 4. Furthermore, the RANS–Nmodel does not suffer from the “double” ABL height problem that can occur with the RANS–2model because the RANS–Nmodel does not require an explicit inversion height. The RANS–Nmodel behaves similarly to the RANS–`max model in terms of obtaining an ABL height implicitly using a single parameter; instead of an ABL length scale arising from `max (i.e., zi≃`0.6 max(G/f )0.4as in van der Laan et al., 2020), the depth is determined by the constant NABL. We note that one can also translate NABL to an ABL length scale using G/NABL. The latter defines an ambient turbulence length scale above the ABL: `amb =Camb G NABL ,(13) with Camb =10−7and Iamb =10−5. If NABL =0, then εamb is set to zero. Since the RANS–Nmodel does not use the global length-scale limiter of Apsley and Castro (1997) (C∗ ε,1=Cε,1), the model does not artificially limit the turbulence length scale in a 3D RANS simulation. The remaining constant, Cε,3, is set the same as the RANS–2model (Eq. 10). 2.4 Similarity The RANS ABL models discussed here ultimately depend on four or five dimensional parameters, but their nondimensional numerical solutions can be described by two or three dimensionless numbers (following the Buckingham pi theorem), as summarized in Table 1. The first dimensionless number is the surface Rossby number, Ro0≡G/(|fc|z0), and can be obtained by writing the 1D momentum equations (Eq. 1) in a complex form using W≡(U−UG)+i(V−VG), with i≡√−1, followed by a substitution of the normalized variables, z0≡z/z0,W0≡ Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs 1989 Table 1. Dimensional and non-dimensional input parameters of RANS inflow models. Model Dimensional input Non-dimensional input RANS–`max G,fc,z0,`max Ro0,Ro` RANS–2 G,fc,z0,zi,d2 dzc,θ0Ro0,Rozi, Nf RANS–N G,fc,z0,NABL Ro0, Nf W/G and ν0 T≡νT/(z0G): Ro0 d dz0ν0 T dW0 dz0=iW0.(14) All models that solve the momentum (Eq. 14) follow a Rossby similarity. The other dimensionless numbers are related to the turbulence model (Eq. 2), which can be written in a non-dimensional form using k0=k/G2,ε0=εz0/G3: d dz0ν0 T σk dk0 dz0+P0+B0−ε0=0, d dz0ν0 T σε dε0 dz0+C∗ ε,1P0−Cε,2ε0+Cε,3B0ε0 k0=0,(15) with P0≡Pz0/G3and B0≡Bz0/G3. Here, the small ambient source terms are neglected. The additional dimensionless numbers are obtained from non-dimensionalizing either C∗ ε,1 (Eq. 5) or B0(via Eqs. 3, 8, 11): RANS −`max : B0=0, C∗ ε,1=Cε,1+Cε,2−Cε,1C3/4 µ k03/2 ε0 Ro` Ro0 , RANS −2: B0=−ν0 T σθNf Ro021 2+1 2tanhz0Rozi/Ro0−1 zT/zi, C∗ ε,1=Cε,1, RANS −N: B0=−ν0 T σθNf Ro02 , C∗ ε,1=Cε,1,(16) where Ro`≡G/(|fc|`max) and Rozi≡G/(|fc|zi) are Rossby numbers based on different ABL length scales, namely, `max and zi, respectively. In addition, Nf≡N/|fc| is the Zilitinkevich number using the Brunt–Väisälä frequency from Eq. (12) using a constant gradient of temperature (representing the inversion or the entire ABL for the RANS–2and RANS–Nmodels). Note that one could also replace Nfby a Richardson number, in the form of (Nf/Ro0)2. The similarity of the RANS–`max and RANS–2 models has been shown through numerical experiments in previous work (van der Laan et al., 2020, 2023b). The similarity of the ABL models can be employed to create an ABL library numerically for all possible solutions, which can be used to obtain an ABL profile with a desired turbulence intensity and wind speed at a reference height using Gand NABL (in the case of the RANS–Nmodel) as free parameters, for a given fcand z0. The proposed RANS–N model has one fewer dimensional number compared to the RANS–2model, which reduces the input parameter space.1In addition, all three RANS models can be used to satisfy Reynolds number similarity by keeping their non-dimensional numbers constant. This is an advantage when running wind speed inflow cases consecutively to reduce the total number of required iterations for wind farm AEP simulations (van der Laan et al., 2019, 2022). 3 Numerical methodology The RANS simulations of the inflow and single-turbine wake are carried out with PyWakeEllipSys (DTU Wind and Energy Systems, 2024), which is Python framework for wind farm CFD simulations. The underlying CFD solver is EllipSys, which is an in-house finite volume code initially developed by Michelsen (1992) and Sørensen (1994). The numerical domain and boundary conditions of the 1D inflow precursor and 3D wind turbine simulations are depicted in Fig. 1 and are further discussed in Sect. 3.1 and 3.2. 3.1 Inflow The RANS inflow models are solved numerically with EllipSys1D (van der Laan and Sørensen, 2017). A 1D grid with a height of 100 km, a first-cell height of 0.01 m and 768 cells is employed, as shown in Fig. 1a. A relatively tall domain is employed to be able to simulate all possible ABL solutions, as discussed in van der Laan et al. (2020). A rough wall boundary condition from Sørensen et al. (2007) is employed at the ground and depends on the roughness length z0. At the top, a Neumann condition is applied. Since the 1D RANS equations are stiff, we solve them transiently with a fixed time step of 1t =1/fcuntil a steady-state has been achieved. 3.2 Single wake The RANS inflow models are applied to RANS singlewake simulations, performed with EllipSys3D. The numerical setup follows a very similar approach to that performed in previous work (van der Laan et al., 2015c) and solves the 3D form of Eqs. (1)–(2). We aim to compare the RANS simulations (both inflow and turbine wakes) with the results of two LES models from Hodgson et al. (2023). These LES models employ an actuator disk (AD) model based on airfoil data to represent the forces of the SWT-2.3-93 turbine (proprietary to Siemens Gamesa Renewable Energy), which 1It could appear that the RANS–2model further includes the parameter zT, but this may be eliminated by relating NABL to N(z) and Ncfollowing Kelly et al. (2019); however this is beyond the scope of the current work. https://doi.org/10.5194/wes-9-1985-2024 Wind Energ. Sci., 9, 1985–2000, 2024 1990 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs Figure 1. Numerical grid and boundary conditions of the 1D inflow precursor (a) and 3D wind turbine simulations (b–c). The cyan rectangle marks the refined domain around the turbine, and every eighth cell is shown. has a rated power of 2.3 MW; a rotor diameter, D, of 93 m; and a hub height, zH, of 68.5 m. Our RANS simulations use the same turbine type, but we employ an AD (Réthoré et al., 2014) including the analytic blade force distribution model of Sørensen et al. (2020), which has shown to compare well with an AD based on airfoil data. In order to perform a fair comparison between the LES and RANS models, we have rerun the LES wake simulations using the same AD model as applied in RANS; see Sect. 3.2.1 for more details. The AD model includes effects of rotor rotation and nonuniform inflow as wind shear and wind veer. In addition, we use a 1D momentum controller (Calaf et al., 2010) similar to one of the LES models from Hodgson et al. (2023) based on the same inputs: a tip speed ratio of 7.75, a thrust coefficient of 0.73, and a power coefficient of 0.45. Note the actual values can differ because a 1D momentum controller typically overestimates the freestream wind speed and thrust force, as shown in previous work (van der Laan et al., 2015b). The effective values of the power and thrust coefficients based on the disk-averaged streamwise velocity are set as 1.026 and 1.264, respectively. A Cartesian domain is employed with dimensions 234D× 203D×30Dfor the streamwise (x), lateral (y), and vertical (z) directions, respectively, as depicted in Fig. 1a–b. The large domain extent is used to minimize the effect of numerical blockage. An inner domain around the turbine, located at (x,y,z)=(0,0,zH), is used to resolve the wind turbine wake with a fine uniform spacing of D/32 (cyan rectangle in Fig. 1a–b). The inner domain has the following horizontal dimensions: −4D < x < 30Dand −1.5D < y < 1.5D. Vertically, the cell sizes are growing with zusing a first-cell height of D/200, a maximum cell size of D/32 at z=3D, and a maximum expansion ratio of 1.2. Above z=3D, the cells continue to grow with a similar expansion ratio. The total number of cells is 43.6 million. The effect of coarser grid spacing is shown in Appendix B. An inlet boundary condition is at the inflow boundary (x=−104D) and at the top of domain (z=25D). The bottom boundary is a rough wall boundary condition (Sørensen et al., 2007). The lateral boundaries (y=±101.5D) are periodic because of the presence of wind veer. A Neumann condition is set at the outflow boundary (x=130D). More details of the numerical setup are discussed in van der Laan et al. (2015c) with the exception of the lateral boundaries, which are set to periodic boundary conditions because of the presence of wind veer. 3.2.1 LES The LES results of Hodgson et al. (2023) are used to compare to our RANS models results for both the inflow and singlewake cases. Hodgson et al. (2023) employed two different LES models: WiRE (Albertson and Parlange, 1999; PortéAgel et al., 2000; Wu and Porté-Agel, 2011) and EllipSys3D (the same solver used for the RANS simulations), here labeled as LES–EPFL and LES–DTU, respectively. In order to provide a fair comparison with the RANS models, we have rerun the LES–DTU single-wake cases following the same methodology as Hodgson et al. (2023) but using a finer grid Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs 1991 spacing of D/32 around the AD instead of D/16. In addition, we have extended the refined domain around the AD in the streamwise direction to 30Dfor the SBL inflow case, such that we can compare LES–DTU results of the far wake with RANS. Furthermore, we have extended the precursor simulation by an additional 2 h such that the LES–DTU stable boundary layer (SBL) single-wake simulation can be averaged over 3 h in order to obtain converged statistics in the far wake. Finally, we employ the same AD model as used in the RANS simulations including a 1D momentum controller for both inflow cases. 3.3 Wind turbine row with SBL inflow The SBL inflow case is also applied to a small wind farm consisting of a row of five SWT-2.3-93 turbines (the same turbine used for the single-wake cases) with 5Dspacing. The RANS wind farm domain is similar to the domain used for the single-wake cases (as depicted in Fig. 1). However, a larger inner domain is used with the following horizontal dimensions: −4D < x < 40Dand −3.5D < y < 3.5D, leading to a total number of 94.4 million cells. The wind turbine row subjected to the SBL inflow case is also simulated with the LES–DTU model using the same extended domain as used for the SBL single-wake case, as discussed in Sect. 3.2.1. 4 Results and discussion: a comparison with LES 4.1 Inflow The two existing and proposed RANS inflow models from Sect. 2 are applied to two ABL cases based on LES results from Hodgson et al. (2023), who used two different LES models. The ABL cases represent a CNBL and a stable ABL (SBL) inspired by the LES intercomparison study from Beare et al. (2006). The LES models from Hodgson et al. (2023) are employed with fc=1.185×10−4s−1and z0=0.001 m. The values of the Coriolis parameter correspond to a latitude of 54.3°, and it is based on the location of the Danish offshore wind farm, Rødsand II. While we adopt the value of fcfrom Hodgson et al. (2023), a lower roughness length is used in the RANS models. This is because the RANS models use Cµ=0.03, while the LES models imply a higher effective Cµbased on the turbulent kinetic energy and friction velocity near the wall, as shown in Baungaard et al. (2024). This is compensated for using a lower roughness length of z0=0.0002 m in the RANS–`max and RANS–Nmodels. Note that if a higher Cµwere set in the RANS models, then the other turbulence model constants would need to be adjusted and calibrated, which is not within the scope of the present article. The RANS inflow models use Gand an additional parameter to obtain the turbulence intensity based on k;IH; and wind speed, UH, at the reference height of 68.5 m – namely, `max,z0, and NABL for RANS–`max, RANS–2, and RANS–N, respectively. The LES-derived input parameters and fitted RANS inflow model parameters are listed in Table 2. The RANS–`max and RANS–Nmodels use precalculated libraries of all possible ABL solutions that depend on two non-dimensional numbers (as discussed in Sect. 2.4) to look up the values for Gand an ABL scale (`max or NABL) for a given set of IHand UH. The RANS–2model uses an optimizer to find the values of Gand z0for the CNBL case. LES-diagnosed values of θ0,zi, and d2/dz|care not necessary for the SBL case because we do not employ the RANS–2model for this case. The RANS inflow model results are compared with the two LES models from Hodgson et al. (2023) for the CNBL and SBL cases in Fig. 2. The results of the LES models compare well with the results of the RANS–`max and RANS–N models in terms of wind speed and turbulence intensity based on k(Ik=√2/3k/√U2+V2) for both ABL cases (Fig. 2a, c, g and i). The good fit around the hub height is expected since the RANS models are tuned for the LES-predicted values of IHand UH. The RANS-predicted wind veer as shown by the relative wind directions in Fig. 2b and h also compares well with both LES models for the CNBL case. However, the RANS–`max and RANS–Nmodels predict stronger wind veer over the rotor area (11.1 and 12.7°, respectively) compared to the LES models (8.8 and 9.9° for DTU and EPFL, respectively) for the SBL case. The main differences between the RANS models are the profiles of turbulence length scale (Fig. 2d, j) and eddy viscosity (Fig. 2e and k), where the RANS–`max model predicts taller ABLs compared to the RANS–Nmodel; the veer difference is due in part to different effective ABL heights (Kelly and van der Laan, 2023). Note that it is not trivial to postprocess an eddy viscosity or turbulence length scale from the LES data that can be directly compared to the RANS models. This is because one would need additional modeling to obtain an eddy viscosity implied by the LES data. Furthermore, the RANS turbulence length scale is a model definition, while an LES-derived turbulence length scale can be ambiguous and is only qualitatively comparable (van der Laan and Andersen, 2018) unless non-Boussinesq contributions are accounted for (e.g., Large et al., 2019). For the SBL case, it is clear that the turbulence length scale in the RANS–`max model is limited to a maximum value of 3.4 m, while the turbulence length scale of the RANS–Nmodel results in a smoother profile that has a higher value in the surface layer but a lower value around the ABL height. As a result, the profiles of wind speed and direction around the ABL height are more diffused in the RANS–`max model compared to the RANS–Nmodel, which is most visible for the SBL case (Fig. 2g and h) around z= 0.2 km. In other words, RANS–Nhas a more pronounced Ekman layer. The RANS–Nmodel predicts a lower ABL height compared to the LES models for both ABL cases, but this could be improved by lowering the applied roughness length. The latter is not performed in order to provide a fairer comparison between the RANS–Nand RANS–`max models https://doi.org/10.5194/wes-9-1985-2024 Wind Energ. Sci., 9, 1985–2000, 2024 1992 M. P. van der Laan et al.: A simple steady-state inflow model for neutral and stable ABLs Table 2. LES-derived input from ABL cases and fitted parameters of RANS inflow models. LES-derived input RANS–`max RANS–2RANS–N IHUHθ0zid2/dz|cG `max G z0G NABL Case [%] [m s−1] [K] [m] [km−1] [m s−1] [m] [m s−1] [m] [m s−1] [s−1] CNBL 5.3 8.4 277.3 650 3.75 ×10−39.67 30.7 9.31 9.31 ×10−59.56 3.90 ×10−3 SBL 3.1 8.8 – – – 9.58 3.38 – – 9.85 2.71 ×10−2 Figure 2. RANS-simulated ABL inflow compared to LES model results from Hodgson et al. (2023) for CNBL (a–f) and SBL inflow cases (g–l). Horizontal dashed lines represent the rotor-swept area of the SWT-2.3-93 wind turbine. using the same roughness length. The RANS–2model compares well with the LES models for the CNBL case but shows a larger turbulence length scale and eddy viscosity compared to the RANS–Nmodel due to zero turbulent buoyancy in the surface layer (Fig. 2d and e). The RANS–2model is not applied to the SBL case because the RANS–2model cannot represent an SBL or a shallow CNBL, as discussed in Appendix A. Results of the implied temperature profile of the RANS–Nmodel, 2(z)/θ0=1+zN2 ABL/g, are shown in Fig. 2f and l. It is clear that the temperature gradient employed is larger in the RANS–Nwith respect to the LES models, although a direct comparison with LES in terms of a temperature profile may not be fair due to the simplicity of the RANS–Nmodel. In addition, the choice of the turbulent Prandtl number in Eq. (11), here σθ=1, will also determine the implied temperature gradient of the RANS–Nmodel because it influences the obtained value of NABL. One could match a constant temperature gradient to the LES results, however, it is not guaranteed that the RANS–Nmodel will compare well with the LES results in terms of wind speed, direction, and turbulent intensity profiles. 4.2 Single wake The RANS inflow models are applied to single-turbine wake simulations, and the results of velocity deficit magnitude and wake-added turbulence intensity are compared with results from the two LES models of Hodgson et al. (2023) in Fig. 3. The wake results are normalized by the simulation results without a turbine. The RANS–2model is only applied to the CNBL case and not to the SBL case because the model cannot represent a shallow ABL, as discussed in Appendix A. The CNBL case shows that all three RANS inflow models predict similar velocity deficits that follow the trends of the LES models (Fig. 3b–d). The differences between the RANS and LES models in the near wake at x=1D(Fig. 3a) are expected, following a previous study (van der Laan et al., 2015c). The difference in velocity deficit between the RANS and LES models for the SBL case is larger than that for the CNBL case. The largest difference between the RANS–`max and RANS–Nmodels is observed at the far wake at x=25D for the SBL case, which is depicted in Fig. 4. Figure 4a shows that the RANS–`max model does not allow the turbine wake to recover vertically with respect to the RANS–N model due to the global length-scale limiter. The LES results at x=25Dsuggest that the RANS–Nmodel predicts the Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024 M. 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Wind Energ. Sci., 9, 1985–2000, 2024 https://doi.org/10.5194/wes-9-1985-2024