STAR-CCM+ is a general-purpose simulation software package best known for computational fluid dynamics. Within STAR-CCM+, DNV customized a steady-state RANS model for simulation of wind farm flows. The turbulence closure is standard k-ε with modified coefficients. The direct influence of buoyancy on the mean flow is simulated via a shallow Boussinesq formulation; extra terms in the closure equations represent the influence of buoyancy on turbulence. Coriolis terms are in the momentum equation. The turbines are represented with a simple actuator disk model where the body forces are functions of the average axial component of velocity across the disk. These functions are derived from the IEA 15 MW power and thrust curves, defined as functions of hub height freestream wind speed, using the procedure described in . More information on this flow model can be found in .

The simulations in this study were run within a domain of size 66 km × 66 km × 17 km. The wind farm is located 40 km downstream of the inflow boundary. The mesh spacing is 12 m around each actuator disk and 24 m around the wind farm.

Vertical inflow profiles are generated using a steady-state 1D single-column precursor simulation with the input potential temperature profile frozen. After the steady-state simulation converges, the potential temperature is unfrozen, and the 1D solution is marched in time to confirm that the full set of profiles are in quasi-equilibrium.

EllipSys3D solves the incompressible Navier–Stokes equations in general curvilinear coordinates using a finite-volume method in a multi-block structure . Rhie–Chow interpolation is applied to prevent pressure decoupling, which is solved using an improved version of the SIMPLEC algorithm . The convective terms are discretized using a fourth-order central difference scheme, which includes an artificial viscosity term to suppress numerical wiggles , and time stepping is performed using a second-order scheme with subiterations. Several RANS and large-eddy simulation (LES) turbulence models are implemented in EllipSys3D, where the anisotropic minimal dissipation (AMD) model has been utilized in the current simulations. Rayleigh damping is applied at high altitudes (>1000 m).

Initially, a precursor is simulated to spin up the CNBL. The precursor is performed in a domain Lx×Ly×Lz=10240 m ×10240 m ×3000 m with a total of Nx×Ny×Nz=512×512×384≈100×106 cells corresponding to a mesh resolution of Δx×Δy×Δz=20 m ×20 m ×5 m in the streamwise, lateral, and vertical directions. The equidistant mesh in the vertical is maintained at an altitude of 1500 m, after which the cells are stretched. Cyclic boundary conditions are imposed in the streamwise and lateral directions, while a wind direction controller is imposed to continuously adjust the wind direction at z=150 m to ensure that the flow direction is aligned with the wind turbines at hub height . The precursor is initially spun up for 20 h, after which cross-stream planes are extracted for a total duration of 2 h.

Subsequently, a mesh is built for the successor, which is Lx×Ly×Lz=30000 m ×30000 m ×3000 m with a total of Nx×Ny×Nz=512×448×192≈44×106 cells. The mesh has a central equidistant region of Lx,equi×Lx,equi×Lx,equi=13530 m ×12120 m ×1500 with Δx×Δy×Δz=30 m ×30 m ×10 m in the streamwise, lateral, and vertical directions with cells stretched to the boundaries. The precursor planes have been repeated to cover the extended domain of the successor simulations. The wind turbines are modeled by applying body forces in EllipSys3D, which is fully coupled to the aero-elastic tool HAWC2 through the Dynamiks interface (https://dynamiks.pages.windenergy.dtu.dk/dynamiks/index.html, last access: 10 February 2026). Velocities are transferred from EllipSys3D to HAWC2, which calculates aerodynamic forces and deflections, which are transferred back to EllipSys3D . HAWC2 also contains a dynamic torque controller, which enables the turbines to respond to the dynamically changing inflow by dynamically updating pitch and rotational speed, but it does not yaw the turbines. The impact of realistic and dynamic wind turbine controllers has been shown to have a significant influence on power production for wind farms . Turbines can be modeled as actuator lines or as actuator disks , which is used in this study. The simulations are run for 2 h, where the initial 1 h transient is discarded as the flow is still developing.

The CFD code code_saturne, primarily developed by EDF, is an open-source, free-to-use finite-volume CFD solver for the Navier–Stokes equations. It can manage scalar transport for various types of flows – 2D, 2D axisymmetric, 3D, steady, unsteady, laminar, turbulent, incompressible, dilatable, weakly compressible, or isothermal. Code_saturne comes with modules specifically designed for certain physics, such as atmospheric flows. An extensive explanation of its modeling capabilities, including the atmospheric module, can be found in code_saturne’s v8.0 online theory guide (https://www.code-saturne.org/documentation/8.0/theory.pdf, last access: 10 February 2026).

Reynolds-averaged Navier–Stokes (RANS) with linear production k-ε closure is used to model turbulence. The presence of wind turbines is accounted for using a non-rotating ADM with a constant body force function of the disk-averaged velocity and yaw control.

Wind farm simulations with code_saturne are performed in two steps. The first step consists of a 1D bi-periodic single-column precursor simulation to generate quasi-steady inflow profiles for the velocity, temperature, and turbulent quantities. The second step consists of the full 3D farm simulation in a circular domain with a refined grid in the farm and around turbines. The numerical domain is 25 km high, and a diameter as large as 4.7 times the length of the longest diagonal of the farm was shown to be sufficient to avoid confinement effects. Damping layers at the top and at lateral boundaries are implemented to prevent the reflection of gravity waves.

The SP-Wind flow solver is in-house software developed over the past 15 years at KU Leuven . In the current study, we use this software to solve the filtered Navier–Stokes equations with a Boussinesq approximation coupled with a transport equation for the potential temperature to investigate the flow in and around a large-scale wind farm . Here, we adopt the same solver version used by , which is described below.

The governing equations are integrated in time using a classic fourth-order Runge–Kutta scheme, with the time step determined by a Courant–Friedrichs–Lewy (CFL) number of 0.4. The streamwise (x) and spanwise (y) directions are discretized using a Fourier pseudo-spectral method. This approach involves discretizing all linear terms in the spectral domain while performing non-linear operations in the physical domain, which reduces the computational cost of convolutions from quadratic to log-linear . Additionally, the 3/2 dealiasing technique from is employed to prevent aliasing errors. For the vertical dimension (z), an energy-preserving fourth-order finite-difference scheme is utilized . The impact of subgrid-scale motions on the resolved flow is modeled using the stability-dependent Smagorinsky model , with the Smagorinsky coefficient Cs set to 0.14. Near the wall, this coefficient is damped using the function proposed by . Continuity is maintained by solving the Poisson equation at each stage of the Runge–Kutta scheme. We refer to for more details on the discretization of the continuity and momentum equations, while the implementation of the thermodynamic equation and subgrid-scale model are explained in detail in .

The flow solver adopts two numerical domains simultaneously marched in time: the precursor and main domains. The precursor domain, which does not contain turbines, has the function of generating a fully developed, statistically steady turbulent flow. This flow is then used to drive the simulation in the main domain. Following the approach in and , the precursor domain dimensions are set to Lxp=Lyp=10 km and Lzp=3 km. The wind farm is situated in the main domain, which must be sufficiently large to avoid artificial effects from domain boundaries. have shown that the width of the numerical domain can significantly alter the numerical results. To this end, we fix the main domain size to Lx×Ly=50×40 km², which leads to a domain-to-farm width ratio of 3.51. Consistent with previous studies, the main domain height is set to Lz=25 km . This vertical extent allows gravity waves to dissipate and radiate energy outward, minimizing reflectivity. After completing the precursor spin-up phase, the precursor domain width and height are extended to match the main domain dimensions, using the method described in and . To reach a statistically steady state, the flow fields in the precursor simulation are marched in time for 20 h. These flow fields are used to drive the main domain, where a second spin-up phase of 1 h takes place so that the flow adjusts to the presence of the turbines. Next, the wind-angle controller, which keeps the flow aligned with the streamwise direction at hub height, is switched off, and statistics are collected over a time window of 2 h.

In regard to the grid resolution, we fix Δx=31.25 m and Δy=21.74 m in the streamwise and spanwise directions, respectively. This leads to Nx=1600 and Ny=1840 grid points for the main domain and Nxp=160 and Nyp=230 points for the precursor domain. In the vertical direction, we adopt a stretched grid, which corresponds to the one used in , i.e., with a resolution of 5 m within the first 1.5 km and stretched above, for a total of 490 grid points. The combination of precursor and main domains leads to a total of roughly 6.92×109 degrees of freedom (DOFs).

At the top of the domain, we use the Rayleigh damping layer (RDL) to minimize gravity-wave reflection . To avoid periodicity in the streamwise direction, we adopt the wave-free fringe-region technique developed by . The buffer layer setup corresponds to the one previously used by . Hence, the RDL is 10 km thick and is located between z=15 and z=25 km. Moreover, νra=5.15 and sra=3 are parameters used in the RDL. The fringe region is 5.5 km long and is located at the end of the main domain. Further, we set xsh=Lx-Lxfr, xeh=Lx-2.8 km, and δsh=δeh=0.4 km, while xsd=xsh, xed=Lx, δsd=2.5 km, and δed=3 km. Finally, we fix the strength of the forcing to hmax=0.3 s⁻¹.

The turbine drag force is represented with the non-rotating actuator disk model , where the power is computed as the product between the thrust force and the turbine disk velocity. We use a constant thrust coefficient value of 0.778, which corresponds to a CT′ of 1.44. A simple yaw controller is implemented to keep the turbine rotor disks perpendicular to the incident wind flow measured 1 rotor diameter upstream.

The Simulator fOr Wind Farm Applications (SOWFA) was developed by NREL . It is built on the OpenFOAM software, an open-source finite-volume solver that can be coupled to an aeroelastic code for turbine load and control study. Turbulent winds and wakes are modeled using LES with the one-equation eddy viscosity subgrid-scale model .

The domain extent is set to Lx×Ly×Lz=26400 m ×32000 m ×6000 m with a mesh resolution of Δx×Δy×Δz=20 m ×20 m ×8 m up to a height of 720 m. The mesh generation was carried out by dividing the domain into four horizontal layers, with the mesh vertically stretched and a larger expansion ratio applied to the upper layers to reduce computational cost. This results in approximately 235 million cells in the domain.

There are two steps in the simulations. The first step is the precursor simulation, in which the turbulent ABLs are generated. A periodic boundary condition is applied to all lateral boundaries for an empty flow domain, where the horizontal driving pressure gradient is adjusted at every time step to control the mean wind speed and direction at the hub height . This is done by determining the source term from the error between the actual planar-averaged velocity at the specified height and the desired velocity. This approach is suitable for turbine engineering analyses in which the hub height wind speed can be prescribed. However, it is acknowledged that this method cannot simultaneously achieve both the desired geostrophic wind vector and the hub height wind direction. The Schumann wall shear stress model is used for wall modeling at the bottom surface, while the top boundary is a free-slip wall. The precursor simulation is performed until the turbulent flow reaches a quasi-steady state before the flow data on a cross-flow plane are recorded to be used as the inflow for the wind farm simulations.

The second step is the wind farm simulation, in which the streamwise boundaries are changed to inlet and outlet boundary conditions. The bottom and top boundaries are identical to those of the precursor simulation. The turbines are modeled using an actuator disk method with a simple controller in which the rotor speed and pitch angle are functions of the average axial velocity across the rotor . The aerodynamic forces and power of the rotors are calculated using the blade element method. A simple yaw controller is implemented to keep the rotor facing local wind directions. The statistical calculations for the turbulent flow and turbine output are performed over the last hour of the simulation time after the initial pass.

As mentioned in Sect. 2, each participant uses different approaches to obtain atmospheric flows. Figures – illustrate the vertical inflow profiles generated by the different solvers, where u and v are velocity components in the streamwise and spanwise directions, respectively; M corresponds to the magnitude of horizontal wind velocity vectors; Φ denotes the wind angle between the wind vector and the x axis; Θ is potential temperature; and e corresponds to the turbulent kinetic energy (TKE).

In general, all solvers produce similar inflow profiles near the rotor height. The velocity profile of the H150 case exhibits a low-level jet-like shape where the peak of supergeostrophic wind speeds is observed close to 200 m height. Wind veering is also significant in the H150 case, where the wind direction difference between the top and bottom of the rotor is almost 15°, while it is less than 5° in the H500 and H500-dh500 cases.

It should be noted that UU may not achieve the geostrophic wind speed of 10 m s⁻¹ because the mean wind speed and direction are controlled at the hub height, as presented in Sect. 2.5. Furthermore, it was challenging to achieve quasi-steady state for UU due to the initial oscillation of the wind speeds above the capping inversion. The main deviations considering veer, potential temperature, and turbulence kinetic energy are in Fig. f, where the SOWFA setup differs due to its inability to resolve the smallest scales because of the required numerical cost. However, this problem does not affect the results in Fig. f, where the deeper boundary layer has less wind shear. There are no significant differences for the velocities and TKE profiles between the H500 and H500-dh500 cases; only the potential temperature profile differs due to the initial capping inversion thickness (Figs.  and ).

Another notable outlier is the DNV potential temperature profile for the H150 case. As can be seen in Fig. e, the inversion in the DNV profile is thicker and starts at a lower height compared with the other simulations. The precursor simulations for all the models started from the same potential temperature profile, with the inversion starting at 150 m and a ground potential temperature of 288.15 K, but the DNV precursor approach preserves the initial potential temperature profile to a greater degree than the other approaches, resulting in material differences in the conditions at the simulated wind farms. The DNV created a second set of inflow profiles from a precursor simulation using a potential temperature profile that is more consistent with the inflow profiles from the other four contributors; this second set of H150 profiles (not shown) is much more similar to the other profiles, particularly the wind speed and wind direction profiles.

The mean streamwise velocity, averaged across the wind farm width at hub height, is presented in Fig.  from 25D upstream to 65D downstream of the first row of turbines. Overall, all solvers indicate a more significant decrease in the wind speed in the H150 case compared to the H500 and H500-dh500 cases. To quantify and compare wake recovery, mean wind speeds are normalized by the wind speed 25D upstream of the wind farm, as shown in Fig. . For the H150 case, EDF, KUL, DTU, and UU predict a similar wake recovery rate, with velocity reductions of approximately 30 %–40 % relative to the freestream winds, while DNV estimates a reduction over 40 %. For the H500 case, UU overpredicts wake recovery compared to EDF, KUL, and DNV, while DTU switches between following the trend of UU and that of the other models. The H500-dh500 case shows similar trends to the H500 case, which thus suggests that the capping inversion thickness does not significantly affect the wind farm wake flows.

Figures – show the streamwise velocity contours for the H150, H500, and H500-dh500 cases, respectively, in an x–y plane at hub height. In the H150 case (Fig. ), the shallow boundary layer restricts the flow above the wind farm, and this leads to a stronger deflection of the wakes at the edges of the farm compared to the 500 m BLH cases shown in Figs.  and .

The stronger wind veering in the shallow boundary layer results in more pronounced skewed wakes, as illustrated in a cross-flow (y–z) plane 10D downstream of the last row of turbines (Figs. , , and ).

Further wind speed comparisons on the x–z and y–z planes can be found in Appendix .

The performance of the wind farm is quantified and compared using power output and turbine yaw angle. Figures – illustrate the row-averaged power output for each case, including absolute power output, power normalized by the first row (P/P1), and power normalized by an isolated turbine output (P/P∞). It is noted that the isolated turbine output data for all solvers can be found in Appendix . For the H150 case, all solvers give a similar row-averaged power output trend where the power reduces by approximately almost 20 % in the second row. For instance, the P/P∞ of the second rows (Figs. c, c, and c) are approximately 60 % for the H150 case and more than 70 % for the H500 and H500-dh500 cases. Even though this is a staggered-layout wind farm where the second row is not directly in the wake of the first row, this significant power reduction in the second row indicates a strong blockage effect in the H150 case compared to the other cases. The power drops further in the third row before it recovers in the fourth and fifth rows. Since DTU and UU did not simulate the H500-dh500 case, fewer results are available for comparison, which makes the results in Fig.  appear less distinct than those in Figs.  and .

The power distribution in the farm is depicted in Fig. . The left, middle, and right columns represent the H150, H500, and H500-dh-500 cases, respectively. In all cases, the turbines near the edges of the front rows generate more power than the turbines in the middle.

The averaged turbine yaw angles are presented in Fig. . Each turbine in the farm responds to the local wind direction by yawing the rotor to maximize the power output. For the H150 case, the more pronounced spanwise flows cause the turbines close to the sides of the farm to yaw more significantly where the averaged yaw angles are more than 10°.

Wind farm power production losses due to blockage and wake interactions are quantified by the following definitions as introduced by . Losses due to wake interactions or wake efficiency (ηw) can be expressed as 1ηw=PtotNP1, where Ptot describes the total wind farm power output, N is the number of turbines in a farm, and P1 is the power of front-row turbines. The losses due to non-local effects, i.e., the blockage effect, can be expressed as 2ηnl=P1P∞, where P∞ is obtained from single-turbine simulations. The total wind farm efficiency (ηfarm) is then the product of losses introduced by non-local and wake effects: 3ηfarm=ηwηnl.

The efficiencies calculated from Eqs. (), (), and () are summarized in Tables , , and . The results indicate that wind farm efficiency is highly dependent on the depth of the boundary layer. In the H150 case, losses from wake interaction can reach nearly 40 %, while in the H500 case, they are around 20 %. Furthermore, due to the blockage effect, the front-row turbines can generate approximately 70 % and 80 % of the output of an isolated turbine in shallow and deeper boundary layers, respectively.

The efficiencies also reveal the influence of capping inversion thickness on wind farm performance. Results from DNV, EDF, and KUL show that the case with a thicker capping inversion leads to a reduced blockage effect but slightly larger wake losses compared to the thinner capping inversion.

The DNV H150 results are an outlier relative to the other models, with lower relative wind speeds throughout the wind farm (Fig. a) and lower wind farm efficiency (Table ). The gap between the DNV wind farm calculation and the other calculations for this case is simply a consequence of simulating a different potential temperature profile, which corresponds to a thinner boundary layer. When the DNV H150 case is rerun with inflow profiles that are more consistent with those of the other contributors, the wind speeds throughout the farm (not shown) and the calculated efficiencies (ηnl=0.695, ηw=0.620, and ηfarm=0.431) are more like the results from the others. The level of agreement is similar to how the DNV results compare with other results for the H500 and H500-dh500 cases. Thus, the outlier wind farm predictions from DNV presented for the H150 case have more to do with inflow conditions than differences between the models.

We here would like to highlight that the level of non-local blockage depends on the simulated conditions and selected dense wind farm layout. In this case, the power density is approximately 12 MW km⁻². This is in the range of the Belgium offshore economic zone and the German Bight but is denser compared to Danish and UK offshore cases.

In addition, the calculated efficiencies are a lot lower than would be calculated in an energy yield analysis, which would involve simulations over a much broader range of conditions, including wind directions and wind speeds where the efficiencies are at or very close to 1.0. However, our focus is not on the overall AEP calculations but rather on the quality and comparison of codes to represent wind farm flows under strong-blockage conditions.