We perform a free-flow large-eddy simulation (LES) using the Fortran code PALM developed at Leibniz Universität Hannover . PALM utilizes a staggered Arakawa C grid: the velocity components are defined at the grid cell edges and are shifted by a half-grid spacing; the scalar variables are defined at the center of a grid cell. The subgrid-scale fluxes are resolved via the Deardorff 1.5-order closure model.

By default, PALM solves prognostic equations for the velocity components u, v, and w and potential temperature θ. If the stability condition is set to true neutral, the temperature is considered constant, and the corresponding equation is not solved. Buoyancy terms are also not considered in a true neutral simulation

A nested simulation in PALM consists of at least one child domain inside a parent domain. Each child domain can simultaneously be a parent domain for another child domain, thus forming a cascading self-nested structure. The top-level parent domain is further referred to as the root domain to make a distinction from inner parent domains. Overall, PALM supports simulation of one root domain and up to 63 child domains.

The nesting algorithm is constructed in a way to optimize computational time for multiple child domains . The nested domains communicate via interpolation which is performed just before the pressure-correction step, so that the time-consuming pressure solver is run only once per the time step. The solution at the nested boundaries of a parent domain – velocity components and scalar quantities, e.g., temperature and humidity – is linearly interpolated to all nested boundaries, except the bottom surface, as boundary conditions. The bottom surface is always located at a zero level as in the root domain and utilizes Dirichlet or Neumann boundary conditions as prescribed in the corresponding child domain input files.

After the interpolation, the prognostic equations are solved for a child domain. In the case of cascading nesting, the procedure is repeated until the solution is found for all nested domains at the current step. In a one-way nesting case, the simulation proceeds to the pressure-correction step, so the solution in parent domains remains unaffected by the solution in child domains. In a two-way nesting case, PALM uses an anterpolation scheme – a term suggested by and first described by . The technical details behind the implementation in PALM are explained in . Each child domain anterpolates its solution via first-order integration to the respective parent domain before the pressure-correction step. Therefore, the two-way nested solution remains similar in the nested area, while the one-way nested solution may eventually diverge for parent and child domains.

One of the ways PALM can simulate a turbulent flow is a precursor scheme, which does not require complex dynamic input data and effectively reduces the domain size required for turbulence development . First, a small precursor domain is simulated with cyclic boundaries until the flow reaches a steady state. The resulting mean wind speed and temperature profiles are then copied over the larger main domain to set up an initial non-cyclic flow with a developed turbulence. Provided that the main run is simulated with the same forcing as the precursor, the mean profiles in the main run remain stationary.

The size of the precursor domain is usually smaller than for the main run, and the y-shift procedure is performed at left/right cyclic boundaries to avoid non-physical regularity of the flow . The y-shift procedure is also applied in the main run for an additional disruption of regularity. Using the precursor scheme also ensures that an idealized input flow remains the same within a stability case regarded.

The grid characteristics of the root and innermost child domain in the PALM simulation were selected to closely match the SOWFA simulation in . The ratio between the parent and child domains' grid spacing, thus, would reach 8 (from 10 to 1.25 m for NBL and CBL cases) or 4 (from 5 to 1.25 m for SBL case). As shown by , the discrepancy with a fine-grid simulation in PALM increases if the grid spacing ratio is 4 or higher. Therefore, we add intermediate child domains and reduce the grid spacing by a factor of 2 until the desired refinement is reached. Hence, NBL and CBL simulations contain three child domains, while the SBL simulation has two (Tables  and , Fig. ).

We perform one-way and two-way nested simulations. To evaluate the nesting effect, we also simulate domains without nested grids using the same precursor flow. Due to high computational time and memory requirements, we only simulate non-nested domains for the grid spacing of Δx = 10 and 5 m.

The precursor profiles undergo development during a simulation and thus may deviate from the initial profiles. The precursor's input parameters are then selected so that the resulting steady-state profiles of mean wind speed and turbulence intensity follow the values estimated from the measurements, particularly the wind speed at the reference height. The Coriolis force is switched off; hence the required wind speed and turbulence intensity profiles in the precursor run are enforced by a combination of the parameters: the initial mean wind U‾0, the pressure gradient forcing dp/dx, and the roughness length z0. The NBL case is run as the true neutral flow with no heat flux. The CBL case is defined via the positive heat flux w′θ′‾ in addition to the parameters mentioned above. The SBL case uses surface cooling over time dTs/dt instead of the heat flux . NBL and SBL cases start with zero temperature gradient; the CBL case has an initial temperature gradient of 1 K (100 m)-1. The surface temperature Ts is varied to match the conditions observed during the reference meteorological measurements at FINO1. The precursor domain characteristics and input parameters are listed in Tables –.

During the precursor simulation, the initial profiles are altered due to the influence of pressure forcing and heat fluxes. The resulting precursor profiles are provided in Table ; the same profiles are used to initialize the main run.

We run main simulations for 3 h with a dynamic time step selected by the model. The simulation is then continued for another hour with the fixed time step of Δt = 0.05 s to obtain a high-frequency output. Then, we probe time series of each wind speed component at the center of the innermost child domain and the corresponding points of the parent domain (Fig. ). The high-frequency time series are further used to compare turbulence statistics with the measurements. Spatial averages (cross-sectional flows, profiles) are calculated for 10 min periods.

We evaluate the model performance based on turbulence characteristics: power spectrum, coherence, co-coherence, and phase. The coherence represents a correlation between time series a(t) and b(t) at two points separated by a certain distance δ and is calculated as follows 4Cohab=SabSaaSbb, where Saa and Sbb are the spectral densities of a(t) and b(t), while Sab is the cross-spectrum of the same series.

The co-coherence represents the real part of the coherence 5Coab=ReCohab=ReSabSaaSbb.

The phase ϕab shows the level of synchronicity between time series a(t) and b(t) 6ϕab=arctanReCohabImCohab.

Since the measurement time series are available only for three levels, 40, 60, and 80 m, the spectra are calculated and compared at h=80 m for the total horizontal U=u2+v2 and vertical w wind speed. The co-coherence is calculated for two vertical separations of δ = 20 m (between levels 60 and 80 m) and δ = 40 m (between levels 40 and 80 m). The sampling frequency for the LES time series matches the output frequency fsLES=1/0.05 s = 20 Hz, and the segment length is chosen as 60 s. The sampling frequency for the measurement time series is lower, fsmast=1/0.1 s = 10 Hz, although the segment length is left the same.

We also review flow characteristics relevant to the turbine performance analysis: power law coefficient and turbulence anisotropy.

The power law is commonly applied to assess wind resources at the hub height from near-surface wind speed measurements. 7U(z)=U‾10z10α, where U‾10 is the wind speed at z=10 m and α is the power law exponent. The power law exponent is sensitive to atmospheric conditions and is usually approximated with a constant; e.g., α=1/7 is applicable to neutral onshore sites but not other stabilities . Often, the approximations do not reflect seasonal and diurnal variations in mean wind profiles . Hence, simulating a long time series with the LES gives a possibility to study wind profiles in detail.

The anisotropic turbulence naturally develops in a simulation with an anisotropic grid resolution but may also occur in isotropic grids, such as those used in this study. The anisotropic turbulence affects wind turbine loads, particularly fatigue loads. Therefore, it is important to evaluate its strength in the simulation . We estimate turbulence anisotropy by comparing spectra of velocity components for the normalized frequency fn=fz/Uz, where z=80 m and Uz is the horizontal velocity at this level. We compute ratios Svv/Suu and Sww/Suu for all regarded cases at fn≈1. The closer both ratios are to the theoretical value of 4/3=1.333, the more isotropic the simulated turbulence is .

All LESs are run at 1024 cores for each case with a time step of Δt = 0.05 s; the required simulation times for each scenario are summarized in Table . Since the domains vary in size and number of grid points, we compare not the total CPU time but the CPU time per second of the simulated time. The non-nested coarse domain (Δx = 10 m) is not computationally demanding, regardless of the stability case. However, the required CPU time gradually increases if the grid spacing is reduced globally for the whole domain. As could be seen for the NBL case, the CPU time per second of the simulated time increases from 5.1 s for Δx=10 m to 31.7 s for Δx=5 m, respectively. Refining the grid locally by adding child domains increases the CPU time compared to the coarse reference non-nested grid (Δx = 10 m). Still, the nested simulation finishes faster than the globally refined non-nested simulation (Δx = 5 m), while allowing better a local grid refinement up to Δx = 1.25 m.

Both NBL and CBL simulations have the same domain structure and grid spacing (Table ). However, CBL simulations require more CPU time compared to the respective NBL (true neutral) simulations due to solving the temperature equation. SBL simulations use CPU time comparable to NBL simulations due to having one child domain less and a smaller root domain size – and thus a lower overall number of the grid points (Table ).

Two-way nested simulations require additional ∼ 2–3 s of the CPU time per simulated time step to anterpolate the child domain solution back to the parent domain. This results in about 10 % increase in the CPU time compared to one-way nesting.

It should be noted that, unless obtaining high-frequency time series is the main goal of a simulation, the time step can be gradually increased for non-nested runs in order to speed up the computation. The computational time will, nevertheless, increase in a similar proportion with the global grid refinement. The time step in nested runs is still limited by the lowest grid spacing in child domains. For example, the dynamic step in the regarded configuration does not exceed 0.075 s to satisfy Courant–Friedrichs–Lewy condition.

Depending on the simulation conditions, LES produces different results in the nested area. If the true neutral case is defined in PALM explicitly via setting a corresponding flag, the one-way and the two-way nested simulations behave similarly with respect to grid spacing and feedback between domains (Fig. ). Switching on the true neutral flag means that the temperature equation and buoyancy terms are not considered in the calculations. As long as those terms are introduced for non-neutral simulations, the two-way nested simulation results in a decreased flow speed in child domains.

Since the child domains anterpolate their solution back to the parent domain, the area of reduced flow speed spreads to the root domain. While the effect is less prominent for the instantaneous fields, it becomes apparent in the 10 min averaged flow (Fig. ). The induction of downward vertical wind in two-way nested simulations was already described by for the 5 h averaged buoyancy-driven flow in PALM. argued that the effect of the secondary circulation described by was caused solely by the insufficient domain size and explained it with the different grid spacing and subsequent divergence of the vertical heat flux in the parent and child domains. The researchers hypothesized that the secondary circulation was an inevitable side effect of the two-way nesting solution due to the better resolution of the turbulence mixing in child domains. In the case of the shear-driven flow, we observe that the slowing effect is more prominent and develops faster. The effect emerges in the beginning of the simulation within 20 min – an approximate time required for the precursor flow to pass the main run domain. In addition, some of the quantities of a shear-driven flow, mainly the vertical velocity w, are not uniformly distributed inside the child domains (Fig. ).

LES resolves scales larger than the grid spacing directly but approximates smaller scales. In a well-resolved flow, the unresolved (subgrid) scales should not exceed the resolved ones. This relation holds for all simulations performed, implying that the grid spacing of Δ=10 m is already small enough for the given flow (Fig. ). The grid refinement does not strongly affect momentum fluxes, except for the CBL case (Fig. b), where turbulent eddies are generally larger than in the NBL and SBL cases. The effect from the nesting mode is also the most pronounced in CBL simulations (Fig. b). The resolved wu‾ and wv‾ fluxes remain stationary in the one-way nesting mode but decrease over time in the two-way nesting mode and eventually merge.

The subgrid-scale fluxes consistently remain near zero for all levels except near-surface cells, where the turbulence intensity is expected to be high due to the surface influence (Fig. ). Consequently, the near-surface subgrid-scale fluxes are comparable to resolved-scale fluxes. However, the subgrid-scale fluxes at lower levels tend to zero faster as the grid spacing is refined. Unlike the one-way nesting mode, the resolved fluxes in the two-way nesting mode show a non-monotonic behavior near the surface in the intermediate child domains. The effect is observed in all two-way simulations, including true neutral conditions. Therefore, it cannot be solely caused by the flow difference in the nested and non-nested areas, despite the flux profiles being time and spatial averages. The occurring non-monotonic behavior can be rather attributed to the way PALM performs anterpolation from a child to the parent domain.

Since the flow is driven by the pressure gradient instead of the Coriolis force, the flow is aligned with the x axis, and the wind direction remains nearly constant. The fluctuations of the lateral component v are stronger for the measurement time series. Therefore, we compare turbulence statistics of the horizontal wind speed u from the LES results to the total horizontal flow in the measurements U=u2+v2 and omit the lateral component v for the LES data.

In one-way nested simulations, the parent domain does not receive feedback from the child domain. Consequently, the spectral characteristics of non-nested domains with the grid spacing of Δx=10 m (NBL and CBL) and 5 m (SBL) match the characteristics of the corresponding domain in a one-way nesting simulation (Figs.  and ). The individual spectra of the nested domains lie apart from each other but show improvement as the grid spacing is reduced. The inertial subrange resolved by LES widens as the grid becomes more refined; however, it is not fully resolved despite the grid spacing being reduced to Δx = 1.25 m.

The two-way nesting mode ensures feedback between the nested domains. Therefore, the root and child domain spectra lie closer to each other and to the one-way spectra of the most refined child domain (Δx = 1.25 m). Despite the exchange between domains in the two-way nested case, the spectral characteristics do not coincide perfectly. The inertial subrange being shorter for Δx=10 m than for the refined domains implies that the grid resolution is the limiting factor, and the solution for the root domain cannot be improved further even in the two-way nesting case.

Despite the NBL case being simulated as a true neutral condition, it showed good agreement with the measurements on par with the CBL case. The result suggests that it is possible to omit a weak heat flux in neutral cases to save computational time and avoid secondary circulation in the two-way nesting mode.

The SBL simulations largely overestimate the energy contained in low-frequency eddies. The inertial subrange of the corresponding measurement time series also starts at higher frequencies, unlike in the NBL and CBL cases. The LES does not fully resolve high frequencies despite gradually reduced grid spacing. Hence the overall agreement for the SBL case is worse than for NBL and CBL. When comparing available measurement profiles for the specific period of SBL time series, we did not observe anomalies or irregularities, such as reported by , which could be studied as a possible cause of a discrepancy. The existing studies on SBL simulations with PALM do not compare simulated spectra against measurements but evaluate other aspects, such as fluxes and grid resolution influence. Hence, simulating SBL in PALM may require additional studies focusing on turbulence characteristics.

In order to match the SBL spectra shape, we performed a short SBL simulation with lower forcing, which led to a decreased turbulence intensity but stronger mean profile shear. The results are provided in Appendix.

The coherence, co-coherence, and phase are plotted against the reduced frequency: 8fr=fδu‾,

where f is the original frequency, δ is the vertical separation distance, and u‾ is the mean wind speed of the two regarded levels, 60 and 80 m for δ=20 m or 40 and 80 m for δ=40 m.

The coherence and co-coherence calculated for NBL and CBL coarse domains (Δx = 10 m) and δ=20 m show strong deviation from the measurements for the one-way and non-nested simulations at fr>1 (Figs. a and a). The tendency to the coherence/co-coherence value of 0.5 suggests that the time series at points separated by δ=20 m remain partially correlated in the coarse grid, which is not the case for the corresponding measurements. While the most refined child domain (Δx = 1.25 m) shows a good match between the LES and measurement series (Figs. b and b), the agreement already improves for Δx = 5 m, and the correlation falls to zero for fr>0.5. The SBL case shows better agreement for the root domain because of the lower initial grid spacing Δx = 5 m. Nevertheless, the coherence is noticeably overestimated for low fr compared to the measurements (Fig. a, b). The time series are generally uncorrelated for the vertical separation of δ=40 m both for the LESs and measurements (Figs. c, d and c, d). However, the NBL case does not capture the high coherence value at fr=0 observed in the measurements.

The phase plots are in line with the coherence. The time series are in phase for fr<0.1, where the coherence is above zero. The effect is strong for the low vertical separation of δ=20 m (Fig. a, b) and is in good agreement with the measurements. The phase becomes more chaotic as the vertical separation distance increases to δ=40 m (Fig. c, d), while the time series become less correlated (Figs. c, d and c, d).