2.1 The numerical model EULAG

The dry ABL flow through a wind turbine is simulated with the multiscale geophysical flow solver EULAG (Prusa et al., 2008; Englberger and Dörnbrack, 2017). The acronym EULAG refers to the ability to solve the equations of motion either in an Eulerian (flux form) (Smolarkiewicz and Margolin, 1993) or in a semi-Lagrangian (advective form) (Smolarkiewicz and Pudykiewicz, 1992) mode. The geophysical flow solver EULAG is at least of second-order accurately in time and space (Smolarkiewicz and Margolin, 1998) and well-suited for massively parallel computations (Prusa et al., 2008). A comprehensive description and discussion of the geophysical flow solver EULAG can be found in Smolarkiewicz and Margolin (1998) and Prusa et al. (2008).

For the numerical simulations, the Boussinesq equations for a flow with constant density ρ₀ = 1.1 kg m⁻³ are solved for the Cartesian velocity components $\mathbf{v}=(u,v,w)$ and for the potential temperature perturbations $\mathrm{\Theta }{}^{\prime }=\mathrm{\Theta }-{\mathrm{\Theta }}_e$ (Smolarkiewicz et al., 2007),

where Θ₀ represents the constant reference value. Height-dependent states ${\mathit{\psi }}_e\left(z\right)=\left(u_e\right(z),v_e(z),w_e(z),{\mathrm{\Theta }}_e(z\left)\right)$ enter Eqs. (1)–(3) in the buoyancy term, the Coriolis term, and as boundary conditions. These background states correspond to the ambient and environmental states. Initial conditions are provided for u, v, w, and Θ^′ in $\mathit{\psi }=(u,v,w,\mathrm{\Theta }{}^{\prime })$. In Eqs. (1), (2), and (3), d∕dt, ∇, and ∇ ⋅ represent the total derivative, the gradient, and the divergence, respectively. The quantity p^′ represents the pressure perturbation with respect to the background state and g the vector of acceleration due to gravity. The factor G represents geometric terms that result from the general, time-dependent coordinate transformation (Wedi and Smolarkiewicz, 2004; Smolarkiewicz and Prusa, 2005; Prusa et al., 2008; Kühnlein et al., 2012). The subgrid-scale terms 𝓥 and ℋ symbolize viscous dissipation of momentum and diffusion of heat and M denotes the inertial forces of coordinate-dependent metric accelerations. The Coriolis force is represented by the angular velocity vector Ω_C of the Earth's rotation. All following simulations are performed with a turbulent kinetic energy (TKE) closure (Schmidt and Schumann, 1989; Margolin et al., 1999).

The axial F_x and tangential F_Θ turbine-induced forces (F_WT = F_x + F_Θ) in Eq. (1) are parameterized with the blade element momentum (BEM) method as a rotating actuator disc without and with rotation in clockwise and counterclockwise directions (${\mathit{\beta }}_v\in \mathit{\{}\mathrm{0},-\mathrm{1},\mathrm{1}\mathit{\}}$), including a nacelle and excluding the tower. We do not simulate the rotor rotation directly; instead, we exert the rotor forces directly on the velocity fields (Eq. 1). As a clockwise wake rotation is initiated by an counterclockwise blade rotation, due to conservation of angular momentum (e.g. described in Zhang et al., 2012), we define a common clockwise rotor rotation as counterclockwise wake rotation with β_v = 1 and ${\mathit{\beta }}_w=-\mathrm{1}$, an counterclockwise rotor rotation as clockwise wake rotation defined by ${\mathit{\beta }}_v=-\mathrm{1}$ and β_w = 1, and no rotation as β_v = 0 and β_w = 0, with β_u = 1 in each case.

The BEM method accounts for local blade characteristics, as it enables calculation of the steady loads as well as the thrust and the power for different wind speeds, rotational speeds, and pitch angles of the blades. For the airfoil data, the 10 MW reference wind turbine from DTU (Bak et al., 2013) is applied, whereas the radius of the rotor and the chord length of the blades are scaled to a rotor with a diameter of 100 m. A more detailed description of the wind-turbine parametrization and the applied smearing of the forces, as well as all values used in the wind-turbine parametrization is given in Englberger and Dörnbrack (2017, parametrization B).

2.2 Metrics

For the investigation of the dependence of wake characteristics on the wake rotation and wind veer, the following characteristics are calculated from the simulation results: the spatial distribution of the time-averaged discrete streamwise velocity $\stackrel{\mathrm{‾}}{u_{i,j,k}}$, the time-averaged discrete spanwise velocity $\stackrel{\mathrm{‾}}{v_{i,j,k}}$, the time-averaged discrete vertical velocity $\stackrel{\mathrm{‾}}{w_{i,j,k}}$, the streamwise velocity deficit

and the streamwise (x), spanwise (y), and vertical (z) turbulence intensity, e.g. in x direction

with ${\mathit{\sigma }}_{u_{i,j,k}}=\sqrt{\stackrel{\mathrm{‾}}{u_{i,j,k}^{\prime \mathrm{2}}}}$ and $u_{i,j,k}{}^{\prime }=u_{i,j,k}-\stackrel{\mathrm{‾}}{u_{i,j,k}}$. Similar formulas apply for TI_y and TI_z in the y and z directions, respectively. The total turbulence intensity is

The indices of the grid points are denoted by i = 1 … n, j = 1 … m, and k = 1 … l in the x, y, and z directions, respectively. These characteristics are averaged over the last 50 min of the corresponding 1 h wind-turbine simulation. The temporal average is calculated online in the numerical model and updated at every time step according to the method of Fröhlich (2006, Eq. 9.1).

Figure 2Schematic illustration of the top sector, the bottom sector, and the southern and northern sectors, defined from a view looking downwind from west to east towards the wind turbine on the disc. All sectors are 90^∘ with 25 m ≤ r ≤ 50 m.

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In the following, the quantities $\stackrel{\mathrm{‾}}{u_{i,j,k}}$, $\stackrel{\mathrm{‾}}{v_{i,j,k}}$, and $\stackrel{\mathrm{‾}}{w_{i,j,k}}$ are evaluated and discussed downwind of the wind turbine up to 20 D. Further, the rotor area is divided into four sections of 90 ^∘, as shown in Fig. 2, including all grid points with a distance r from the rotor centre $R/\mathrm{2}\le r\le R$, referred to hereafter as the top sector, bottom sector, northern sector, and southern sector, looking from upwind (i = 1) downwind toward the actuator. In this work, we refer to the left sector as the northern sector and to the right sector as the southern sector, defined from a view looking downwind from west to east towards the wind turbine on the disc. This naming results from a zonal wind from west to east representing the streamwise flow and a meridional wind from south to north representing the spanwise flow through the wind turbine in the analysed regimes of this work. However, the results of this work are valid independent of the wind direction of the streamwise and spanwise flow through the wind turbine. In any considered case, a left sector replaces the northern sector and a right sector replaces the southern sector.

2.3 Setup of the wind-turbine simulations

In this work, a wind turbine is placed in two different regimes characteristic of different parts of the diurnal cycle: the EBL and the SBL. Both regimes develop from a diurnal cycle precursor simulation over a homogeneous surface with periodic horizontal boundary conditions (Englberger and Dörnbrack, 2018). The simulation includes 512 × 512 grid points in the horizontal direction with a resolution of 5 m. The vertical resolution is also 5 m in the lowest 200 m, stretching up to the top of the domain at 2 km. The precursor simulation is initialized with a geostrophic wind of 10 m s⁻¹ as zonal (west–east) wind and no meridional (south–north) and vertical wind. The initial potential temperature is set to a constant value of 300 K up to 1 km, increasing above with a lapse rate of 10 K km⁻¹. A sensible heat flux forces the diurnal cycle with a maximum of 140 W m⁻² during the day and a minimum of −10 W m⁻² during the night. The diurnal cycle simulation is initialized at 00:00 UTC. For the EBL regime, a 1 h time period from 18:00 to 19:00 UTC is extracted. During this time period the surface fluxes change sign. For the SBL regime, the selected 1 h time period starts at 00:00 UTC (24 h after initialization). The horizontally averaged zonal (u) and meridional (v) velocity profiles, the vertical potential temperature (Θ) profile, the total turbulence intensity (TI) (Eq. 6), and the turbulence intensities in the x, y, and z directions (TI_x, TI_y, TI_z) (Eq. 5) of the beginning of the corresponding atmospheric regime are presented in Fig. 3.

Three factors distinguish the EBL and the SBL from each other. First, the meridional wind v is zero with $\frac{\partial v}{\partial z}=\mathrm{0}$ in the EBL, whereas the v component in the SBL is larger than zero with $\frac{\partial v}{\partial z}\ne \mathrm{0}$ for z < 100 m (Fig. 3a). This veering wind at night is the main difference to the non-veered wind in the EBL. The resulting wind direction change with height in the SBL is an effect of the Coriolis force acting on the velocity components and friction due to the surface in combination with low levels of turbulence due to radiative cooling of the surface at night. The veering wind profiles correspond to the Ekman spiral with

for a Coriolis parameter $f=\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻¹ and an eddy viscosity coefficient κ = 0.05 m² s⁻¹, which is an appropriate value for an SBL regime (Yamada and Mellor, 1975). This results in the zonal and meridional wind profiles representing the Ekman spiral:

following Stull (1988), with a geostrophic wind u_g = 10 m s⁻¹ in Fig. 3a.

Second, the potential temperature profiles differ between the EBL and the SBL (Fig. 3b). The sensible heat flux of $-\mathrm{10}W{m}^{-\mathrm{2}}$ during the night leads to a maximum cooling close to the surface in the SBL regime, whereas $\frac{\partial \mathrm{\Theta }}{\partial z}$ ≈ 0 in the EBL regime when the surface fluxes change sign and are therefore very small.

Finally, the EBL and the SBL also differ in the amount of turbulence in the atmosphere. Shortwave heating of the surface during the day triggers convective turbulence. The EBL still experiences well-mixed turbulence, resulting in higher levels of turbulence intensity in comparison to the SBL, with a minor variation in height of TI, TI_x, and TI_y (Fig. 3c). At night, radiative cooling of the surface results in negative buoyancy which damps turbulence. Therefore, the total turbulence intensity and all three individual components of TI are smaller in the SBL regime in comparison to the EBL regime (Fig. 3c). A more detailed description of the diurnal cycle parameters from Fig. 3 is presented in Englberger and Dörnbrack (2018).

Figure 3Vertical profiles of the horizontal wind components u and v in (a), the potential temperature Θ in (b), and the total TI, streamwise TI_x, spanwise TI_y, and vertical turbulence intensity TI_z in (c). The profiles represent the horizontal averages of the EBL and the SBL regimes in the precursor diurnal cycle simulation.

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The EBL regime and the SBL regime represent the two atmospheric regimes in the corresponding wind-turbine simulations. The wind-turbine simulations are performed with periodic boundary conditions in y direction and open boundary conditions in x direction. To integrate a turbulent regime in the wind-turbine simulations with open streamwise boundary conditions, a synchronized coupling between the ABL regime and the wind-turbine simulation is applied. Here, the initial fields of $\mathit{\psi }=(u,v,w,\mathrm{\Theta }{}^{\prime })$ (Fig. 3a and b) at 18:00 or 00:00 UTC are applied as initialization of the corresponding wind-turbine simulation. Their horizontal averages are the background fields ${\mathit{\psi }}_e\left(z\right)=\left(u_e\right(z),v_e(z),w_e(z)$, Θ_e(z)). At each time step, two dimensional y–z slices represent the upstream values of ψ at the leftmost edge of the domain. This technique is similar to those used in Kataoka and Mizuno (2002), Naughton et al. (2011), Witha et al. (2014), and Dörenkämper et al. (2015) and has successfully been applied in Englberger and Dörnbrack (2018), where a more detailed description is also provided.

Further, the wind-turbine simulations are performed on $\mathrm{512}\times \mathrm{512}\times \mathrm{64}$ grid points with a horizontal resolution of 5 m and a vertical resolution of 5 m in the lowest 200 and 10 m above. The rotor with R = 50 m and z_h =100 m is located at 300 m in x direction and centred in y direction. The wind-turbine simulations are performed with a rotation frequency of Ω = 0.117 s⁻¹ for 10 min and after restart continued for 50 min.

Three different working conditions of the rotor are considered: common clockwise-rotating rotor (CR), an counterclockwise-rotating rotor (CCR), and no rotation of the rotor (NR), defined from a view looking downwind on the turbine (Fig. 2). In the simulations of the considered working conditions in the same atmospheric regime, only the prefactors ${\mathit{\beta }}_{\mathit{v}}\in \mathit{\{}\mathrm{0},-\mathrm{1},\mathrm{1}\mathit{\}}$ (Eq. 1) differ, as explained above.

4.1 Impact on spanwise and vertical velocity

Figure 5y–z cross sections of the EBL wind-turbine simulations for CR and CCR at x = 3 D and 5 D. The first row (a–d) presents the (v, w) vectors in the y–z plane, the second row (e–h) the meridional wake velocity v, and the third row (i–j) the vertical wake velocity w. The blue contour represents the circumference of the actuator. The streamwise flow is from the west, and these cross sections correspond to a view towards the east downwind through the turbine with north on the left and south on the right.

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As suggested by the analysis of Sect. 3, the rotational direction impact is limited to the change of the sign in Eq. (17) when the inflow lacks veer, as in the EBL. This behaviour is investigated by assessing the wake characteristics of an EBL wind-turbine simulation, which are presented in the y–z plane x =3 D or 5 D in Fig. 5. The top row (Fig. 5a–d) represents the vectors (v, w). As expected, the rotational direction of the flow in the wake inside the rotor region is dictated by the rotational direction of the rotor itself and is opposite to the rotor rotation. Approaching downwind from x =3 D to 5 D, the rotational direction imprint of the wind-turbine wake decreases.

The interactions between the wake rotation and the inflow are embodied in the cross-stream and vertical velocities. The evolution v and w dictated is represented in the second row (e–h) for v and in the third row (i–l) for w in Fig. 5. As predicted by the analysis of Sect. 3 (Eq. 15), the signs of v are opposite in the upper and the lower rotor halves for CR and CCR, respectively (compare Fig. 5e to g and f to h). The same is valid for the signs of w (Eq. 10) (compare Fig. 5i to k and j to l). This difference between CR and CCR is pronounced at a downwind distance of x = 3 D. At x = 5 D, the difference is smaller as v and w approach the inflow conditions of ≈ 0 m s⁻¹ (Fig. 3a).

Figure 6y–z cross sections of the SBL wind-turbine simulations for CR and CCR at x = 3 D and 5 D. The first row (a–d) presents the (v, w) vectors as the difference between the vector at x = 3 D (or 5 D) and the inflow region, the second row (e–h) the velocity vector in the y–z plane, the third row (i–l) the meridional wake velocity v, and the fourth row (m–p) the vertical wake velocity w. The blue contour represents the circumference of the actuator. The streamwise flow is from the west, and these cross sections correspond to a view towards the east downwind through the turbine with north on the left and south on the right.

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In contrast to the no-veer case, the effect of the veering inflow on the wind-turbine wake will not be limited to the change of the sign; also, the wake characteristics will be different in CR and CCR, as suggested by the analysis of Sect. 3. Figure 6 represents the wake characteristics for a veering inflow in the SBL wind-turbine simulations. Here, the top row (a–d) represents the (v, w) vectors in the y–z plane as the difference between the quantities at x = 3 D (or 5 D) and upstream at $x=-\mathrm{2}$ D to emphasize the effect of the rotating actuator. At a downwind distance of x = 3 D, the rotational direction of the rotor determines the wake rotation in both cases. Differences emerge by 5 D downwind. The wake rotation in CCR is the same as at x = 3 D, but only the magnitude of the (v, w) vectors is smaller. In CR, however, the magnitude of (v, w) approaches zero and the rotational direction of the rotor no longer determines the wake rotation. Further, the applied vortex leads to an inflow into the lower northern rotor part, resulting in rising motion in the southern rotor part in CR (Fig. 6a), whereas it ascends in the northern rotor part in CCR (Fig. 6c).

The second row (Fig. 6e–h) corresponds to the first row of Fig. 5, representing the vectors (v, w) at x = 3 D and 5 D. The magnitude is weaker in CR (Fig. 6e) in the upper and lower rotor parts in comparison to CCR (Fig. 6g), resulting in larger and more organized (v, w) wind components in CCR. At a downwind distance of x = 5 D, in CCR (Fig. 6h), the rotational direction of the wake flow is the same as at x = 3 D; only the strength is weaker. In the case of CR (Fig. 6f), however, the flow direction in the upper rotor half has changed in comparison to x = 3 D (Fig. 6e). Now it flows from north to south, resulting in a wake which is clearly dominated by the inflow.

To distinguish both contributing parameters of the (v, w) vectors, the v cross sections are presented in the third row (i–l) of Fig. 6. The positive and negative perturbations in v are the opposite in CR (Fig. 6i) and CCR (Fig. 6k) in the corresponding rotor sector at x = 3 D, with larger $\left|v\right|$ values in the upper and lower rotor sectors in the case of CCR. This pattern is weakening downwind and persists at x = 5 D (Fig. 6j, l).

Considering the w cross section in the fourth row (m–p) of Fig. 6, the upward and downward orientations of w differ at x = 3 D (Fig. 6m, o) in the northern and the southern sectors and also for CR (Fig. 6m) and CCR (Fig. 6o) comparing the same sectors. In comparison to the EBL wind-turbine simulations (Fig. 5i–l), the veering inflow has an additional effect on the vertical velocity in the wake at x = 3 D (Fig. 6m and o). The veering inflow leads to an intensified wake entrainment at the lower northern rotor part, which is independent of the rotational direction (see inflow pattern of the (v, w) vectors in the lower northern rotor part in Fig. 6a and c). At this location the flow is directed upward. In the case of CCR, it overlaps with the updraught in the northern sector, resulting in a rising motion of w in the northern part at x = 3 D (Fig. 6o). In the case of CR, a new updraught region manifests, weakening the downward region in the northern rotor part. This interaction results in an updraught in the southern rotor part at x = 3 D (Fig. 6m). At x = 5 D (Fig. 6n, p), the flow pattern is the same as at x = 3 D; only the strength is weaker.

4.2 Impact on streamwise velocity and total turbulence intensity

The rotational direction of the rotor modifies the spanwise and vertical velocity components in the wake under veering inflow. Here, we investigate their effect on the streamwise velocity and the total turbulence intensity.

Figure 7Coloured contours of the time-averaged zonal velocity $\stackrel{\mathrm{‾}}{u_{i_{*},j,k}}$ in metres per second at a downward position of x = 5 D with index i_* behind the rotor of the non-rotating EBL wind-turbine simulation without wind veer in (a) and the non-rotating SBL wind-turbine simulation with veering inflow in (b). The blue contour represents the circumference of the actuator.

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The inflow profiles from Fig. 3a predict a different wake behaviour under veering inflow in comparison to no wind veer, regardless of the rotational direction. Therefore, different wake characteristics should prevail in both non-rotating simulations NR. Figure 7 represents y–z cross sections of the streamwise velocity at x = 5 D for the non-rotating wind-turbine simulations with a non-veered inflow in the evening in (a) and a veering inflow in a stable regime in (b). In the case of no veer, the wake at x = 5 D retains the shape of the rotor (Fig. 7a). In the case of a veering inflow, the wake in the lower rotor half shifts to the north (Fig. 7b). The striking difference in the lower rotor part corresponds to the inflow profiles of Fig. 3a, where a veering inflow is characterized by a southern component for z < 100 m in the SBL, whereas the meridional inflow velocity is ≈ 0 m s⁻¹ in the EBL in all rotor heights. The skewed wake is only due to the veering inflow, as it also occurs without a rotating rotor. This simulated structure resembles those of the simulations of Abkar and Porté-Agel (2016), Vollmer et al. (2017), Bromm et al. (2017), Churchfield and Sirnivas (2018), and Englberger and Dörnbrack (2018).

Figure 8Coloured contours of the streamwise velocity $\stackrel{\mathrm{‾}}{u_{i,j,k_h}}$ in metres per second at hub height k_h, averaged over the last 50 min, for EBL_CR in (a), SBL_CR in (b), EBL_NR in (c), SBL_NR in (d), EBL_CCR in (e), and SBL_CR in (f). The black contours represent the velocity deficit ${\mathrm{VD}}_{i,j,k_h}$ at the same vertical location.

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Figure 9Coloured contours of the streamwise velocity $\stackrel{\mathrm{‾}}{u_{i,j,k_{*}}}$ in metres per second at z_h = 125 m with index k_*, averaged over the last 50 min, for the same cases as in Fig. 8. The black contours represent the velocity deficit ${\mathrm{VD}}_{i,j,k_{*}}$ at the same vertical location.

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Figure 10Coloured contours of the streamwise velocity $\stackrel{\mathrm{‾}}{u_{i,j,k_{*}}}$ in metres per second at z = 75 m with index k_*, averaged over the last 50 min, for the same cases as in Fig. 8. The black contours represent the velocity deficit ${\mathrm{VD}}_{i,j,k_{*}}$ at the same vertical location.

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The evolution of the wake at specific altitudes also shows the effect of veer. The streamwise velocity appears at hub height of z_h = 100 m in Fig. 8, in the upper rotor part at z = 125 m in Fig. 9, and in the lower rotor part at z = 75 m in Fig. 10 for the EBL (no veer) wind-turbine simulations in the left column (a, c, e) and the SBL (veered) wind-turbine simulations in the right column (b, d, f) for a clockwise-rotating rotor CR in the first row, no rotation of the disc NR in the second row, and an counterclockwise-rotating rotor CCR in the third row.

According to the analysis of Sect. 3, disregarding the sign in Eq. (17), no significant wake differences are expected in the non-veering simulations of the EBL for CR and CCR. This is in agreement with the situation at hub height (Fig. 8a vs. e). Above and below hub height, differences emerge in the near wake (Figs. 9a, e, 10a, e), resulting from the opposite sign in CR and CCR in the upper and lower rotor parts (Eq. 17). The near wake deflects towards the north in the upper rotor half (Fig. 9a) and towards the south in the lower rotor half (Fig. 10a) in CR, whereas it deflects towards the north in the lower rotor part (Fig. 10e) and towards the south in the upper rotor part (Fig. 9e) in CCR. In contrast, the non-rotating EBL simulation NR does not show any near-wake deflection at any height (Figs. 8c–10c). The effect in the rotating actuator simulations is therefore caused by the rotation of the rotor and can be explained by a transport of higher momentum air counterclockwise (clockwise) in CR (CCR), with the opposite situation prevailing at 75 m.

Wake elongation also exhibits the impact of the rotor's rotation. Comparing NR to CCR and CR of the EBL regime, there is further a difference in the wake elongation, with a larger velocity deficit in NR at hub height (Fig. 8c) at the same downwind position in comparison to CCR and CR (Fig. 8e and a). The faster recovery is related to enhanced entrainment in the CR and CCR simulations as the rotation itself acts as a source of turbulence.

In the case of a veering inflow, the analysis of Sect. 3 predicts significant wake differences in the CR and CCR wind-turbine simulations of the SBL regime, which are not limited to the change of the sign in CR and CCR. The wake structures at hub height, presented in Fig. 8b and d for the CR and the NR simulations, are comparable regarding the elongation, the width, and the deflection angle. The wake structure in CCR (Fig. 8f) differs. Considering the black velocity deficit contours (Eq. 4), the wake recovers more rapidly and, especially in the far wake, it is broader with a slightly larger wake deflection angle towards the dashed centre line.

The same situation occurs in the upper rotor part at 125 m (Fig. 9). Under veering wind conditions, the wake characteristics are also comparable in CR (Fig. 9b) and NR (Fig. 9d), with a slightly less rapid wake recovery in NR due to no additional turbulence generated by disc rotation. Comparing CR (and NR) to CCR (Fig. 9f), the wake recovers much more rapidly in CCR, with a smaller velocity deficit in the near wake with VD_max≈ 0.4 in comparison to VD_max ≈ 0.6 in CR and NR. A comparison of NR EBL (Fig. 9c) to NR SBL (Fig. 9d) shows a wake deflection angle in the NR SBL simulation, resulting from a meridional wind component ≤ 0 m s⁻¹ in the upper rotor part (Fig. 3a). Comparing the rotating SBL cases (Fig. 9b, f) to NR (Fig. 9d), the wake deflection angle decreases in CR, whereas it increases in CCR. This deflection results from the counterclockwise transport of higher, momentum air in CR and the clockwise transport in CCR. Interestingly, although veer is constrained to the lower half of the rotor disc, these differences in the upper half of the wake still occur.

According to the analysis of Sect. 3, the effect of the rotational direction on v and w is predicted to be much larger in the lower rotor part, and this is also expected for u. Figure 10b, d, and f represent the situation in the lower rotor part at 75 m. CR (Fig. 10b) and NR (Fig. 10d) again show similar wakes; however, comparing CR (and NR) to CCR (Fig. 10f), the velocity deficit maximum, the elongation, the width, and the deflection angle of the wakes differ. Compared to the difference in the wake deflection angle in the near wake for CR and CCR in the EBL wind-turbine simulations (Fig. 10a and e), the wake deflection angle in the SBL wind-turbine simulations (Fig. 10b and f) is also influenced by the inflow wind direction angle. In the case of CR, the wake deflection angle decreases (Fig. 10b), whereas in CCR, it increases (Fig. 10f), similar as in the upper rotor part (Fig. 9b and f).

Figure 11Vertical and spanwise profiles of u (first two rows) and TI (last two rows) through the centre of the rotor at y = 0 D and z = 100 m for the SBL and the EBL wind-turbine simulations at x = 3 D (first column), 5 D (second column), 7 D (third column), and 10 D (fourth column) downwind and for both rotational directions CR and CCR.

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Figure 12Spanwise profiles of u (first two rows) and TI (last two rows) at z = 75 m in the first and third rows and at z = 125 m in the second and last rows for the SBL and the EBL wind-turbine simulations at x = 3 D (first column), 5 D (second column), 7 D (third column), and 10 D (fourth column) downwind and for both rotational directions CR and CCR.

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For a more qualitative investigation of the impact of the rotational direction on the wake under veering inflow, Fig. 11 shows vertical and spanwise profiles through the rotor centre of the streamwise velocity in the first two rows (a–h) and also of the total turbulence intensity (Eq. 6) in the last two rows (i–p) at x = 3 D, 5, 7, and 10 D for CR- and CCR-rotating turbines in the SBL (solid red (CR) and blue (CCR) lines) and the EBL (dashed red (CR) and blue (CCR) lines).

Without veering inflow, the difference in u and TI between CR and CCR is trivial. Further, the profiles are almost symmetric to the rotor centre lines of z = z_h and y = 0 D. However, under veering wind conditions in the SBL, significant differences emerge. Considering the upper rotor part up to z = 150 m in the vertical profiles of u (Fig. 11a–d) and TI (Fig. 11i–l), the downwind velocity in the case of CCR is larger in comparison to CR, whereas the total turbulence intensity is larger up to x = 7 D and smaller at x = 10 D. In the top-tip region of the rotor and for z ≥ 150 m, the total turbulence intensity is larger in the case of CR in comparison to CCR for x ≥ 5 D. To investigate the impact in the lower rotor part, a vertical profile through the rotor centre is not appropriate due to wake deflection. Therefore, spanwise profiles of u and TI at 75 m are shown in the first (a–d) and third (i–l) rows of Fig. 12, including the deflected wake towards the north at y < 0 D. Up to at least x = 7 D (Fig. 12a–c), the streamwise velocity in the deflected wake is larger in the CCR case in the southern rotor part with a minor difference between CR and CCR and slightly larger values in CR in the northern rotor part, whereas differences between the profiles have eroded by x = 10 D (Fig. 12d). The lateral position y_* of the wake centre at x =3, 5, 7, and 10 D at z = 75 m is shown in Fig. 10b for CR and in Fig. 10f for CCR. In the case of CCR at x = 3 D, $y_{*}\approx -\mathrm{1}/\mathrm{2}\mathrm{D}$ and at x = 7 D, $y_{*}\approx -\mathrm{1}\mathrm{D}$, with slightly smaller values of y_* in the case of CR, resulting from the smaller wake deflection angle in CR in comparison to CCR (compare Fig. 10b to f). Therefore, comparing u at the corresponding wake centre positions in Fig. 12a–d, u_CCR>u_CR for x≤10 D. The same tendency of a higher streamwise velocity in CCR in comparison to CR is valid in the whole wake region at 125 m (Fig. 12e–h), where the difference also decreases downwind.

Contrasts in turbulence intensity emerge, due to differences in entrainment. The total turbulence intensity around y = 0 D in the lower rotor part (Fig. 12i–l) shows the same relation between CR and CCR as in the upper rotor part (Fig. 12m–p) and at hub height (Fig. 11m–p), with TI_CCR>TI_CR up to x = 7 D. However, considering the difference in the deflected wake at z = 75 m (i–l), TI_CCR<TI_CR at x= 3 D for $y\le y_{*}=-\mathrm{1}/\mathrm{2}\mathrm{D}$ (Fig. 12i). Approaching downwind, the difference in TI between CR and CCR decreases for x ≥ 5 D (Fig. 12j–l). The much larger values of CCR at y = 0 D in the lower rotor half for x ≤ 7 D (Fig. 12i–k) can be explained with a broader wake width approaching the centre line in Fig. 10f in comparison to Fig. 10b representing CR. Approaching x =10 D (Fig. 12l), the profiles for CR and CCR become rather similar, no longer showing a rotational direction impact. Larger TI values in CCR in comparison to CR can be explained by a more rapid wake recovery (Figs. 8 and 10f vs. b) resulting from a larger entrainment rate in CCR in comparison to CR.

Summarizing, the rotational direction has no significant impact on u and TI in the case of no wind veer, represented by our EBL simulation. However, in the case of a veering wind in the SBL, a consistent downwind impact on u and TI emerges at all heights over the rotor, with larger u values and TI values in the case of CCR in comparison to CR.