3.1. Wake recovery, deflection and deficit budgets

To verify the proposed synthetic inflow methodology, we simulate a single wind turbine wake in turbulent inflow. The concurrent-precursor TNBL simulations in Heck and Howland (Reference Heck and Howland2025) are used as a reference, where the Rossby number $Ro_G = G/(f_c D)$ based on the geostrophic wind speed $G$ varies between $100 \text{ and } 1000$ . In the set of synthetic inflow simulations, labelled ‘Synthetic’ throughout this study, the mean wind speed and direction profiles from the TNBL simulations are imposed as inflow conditions in the synthetic inflow method. The rotor-averaged $\mathrm{TI}$ is the target in the $\mathrm{TI}$ controller, which matches the inflow turbulence at the rotor location. At the rotor plane, the wind veer $\alpha _v$ deviates slightly from the prescribed mean profile due to inflow adjustment. A summary of the inflow characteristics between the synthetic inflow and TNBL simulations is given in table 2.

Table 2.

Inflow properties of the verification simulations for the TNBL and synthetic simulations. All simulations are run at a latitude $\phi = {45}^{\circ }$ .

Table 2 Long description

The table compares inflow properties of verification simulations for TNBL and synthetic simulations. It has 6 rows and 13 columns. The columns are labeled as follows: Name, Ro, RoG, U h / G, α v (deg.), TI (%) TNBL simulations, ψ h (deg.), z 0 (m), α v (deg.), TI (%) Synthetic inflow simulations, ψ h (deg.), λ BP / D. The rows are labeled with different simulation names: R1000, R500, R250, R167, R125, R100. Each row contains specific values for each column header. The table provides a detailed comparison of various inflow characteristics between the synthetic inflow and TNBL simulations.

To compare wakes between the concurrent-precursor TNBL inflow and the synthetic inflow, we show the wake recovery and wake deflection in figure 3. The wake recovery is computed as the wake velocity deficit $\langle \overline {\Delta u}\rangle$ averaged within the streamtube. The wake deflection $y_c$ is computed as the geometric centroid of the streamtube position (Heck & Howland Reference Heck and Howland2025). Due to wind-angle drift in the TNBL, the wind angle at hub height ( $\psi _h$ ), measured in the rotor plane, is not exactly aligned with the rotor. Following Heck and Howland (Reference Heck and Howland2025), the linearised advection $\psi _h \cdot x$ from any misalignment between the inflow angle at the rotor hub height is subtracted to distinguish inflow-induced wake deflections from mean advection. Throughout this paper, we define deflections as positive when they deflect anticlockwise as viewed from above ( $y_c \gt 0$ in the chosen coordinate system) and negative when they deflect clockwise as viewed from above ( $y_c \lt 0$ ), following the convention of van der Laan and Sørensen (Reference van der Laan and Sørensen2017). Wake recovery and deflection for the synthetic and TNBL inflow simulations are shown spanning all Rossby numbers in figure 3.

Figure 3.

Comparison of ( $a$ ) the streamtube-averaged wake velocity deficit $\langle \overline {\Delta u} \rangle$ and ( $b$ ) wake centroid between the synthetic inflow simulations (dashed lines) and concurrent-precursor TNBL simulations (solid lines), plotted as a function of the inverse Rossby number $Ro^{-1}$ .

Figure 3 Long description

Two line graphs depict the streamwise evolution of the wake recovery, computed as the streamtube-averaged wake velocity deficit and the wake deflection, computed as the streamtube centroid. Lighter colours correspond with lower Rossby numbers and stronger Coriolis forcing. Panel A: the x-axis, labelled x/D, ranges from −3 to 15, and the y-axis ranges from −0.5 to 0. Panel B: the x-axis, labelled x/D, ranges from 0 to 15 and the y-axis, labeled yc/D, ranges from 0 to 0.2. A black dashed line shows the zero deflection threshold.

Overall, wake recovery agrees well between the synthetic inflow and TNBL inflow simulations across all Rossby numbers. The normalised mean absolute error (NMAE) between the synthetic and TNBL inflow for the streamtube-averaged wake velocity deficit is in the range $1\,\%{-}4\,\%$ , and pointwise discrepancies between the methods can reach $\approx 5\,\%$ of $U_h$ . Additionally, the magnitude and direction of wake deflections are qualitatively similar between the synthetic and TNBL inflow methods, reaching $\approx 0.2D$ at $Ro = 100$ . In both the synthetic and TNBL inflows, decreasing the Rossby number results in increasingly positive wake deflections monotonically, transitioning from positive to negative wake deflections at $Ro \sim 250$ . While the quantitative agreement between the TNBL and synthetic inflow-induced wake deflections is lower than the agreement in wake velocity deficit, we note that there is also a considerably larger spread of wake deflections due to unsteadiness in the inflow turbulence over time scales relevant to the ABL and to wind farms (Churchfield et al. Reference Churchfield, Wang, Scholbrock, Herges, Mikkelsen and Sjöholm2016; Heck & Howland Reference Heck and Howland2025).

While the wake velocity deficit and deflection agree well between the synthetic and TNBL inflows, it is critical to also compare the individual terms in the momentum balance that govern the wake dynamics. For example, error cancellation between terms in the streamwise momentum budget may result in similar wake recovery rates governed by different wake physics. In figure 4, we compare the streamwise momentum deficit terms from (2.3), averaged within the streamtube, as a function of streamwise position $x$ . Budget terms for the $\mathrm{R100}$ and $\mathrm{R500}$ simulations are shown.

Figure 4.

Streamwise evolution of the streamtube-averaged streamwise RANS budget terms for ( $a$ ) relatively weak Coriolis forcing $(\mathrm{R500})$ and ( $b$ ) relatively strong Coriolis forcing $(\mathrm{R100})$ .

Figure 4 Long description

Two line graphs depict the streamwise evolution of the streamtube-averaged streamwise RANS budget terms for different Coriolis forcing conditions. Panel A: The line graph shows the evolution for relatively weak Coriolis forcing. The x-axis is labeled x/D and ranges from 0 to 15. The y-axis is labeled deficit x-RANS term and ranges from -0.04 to 0.04. The graph includes multiple lines representing different terms such as Advection, Pressure gradient, Subgrid stress, Base advection, Coriolis, Residual, x-Reynolds stress, y-Reynolds stress, and z-Reynolds stress. The lines are color-coded and include both synthetic and TNBL inflow conditions. Panel B: The line graph shows the evolution for relatively strong Coriolis forcing. The x-axis is labeled x/D and ranges from 0 to 15. The y-axis is labeled deficit x-RANS term and ranges from -0.04 to 0.04. Similar to Panel A, this graph also includes multiple lines representing different terms and inflow conditions.

Figure 5.

Streamwise evolution of the streamtube-averaged lateral RANS deficit budget terms for ( $a$ ) relatively weak Coriolis forcing $(\mathrm{R500})$ and ( $b$ ) relatively strong Coriolis forcing $(\mathrm{R100})$ . RS, Reynolds stress; PG, pressure gradient.

Figure 5 Long description

Two line graphs depict the streamwise evolution of the streamtube-averaged lateral RANS budget terms for different Coriolis forcing conditions. Panel A: The line graph shows the evolution for relatively weak Coriolis forcing. The x-axis is labeled x/D and ranges from 0 to 15. The y-axis is labeled deficit y-RANS term and ranges from −0.005 to 0.005. The graph includes multiple lines representing different terms such as Advection, Subgrid stress, Base advection, Coriolis, Residual and Combined RS + PG. The lines are color-coded and include both synthetic and TNBL inflow conditions. Panel B: The line graph shows the evolution for relatively strong Coriolis forcing. The x-axis is labeled x/D and ranges from 0 to 15. The y-axis is labeled deficit y-RANS term and ranges from −0.005 to 0.005. Similar to Panel A, this graph also includes multiple lines representing different terms and inflow conditions.

The synthetic inflow method matches the streamwise wake dynamics of the concurrent-precursor TNBL simulations. After the streamwise pressure gradient forcing becomes negligible ( $x/D \approx 3$ ), turbulent momentum entrainment, as noted by the Reynolds stress divergence terms, constitutes ${\gt} 95\,\%$ of the wake recovery. We further split the Reynolds stress divergence into the three component directions to highlight the importance of vertical and lateral mixing in wake recovery. The vertical Reynolds stress gradients match well across all Rossby numbers, which are most strongly influenced by the vertical shear profile in each simulation. However, the agreement in lateral Reynolds stress gradients worsens slightly at lower Rossby numbers. As we show in § 3.2, the lateral Reynolds stress gradients are strongly influenced by the integral length scale of turbulence. The NMAE for the Reynolds stress forcing contributions and advection term is within $10\,\%$ for all cases, except for the $\mathrm{R100}$ lateral Reynolds stress forcing (shown in figure 4 b), which is underestimated by approximately $20\,\%$ , and which we explore further in § 3.2. Nonetheless, the mean advection and, by extension, wake recovery are well predicted in all cases (NMAE $\lt 8\,\%$ ).

Finally, we compare the streamtube-averaged forcing terms in the lateral momentum deficit budget, which influence the wake deflection. Lateral forcing terms as a function of the streamwise coordinate are shown in figure 5. For brevity, only the $\mathrm{R500}$ and $\mathrm{R100}$ budgets are shown. As with the streamwise momentum deficit budgets, agreement between the synthetic inflow and TNBL wakes is also excellent for the terms in the lateral momentum deficit budget. The Coriolis forcing term (purple) in figure 5 is matched because the wake velocity deficit is accurately predicted. The combined pressure gradient forcing and Reynolds stress divergence (red) is also in excellent agreement, and generally opposes the direct Coriolis force. Here, we show the sum of the pressure gradient forcing and Reynolds stress terms because a large component of the lateral pressure gradient forcing balances the lateral Reynolds stress divergence (cf. Pope Reference Pope2000). The agreement in wake budgets between the TNBL and the synthetic inflow indicates that the synthetic inflow method can be used to explore wake–ABL interactions and analyse wake dynamics, with comparable accuracy to the more computationally expensive concurrent-precursor method.

3.2. Turbulence-length-scale effects

In § 3.1, the exact mean wind velocity profiles and hub-height $\mathrm{TI}$ from the TNBL LES were prescribed to the synthetic inflow LES. However, the structure of the synthetic turbulence, which is independent of mean velocity and $\mathrm{TI}$ profiles, remains as a free parameter. By bandpass-filtering the turbulence in the HIT simulation before it is superimposed into the primary and empty simulations, we can explicitly modify and study the effects of the integral length scale of turbulence on the wake evolution. Here, we show two variations of the integral length scale, $L_0^x$ and $L_0^y$ , computed similarly to Vahidi and Porté-Agel (Reference Vahidi and Porté-Agel2024) based on the normalised autocorrelation of the streamwise velocity field in the precursor simulation ${u^B}'(x, y, z_h)$ in the streamwise and lateral directions, respectively. The integration is truncated at the point where the autocorrelation function first reaches zero. A summary of the simulations presented in this section is shown in table 3, alongside streamwise and lateral budget quantities, integrated within the streamtube.

Table 3.

Simulation parameters of a wind turbine at a turbine-diameter-based Rossby number $Ro = 100$ , varying the integral length scale. The TNBL case is the same as the $\mathrm{R100}$ case in table 2.

Table 3 Long description

The table presents simulation parameters of a wind turbine at a turbine-diameter-based Rossby number, varying the integral length scale. It includes inflow properties and integrated x-RANS budget quantities within the streamtube. The table has 5 rows and 8 columns. Column headers are: Case label, alpha_v (deg.), TI (percent), lambda_P/D, L_0^x/D, L_0^y/D, y-Reynolds stress, and z-Reynolds stress. Row 1: TNBL, 5.7, 3.3, blank, 0.42, 0.24, 0.059, 0.044. Row 2: Synth-L1, 5.2, 3.4, pi/2, 0.25, 0.11, 0.045, 0.041. Row 3: Synth-L2, 5.3, 3.3, 2pi/3, 0.32, 0.18, 0.053, 0.045. Row 4: Synth-L3, 5.3, 3.3, pi, 0.46, 0.29, 0.062, 0.043. Row 5: Synth-L4, 5.3, 3.3, 2pi, 0.58, 0.41, 0.067, 0.042.

The integral length scale ( $L_0$ ) impacts both wake recovery and wake deflections, shown in figure 6. The wake recovery rate increases as $L_0$ increases, while the wake deflection magnitude generally decreases as $L_0$ increases. To understand why the wake recovery is enhanced by increasing $L_0$ , we compute the turbulent entrainment of streamwise momentum, integrated within the streamtube from $x\in [0, 15]D$ , shown in table 3. The vertical Reynolds stress gradient (i.e. turbulent entrainment from aloft) is approximately constant as $L_0$ changes because the inflow wind profile is not affected by $L_0$ . However, the lateral Reynolds stress gradient (lateral turbulent entrainment of momentum) increases as $L_0$ increases due to enhanced wake meandering (Bourhis et al. Reference Bourhis, Messmer, Hölling and Buxton2025; Vahidi & Porté-Agel Reference Vahidi and Porté-Agel2024). The closest representation of the wake dynamics in the TNBL is a synthetic inflow simulation that matches $L_0^x$ and $L_0^y$ , which would reside between the Synth-L2 and Synth-L3 cases. While we can prescribe a high-pass cutoff wavelength of turbulence $\lambda _{BP}$ , the resulting integral length scales are a function of the flow as well. For example, the turbulence is elongated in the streamwise direction, resulting in $L_0^x \gt L_0^y$ , despite isotropic bandpass filtering, but the degree of anisotropy varies with $\lambda _{{BP}}$ . We find that $L_0^y$ is more strongly correlated with $\lambda _{{BP}}$ than $L_0^x$ (see table 3). Future work could improve upon the synthetic turbulence generation and modify the spectral content of the imposed fluctuations.

Figure 6.

( $a$ ) Wake recovery and ( $b$ ) wake deflection as a function of streamwise position for the inflow profiles approximating the TNBL $\mathrm{R100}$ reference case, varying the turbulent integral length scale $L_0$ .

Figure 6 Long description

Two line graphs depict the streamwise evolution of the wake recovery, computed as the streamtube-averaged wake velocity deficit and the wake deflection, computed as the streamtube centroid. Lighter colours correspond with larger integral length scales of turbulence. Panel A: the x-axis, labelled x/D, ranges from −3 to 15, and the y-axis ranges from −0.5 to 0. Panel B: the x-axis, labelled x/D, ranges from 0 to 15 and the y-axis, labeled yc/D, ranges from 0 to 0.3. A black dashed line shows the zero deflection threshold.

Wake deflections decrease with increasing $L_0$ for two reasons. First, the direct Coriolis forcing scales linearly with $\overline {\Delta u}$ and contributes to positive (anticlockwise) wake deflections in the northern hemisphere. As a direct result of the enhanced wake recovery, the magnitude of $\overline {\Delta u}$ decreases with increasing $L_0$ , and the direct Coriolis force contributes to smaller positive deflections. Second, because $v^B$ is negative aloft in positive wind veer, larger length scales of turbulence entrain more negatively flowing lateral momentum from higher aloft. As a result, the vertical gradient of $-\Delta \overline {v'w'}$ increases in magnitude as $L_0$ increases, further suppressing positive wake deflections. The net deflection is a summation of these competing forcings in the ABL.

3.3. Wake comparison between synthetic and SBL inflows

In this section, we provide a brief extension and comparison of the synthetic inflow method to wakes immersed in a stably stratified ABL (SBL). The SBL simulations are performed using the initialisation procedure of Shen et al. (Reference Shen, Liu, Lu and Stevens2024) with a grid resolution $(\Delta _x, \Delta _y, \Delta _z) = (20, 12.5, 6.25)$ m and $384 \times 256 \times 256$ grid points. We perform six simulations of cooling rates between $C_r = 0.1$ $\text{and } {0.5}\,\textrm {K h}^{-1}$ . As LES of the SBL is known to be sensitive to grid resolution (cf. Ghobrial et al. Reference Ghobrial, Stallard, Schultz and Ouro2025; Maronga & Li Reference Maronga and Li2022), Appendix B reports a horizontal refinement test for representative stable cases. Although grid refinement slightly alters the SBL precursor flow properties, the central conclusion that the synthetic inflow method reproduces the corresponding wake recovery behaviour is not sensitive to the SBL grid resolution. The SBL simulations have zero free atmosphere stratification following the initialisation from Shen et al. (Reference Shen, Liu, Lu and Stevens2024). All flows are driven by a geostrophic wind speed $G={12}\,\textrm {m s}^{-1}$ with a roughness length $z_0 = {0.1}\,\textrm {m}$ . The spin-up simulation is performed for ${11}\,\textrm {h}$ , followed by a short domain rotation to align the hub-height flow with the $x$ axis, and the concurrent-precursor simulation is run from hours $11$ to $20$ , following Klemmer and Howland (Reference Klemmer and Howland2024). Time-averaged statistics are computed after a two-hour transient period for the wakes to develop. The mean wind speed, direction and temperature profiles are then prescribed in the synthetic inflow simulation, and a Dirichlet boundary condition is used to enforce a time-dependent ground temperature. A summary of inflow characteristics for the SBL simulations and the stratified synthetic inflow simulations is given in table 4, and vertical ABL profiles are shown in Appendix B.

Table 4.

Parameters for SBL and stratified synthetic inflow simulations. All simulations are run at a latitude $\phi =45^\circ$ , and the turbine is placed at $z_h/D=0.625$ .

Table 4 Long description

Table with two columns: SBL simulations and Synthetic inflow simulations. Each column lists parameters for different cooling rates. Parameters include latitude, turbine placement, and various atmospheric conditions such as wind speed, direction, and temperature profiles.

In figure 7(a), we compare wake recovery in the SBL simulations with the wake recovery from the stratified synthetic inflow simulations, quantified with the streamtube-averaged velocity deficit. Overall, the wake recovery agrees between the methods, even at large cooling rates, with NMAE values between $1\,\%$ and $4\,\%$ (pointwise discrepancies are larger, reaching ${\sim} 10\,\%$ of $U_h$ ). One surprising observation is that wake recovery increases with increasing stability (increasing $C_r$ ), despite the suppressed length scales of turbulence $L_0$ and decreasing free-stream $\mathrm{TI}$ as $C_r$ increases (see table 4). We attribute the accelerated wake recovery with increasing stability, which opposes conventional intuition (cf. Abkar & Porté-Agel Reference Abkar and Porté-Agel2015), to enhanced turbulent mixing caused by the increasing wind veer ( $\alpha _v$ ), for which the physical mechanisms are discussed in depth in § 4.1.

Figure 7.

( $a$ ) Wake recovery and ( $b$ ) wake deflection as a function of streamwise distance compared between SBL and stratified synthetic inflow LES for varying cooling rates $C_r$ (in $\textrm {K h}^{-1}$ , see table 4).

Figure 7 Long description

Two line graphs depict the streamwise evolution of the wake recovery, computed as the streamtube-averaged wake velocity deficit and the wake deflection, computed as the streamtube centroid. Lighter colours correspond with lower cooling rates of the stably stratified boundary layer. Panel A: the x-axis, labelled x/D, ranges from −3 to 15, and the y-axis ranges from −0.5 to 0. Panel B: the x-axis, labelled x/D, ranges from 0 to 15 and the y-axis, labeled yc/D, ranges from −0.4 to 0.2. A black dashed line shows the zero deflection threshold.

In contrast with the good agreement in wake recovery between the stratified synthetic simulations and the SBL reference cases, the quantitative predictions of wake deflections are less strong. The qualitative shape and trends are captured, showing an initially positive wake deflection in the near wake, which becomes negative in the turbulent far wake. The quantitative mismatch in wake deflection between the stratified synthetic inflow simulations and the SBL simulations is driven by several factors. First, the magnitude of terms in the lateral momentum budget is much smaller than in the streamwise momentum budget, taking longer to converge. We opt to time-average from hours 13 to 20 in all cases, rather than a full inertial period, due to the transience in the stable boundary layer. As a result, the reference SBL wake deflection contains inherent variability from inertial oscillations. Second, the SBL simulations evolve with time, while the synthetic inflow simulations are statistically quasi-steady. Here, we are averaging over changes in wind direction in the SBL, which drifts due to surface cooling (cf. Shen et al. Reference Shen, Liu, Lu and Stevens2024). By contrast, the synthetic inflow case is statistically stationary, which is in part an advantage as it enables computing well-converged turbulence statistics.

In summary, the synthetic inflow method provides a simplified, controllable framework for investigating stratified ABL effects on wake dynamics, reproducing the flow physics of concurrent-precursor LES. Furthermore, we can use the methodology to study the effects of turbulence integral length scales, which do not affect profiles of wind speed, direction or $\mathrm{TI}$ , on wake dynamics. These results highlight an opportunity for studying stratified wake dynamics in a controlled environment unaffected by SBL transience, which we suggest exploring in future work. Replacing the ABL mean wind profiles from concurrent-precursor simulations with the simplified inflow parametrisation given in (2.4) also captures correct parametric trends in wake recovery and deflection, shown in Appendix C, which we will use to explore a wide range of ABL parameters.

4.1. Wake recovery regimes and streamwise dynamics

To begin, we investigate the effect of the ABL inflow on wind turbine wake recovery. We begin with data from Sweep A in table 1, covering the widest range of wind shear and wind veer. The streamtube-averaged wake velocity deficit is shown in figure 8 as a function of downstream distance. Within the ranges of parameters observed in the field, veer and inflow TI have the largest effect on the wake recovery rate. The influence of wind veer on wake recovery is largest at low $\mathrm{TI}$ . At high $\mathrm{TI}$ , wake recovery is generally faster for all cases than at lower $\mathrm{TI}$ , and the effects of wind veer are still present but diminished. The shaded regions in figure 8(a, b) show the spread in wake recovery due to the full variation in wind shear and $Ro$ .

Figure 8.

Streamtube-averaged wake velocity deficit as a function of downstream distance for ( $a$ ) $\mathrm{TI} = 3\,\%$ and ( $b$ ) $\mathrm{TI} = 8\,\%$ . The shaded region shows the full range of wake recovery across different values of shear and Rossby number. ( $c$ ) Near-wake length $x_0$ computed with the ensemble mean velocity deficit as a function of veer and $\mathrm{TI}$ .

Figure 8 Long description

The image contains three panels, each showing different types of graphs. Panel A and Panel B display line graphs, while Panel C shows a scatter plot. Panel A and Panel B illustrate the streamtube-averaged wake velocity deficit as a function of downstream distance for turbulence intensities of 3 percent and 8 percent, respectively. The x-axis represents the downstream distance normalized by the diameter (x/D), and the y-axis represents the normalized wake velocity deficit. Different lines represent various veer angles, with the shaded region indicating the full range of wake recovery across different values of shear and Rossby number. Panel C shows the near-wake length computed with the ensemble mean velocity deficit as a function of veer and turbulence intensity. The x-axis represents the veer angle in degrees, and the y-axis represents the normalized near-wake length. Different symbols represent different turbulence intensities, with circles for 3 percent and crosses for 8 percent. The graphs collectively demonstrate the influence of turbulence intensity and veer on wake recovery and near-wake length.

As the magnitude of the inflow wind veer increases, two effects synergistically enhance wake recovery. First, the near-wake length $x_0$ (i.e. the $x$ location of minimum wake velocity) decreases with increasing wind veer magnitude, as shown in figure 8(c). The dependence of $x_0$ on wind veer is stronger at low inflow $\mathrm{TI}$ , and is symmetric across positive wind veer and negative wind veer (also called backing, where winds turn to the left with increasing height in the northern hemisphere). As a result, the wake recovery begins closer to the rotor disk, and at a given point downstream, wake recovery is enhanced because the far wake has been turbulent for a longer distance. Although wakes simulated with the actuator line or actuator surface methods would better capture the wake core and tip vortex breakdown in complex inflow conditions (Li & Yang Reference Li and Yang2020), we expect the qualitative trend of decreasing $x_0$ within increasing veer magnitude to remain unchanged.

Second, the rate (slope) of wake recovery increases with increasing wind veer magnitude, which has also been observed in concurrent-precursor simulations (Abkar & Porté-Agel Reference Abkar and Porté-Agel2016; Churchfield & Sirnivas Reference Churchfield and Sirnivas2018). Transforming the wake recovery profiles into log–log space elucidates the accelerated rate of wake recovery due to wind veer. This is shown in figure 9(a, b). Empirically, we can approximate wake recovery rate using a power-law fit: $\langle \overline {\Delta u} \rangle / U_h = a (x/D)^{-n}$ , where $n$ is the power-law decay exponent and $a$ is a scaling factor (Högström et al. 1988; Vermeulen Reference Vermeulen1980; Stevens & Meneveau Reference Stevens and Meneveau2017). Here, the exponent $n$ is fit to the velocity deficit profile beyond the point where the concavity of the velocity deficit profile changes, which has been described as the start of the fully turbulent wake (Fei et al. Reference Fei, Nishino and Vogel2025). We fit to the ensemble mean across all shears and Rossby numbers (solid lines in figure 8 a, b) for simplicity. The resulting power-law decay exponents are shown in figure 9(c), where increasing the power-law wake decay exponent increases symmetrically with increasing veer magnitude. The range of power-law wake decay rates that we fitted from the synthetic inflow simulations falls within the range of reported values from wind tunnel experiments (Chu & Chiang Reference Chu and Chiang2014) and in field campaigns (Högström et al. 1988; Iungo & Porté-Agel Reference Iungo and Porté-Agel2014).

Figure 9.

( $a$ , $b$ ) Negative of figures 8(a) and 8(b) except in log–log axes. Dashed lines are power-law fits to the ensemble mean (across shear and Rossby number) velocity deficit profiles as a function of veer and $\mathrm{TI}$ . ( $c$ ) Power-law decay exponent computed by power-law fits to ensemble mean profiles.

Figure 9 Long description

The image contains three panels, each showing different types of graphs. Panel A and Panel B display line graphs, while Panel C shows a scatter plot. Panel A and Panel B illustrate the streamtube-averaged wake velocity deficit as a function of downstream distance for turbulence intensities of 3 percent and 8 percent, respectively. The x-axis represents the downstream distance normalized by the diameter (x/D) in logarithmic spacing, and the y-axis represents the normalized wake velocity deficit in logarithmic spacing. Different lines represent various veer angles, with the shaded region indicating the full range of wake recovery across different values of shear and Rossby number. Panel C shows the power-law decay exponent computed with the ensemble-mean velocity deficit as a function of veer and turbulence intensity. The x-axis represents the veer angle in degrees, and the y-axis represents the power law decay exponent. Different symbols represent different turbulence intensities, with circles for 3 percent and crosses for 8 percent.

To explain the increasing wake recovery with increasing veer magnitude, we analyse the streamwise momentum deficit budget. Specifically, the wake velocity deficit is the integral of the advection term (Heck & Howland Reference Heck and Howland2025). In figure 10, we show the parametric dependence of the integrated Reynolds stress contributions within the streamtube from the rotor to $x/D=15$ downstream. As the wind veer magnitude increases, the total vertical entrainment of momentum increases (figure 10 a), while the lateral entrainment of momentum only slightly decreases (figure 10 b). The sum of the Reynolds stress divergence terms, shown in figure 10(c), increases with increased inflow wind veer magnitude. Relative to the zero veer case, the increased turbulent flux of momentum increases the mean advection with increasing veer magnitude, shown in figure 10(d). To summarise, the enhanced rate of wake recovery in veered inflow conditions, quantified by the power-law decay exponent $n$ , is caused by an increase in turbulent momentum entrainment from aloft.

Figure 10.

( $a$ – $c$ ) Streamtube-integrated Reynolds stress contributions as a function of wind veer at the rotor from the ( $a$ ) vertical and ( $b$ ) lateral directions, and ( $c$ ) combined. ( $d$ ) Streamtube-integrated advection, relative to the equivalent inflow but without veer, versus mean streamtube aspect ratio. ( $e$ ) Correlation of wind veer and mean streamtube aspect ratio. ( $f$ ) Correlation of the ratio of vertical/lateral Reynolds stress contributions to the mean aspect ratio of the streamtube $y/z$ .

Figure 10 Long description

Panel A: A scatter plot showing the integral of the vertical Reynolds stress contribution as a function of wind veer in degrees. The x-axis is labeled ||Veer|| (deg) and the y-axis is labeled ∫_γ RS_z dγ. The data points are color-coded by the magnitude of shear and use different symbols to represent turbulence intensity (TI) and stable boundary layer (SBL) conditions. Panel B: A scatter plot showing the integral of the lateral Reynolds stress contribution as a function of wind veer in degrees. The x-axis is labeled ||Veer|| (deg) and the y-axis is labeled ∫_γ RS_y dγ. The data points are similarly color-coded and symbolized as in Panel A. Panel C: A scatter plot showing the combined integral of the vertical and lateral Reynolds stress contributions as a function of wind veer in degrees. The x-axis is labeled ||Veer|| (deg) and the y-axis is labeled ∫_γ (RS_y + RS_z) dγ. The data points are color-coded and symbolized as in the previous panels. Panel D: A scatter plot showing the ratio of advection with and without veer as a function of the mean streamtube aspect ratio. The x-axis is labeled ||Veer|| (deg) and the y-axis is labeled Adv. / Adv.(0 Veer). The data points are color-coded by different shades of blue. Panel E: A scatter plot showing the correlation between wind veer and the mean streamtube aspect ratio. The x-axis is labeled ||Veer|| (deg) and the y-axis is labeled Streamtube aspect ratio y/z. The data points are color-coded by different shades of purple. Panel F: A scatter plot showing the correlation between the ratio of vertical to lateral Reynolds stress contributions and the mean aspect ratio of the streamtube. The x-axis is labeled Streamtube aspect ratio y/z and the y-axis is labeled RS aspect ratio RS_y/RS_z. The data points are color-coded and symbolized as in the previous panels, with a dashed line indicating a linear relationship.

It is well established that wakes in veered conditions will elongate and skew as different vertical levels of the wake are exposed to changing directions of inflow (Abkar & Porté-Agel Reference Abkar and Porté-Agel2016; Magnusson & Smedman Reference Magnusson and Smedman1994). The height of the wake remains nearly the same, but the wake is elongated in the lateral direction. Here, we measure the aspect ratio of the wake as the ratio of the maximum extents of the streamtube in the lateral to the vertical directions. Due to wake skewing, which is strongly correlated with the inflow veer (figure 10 e), the elongated wakes have a greater surface area. This allows for increased turbulent entrainment of momentum from above. In figure 10(f), we compare the ratio of integrated vertical to lateral momentum entrainment (labelled $\mathrm{RS}_z/\mathrm{RS}_y$ ) to the streamtube aspect ratio $y/z$ . For the $\mathrm{TI}=3\,\%$ simulations and the concurrent-precursor SBL simulations in § 3.3, the correlation between the geometric aspect ratio of the wake and the ratio of turbulent stresses is nearly 1:1, indicating that the geometric effect of wake skewing enhances wake recovery.

In this exploration, we have observed that wind turbine wake recovery is most strongly influenced by the inflow wind veer and the TI. Additionally, from § 3.2, the integral length scale of turbulence also has a leading-order role in wake recovery. To conclude this section, we explore parameter regimes of wake recovery using the synthetic inflow methodology to sweep over ranges of the three variables with the largest influence on wake recovery: wind veer ( $\alpha _v = 0 {-} 30^\circ$ ), $\mathrm{TI} = 2\,\% {-} 10\,\%$ and integral length scale ( $\lambda _{{BP}} = \pi /3, 2\pi$ ). The exact simulation inputs are given in Sweep B in table 1. Because the inflow properties influence the wake dynamics in several ways (near-wake length, power-law decay constant, etc.), we introduce a simple metric that encodes all of these effects into a practical measure of wake recovery: the distance $L_{{wake}}$ at which the wake velocity deficit $\langle \overline {\Delta u}\rangle$ recovers to $25\,\%$ of its maximum value in magnitude. For cases where the wake velocity deficit magnitude does not reach $25\,\%$ of its maximum value inside the simulation domain, a power-law fit is used to extrapolate the value of $L_{{wake}}$ , which is justified by the goodness-of-fit in figure 9(a, b). Contours of wake recovery regimes are shown in figure 11, where a Gaussian process (GP) regressor is used to interpolate between the LES data points using $\alpha _v$ , $\mathrm{TI}$ and $L_0^y$ as inputs. The LES cases from Sweep A are used in the out-of-sample evaluation for the GP accuracy ( $R^2 = 0.86$ ).

Figure 11.

Regimes of wake length $L_{{wake}}$ (distance where the wake recovers to $25\,\%$ of the maximum deficit magnitude) plotted as a function of veer and hub-height TI, parsed between ( $a$ ) relatively small integral length scale inflow ( $L^y_0/D = 0.17 \pm 0.03$ ), ( $b$ ) out-of-sample testing ( $R^2 = 0.86$ ) and ( $c$ ) large integral length scale inflow ( $L^y_0/D = 0.40 \pm 0.05$ ). Contours are interpolated using GP regression.

Figure 11 Long description

Three contour plots show the dependence of the wake length Lwake, normalised to the turbine diameter D, on the turbulence intensity from 2-12 percent and wind veer magnitude from 0-30 degrees for three turbulence length scales. Darker colours concentrated in the lower-left corner represent longer Lwake up to Lwake/D = 30. Scattered points are also colored with the same colour mapping and represent specific LES simulations.

The comprehensive simulations performed here allow the wake recovery to be mapped across the atmospheric conditions in which wind turbines will operate. In figure 11, we study three key ABL flow variables for wake recovery: (i) $\mathrm{TI}$ ; (ii) integral length scale of turbulence; and (iii) wind veer. At lower $L_0/D$ , the wake length is more sensitive to wind veer and $\mathrm{TI}$ than at high $L_0/D$ . Similarly, at low $\mathrm{TI}$ , wind veer has a larger impact on the wake length than at high $\mathrm{TI}$ . We can contextualise these trends using specific turbine models in varying inflow conditions. For example, a large offshore wind turbine like the IEA 15 MW reference turbine ( $D \sim {240}\,\textrm {m}$ ) may experience a relatively small $L_0/D$ due to its size and operate in relatively low inflow turbulence intensities (cf. Bodini et al. Reference Bodini, Lundquist and Kirincich2020). From figure 11(a), the wake of these conditions is significantly influenced by the magnitude of wind veer in the inflow: varying the veer magnitude from $0^\circ$ to $12^\circ$ halves the wake length. By contrast, a smaller 100 m wind turbine will operate at relatively larger $L_0/D$ , where the effects of wind veer are significantly diminished, regardless of $\mathrm{TI}$ , shown in figure 11(c). The interplay between the turbulence length scale, $\mathrm{TI}$ and wind veer depends not only on the turbine size but also on the instantaneous inflow conditions. Quantifying the impact on wake recovery within the complex, multi-dimensional ABL parameter space elucidates key relationships between the wake evolution and the properties of the incident inflow, which may inform future wind farm design and control.

4.2. Wake deflection regimes and lateral dynamics

In this section, we investigate wake deflections induced by Coriolis effects, which depend parametrically on incident conditions. We simulate wakes for Sweep C inflows (table 1) over shear ( $\alpha _s = -0.1 \text{ to } 0.2$ ), veer ( $\alpha _v = -10^\circ \text{ to } 20^\circ$ ), $\mathrm{TI} =3\,\%, 8\,\%$ and Rossby number ( $Ro = 100, 1000$ ), at higher granularity in shear and veer than Sweep A used in § 4.1. Wake deflections ( $y_c$ ) at a fixed streamwise distance $x/D = 10$ are shown in figure 12 as a function of inflow variables.

Figure 12.

Wake deflections at $x/D=10$ , measured by the centroid of the streamtube, for ( $a$ – $c$ ) $3\,\%$ TI and ( $d$ – $f$ ) $8\,\%$ TI. Columns plot the same wake deflection data against different inflow variables: ( $a{,}\, d$ ) deflection versus veer; ( $b{,}\, e$ ) deflection versus shear; ( $c{,}\, f$ ) deflection versus product of shear and veer. All subplots: light $\circ$ for $Ro = 100$ , dark $\times$ for $Ro = 1000$ .

Figure 12 Long description

Six scatter plots show wake deflections at a downstream distance x/D = 10 on the y-axis, ranging from −0.1 to 0.3, for varying inflow conditions, plotted against different inflow variables. All subplots show scatter plots split based on the Rossby number. Light points denote the strong Coriolis Ro = 100 inflow, and dark crosses denote the weak Coriolis Ro = 1000 inflow. The first row shows LES data from TI = 3% inflows and the second row shows TI = 8% inflows. The x-axis changes between the three columns: column 1 (panels A, D): wind veer, ranging from −10 to 20 degrees; column 2 (panels B, E): wind shear, ranging from −0.1 to 0.2; column 3 (panels C, F): combined shear times veer, ranging from −2 to 4 degrees.

Wake deflections induced by the simulated inflow conditions may be positive or negative in sign. There is a net shift towards increasingly positive wake deflections across all shear and veer conditions from the high-Rossby-number inflows (weak Coriolis, $\times$ markers) to the low-Rossby-number inflows (strong Coriolis, $\circ$ markers). In figures 12(a, b) and 12(d, e), we plot deflections against the wind veer and wind shear, respectively. Notably, a wide spread of wake deflections may be observed for a fixed amount of inflow veer, due to varying inflow shear, particularly in the $\mathrm{TI}=3\,\%$ simulations. The same is also true for shear, where varying the inflow veer also leads to a spread of values $y_c$ for a fixed amount of wind shear. Rather than parsing the effects of shear and veer separately, figures 12(c) and 12(f) show wake deflections at $x/D = 10$ plotted against the product of the wind shear and wind veer ( $\alpha _s \cdot \alpha _v$ ), which elucidates a novel scaling relationship between the inflow and the observed deflection. The observed wake deflections are strongly a function of this combined parameter rather than shear and veer independently, particularly at $\mathrm{TI} = 3\,\%$ . At $\mathrm{TI}=8\,\%$ , we observe that the collapse of wake deflections worsens, and deflections are generally smaller and approximately constant at a fixed $Ro$ across variations in shear or veer.

To investigate the cause of the collapse in the wake deflections, we consider the lateral deficit RANS equations (2.3). As is done in Heck and Howland (Reference Heck and Howland2025), we can reconstruct the individual forcing contributions to the net wake deflection by integrating the forcing terms in the lateral momentum budget ( $f_{y,j}$ ) in a Lagrangian frame of reference:

(4.1) \begin{equation} y_c(x) = \sum _j \int _0^x \frac {1}{\langle \overline {u} \rangle (x')} \int _0^{x'} \frac {1}{\langle \overline {u} \rangle (x'')} \langle \overline {f}_{y, j} \rangle (x'') \,\text{d}x'' \,\text{d}x'. \end{equation}

The observed wake deflection is the integral of the advection term in the full lateral RANS budget $\bar {u}_j \cdot \tfrac {\partial \overline {v}}{\partial x_j}$ . Assuming the advection term in the background flow to be negligible ( $\overline {u_j^B} \cdot \tfrac {\partial \overline {v^B}}{\partial x_j} = 0$ , e.g. horizontally homogeneous flow), the full RANS advection term is equal to the sum of the advection and base advection terms in the deficit RANS budget in (2.3). In figure 13(a), the measured wake deflection at $x/D = 10$ downstream is shown as a function of $\alpha _s \cdot \alpha _v$ and the Rossby number at $\mathrm{TI} = 3\,\%$ , which matches the sum of the advection and base advection terms (labelled ‘Integrated deflection’ in figure 13 b). We show the contributions from the Reynolds stresses, lateral pressure forcing and Coriolis forcing in figure 13(c–e), integrated within the streamtube from the rotor to $x/D = 10$ .

Figure 13.

Lateral wake deflection and dynamics, plotted against the product of the local inflow shear and veer at the rotor $\alpha _s \cdot \alpha _v$ for an inflow $\mathrm{TI}=3\,\%$ and $x/D=10$ . ( $a$ ) Wake deflection computed as the centroid of the streamtube seeded at the rotor. ( $b$ ) The integrated advection and base advection terms equal the observed wake deflection. ( $c$ – $e$ ) Streamtube-averaged budget terms integrated using (4.1) showing individual forcing contributions to the observed deflection from ( $c$ ) lateral Reynolds stresses, ( $d$ ) lateral pressure forcing and ( $e$ ) direct Coriolis forcing.

Figure 13 Long description

Five scatter plots show wake deflections (yc) and contributions to yc from Reynolds stresses, pressure forcing, and the Coriolis force at a downstream distance x/D = 10 on the y-axis, ranging from -0.1 to 0.3, for varying inflow conditions, plotted against different inflow variables. All subplots show scatter plots split based on the Rossby number. Light points denote the strong Coriolis Ro = 100 inflow, and dark crosses denote the weak Coriolis Ro = 1000 inflow. All LES data is for an inflow TI = 3%. The shared y-axis for all subfigures is the contribution to the wake deflection yc, normalised to the turbine diameter D, and ranges from -0.7 to 0.7. Panel A: observed wake deflection. Panel B: sum of forcing terms (Panels C-E). Panel C: Reynolds stress contribution. Panel D: Pressure force. Panel E: Coriolis force.

The explanation for the collapse of the wake deflection to the axis $\alpha _s \cdot \alpha _v$ is given by the individual momentum budget terms. Specifically, the wake deflection contribution from the Reynolds stress divergence, shown in figure 13(c), collapses with an approximately linear scaling when plotted against the product $\alpha _s \cdot \alpha _v$ . The lateral pressure force balances the dominant forcing in the flow: at low $\alpha _s \cdot \alpha _v$ , the pressure opposes the Coriolis forcing, but as $\alpha _s \cdot \alpha _v$ increases, the pressure also collapses with approximately linear scaling as a function of $\alpha _s \cdot \alpha _v$ . That is, the sum of the lateral pressure force and Coriolis force scales with $\alpha _s \cdot \alpha _v$ (not shown), even though the Coriolis contribution to $y_c$ instead scales with veer alone. Overall, this lateral dynamics contributes to the collapse of wake deflections onto the combined variable $\alpha _s \cdot \alpha _v$ , the product of shear and veer, particularly at low $\mathrm{TI}$ .

Finally, we identify regimes of wake deflections as a function of inflow parameters using the synthetic turbulent inflow methodology. Using the wide range of ABL parameters in Sweep A and Sweep C, totalling 540 simulations, we train a GP to estimate the wake deflection as a function of the combined shear parameter $\alpha _s \cdot \alpha _v$ , $Ro$ and $\mathrm{TI}$ . Wake deflection contours are shown at varying turbulence intensities in figure 14. A separate testing dataset (Sweep D in table 1) is used to evaluate the efficacy of the GP ( $R^2 = 0.82$ ), shown in figure 14(b).

Figure 14.

Contours of wake deflection at $x/D=10$ as predicted by the GP regressor with LES points overlaid, for ( $a$ ) $\mathrm{TI} = 3\,\%$ , ( $b$ ) $\mathrm{TI} = 4\,\%$ (out-of-sample) and ( $c$ ) $\mathrm{TI} = 8\,\%$ . Coriolis forcing strength increases as $Ro^{-1}$ increases.

Figure 14 Long description

Three contour plots show the dependence of the wake deflection yc, normalised to the turbine diameter D, on the Coriolis strength, represented by the inverse Rossby number 1/Ro ranging from 0 to 0.015, and the combined shear times veer parameter, ranging from -2 to 6 degrees. Three turbulence intensities (3, 4, 8%) are shown in panels A, B, C, respectively. A diverging colour palette shows positive deflections in red hues and negative deflections in blue hues. The bifurcation region of zero deflection is white, and accented with a bold black line labelled with a zero.

The contours delineate an interface between positive and negative wake deflections. Positive $\alpha _s \cdot \alpha _v$ corresponds to the canonical Ekman spiral; thus ABL winds tend to induce negative wake deflections in the northern hemisphere. As the Rossby number decreases, direct Coriolis forces become increasingly relevant, and wake deflections become increasingly positive. The critical Rossby number, which delineates the boundary between positive and negative deflections, depends on the incident flow. In general, higher wind shear and wind veer decrease the critical Rossby number. Note, however, that negative combinations of $\alpha _s \cdot \alpha _v$ also contribute to positive wake deflections. That is, if either the wind shear $\alpha _s$ is negative or the wind veer $\alpha _v$ is negative – but not both – then the shear- and veer-driven deflection can work with the direct Coriolis forcing, resulting in large positive deflections. Finally, at zero wind shear or zero wind veer, and for $Ro^{-1} \to 0$ (i.e. negligible Coriolis forcing), no wake deflections are expected. That is, wind veer in the absence of wind-speed shear (or vice versa) is insufficient to create inflow-induced wake deflections. At elevated $\mathrm{TI}$ , the collapse of wake deflections with the combined shear parameter $\alpha _s \cdot \alpha _v$ is weaker, as highlighted by figure 12(f). However, for brevity, we do not include the separated effects of wind shear and wind veer on wake deflections here.

The wake deflection regimes identified by the synthetic inflow methodology may lead to an improved understanding of wake deflection mechanisms and modelling. Coriolis-induced lateral wake deflections can cause an asymmetry in the power production of downwind turbines (Bromm et al. Reference Bromm, Vollmer and Kühn2017) and also affect the optimal wind farm control strategy (Nouri et al. Reference Nouri, Vasel-Be-Hagh and Archer2020). However, modelling inflow-induced deflections is challenging due to the influence of competing forces, including pressure gradients and turbulence entrainment. Simplified scaling relationships, such as the collapse of wake deflections onto the product of shear and veer $\alpha _s \cdot \alpha _v$ , are useful for guiding improvements in wake modelling that can inform decisions on wind farm design and control.