The experiments are performed in the W-tunnel at TU Delft Aerospace Engineering Laboratories. The W-tunnel is an open-jet wind tunnel with an adjustable test section at the outlet. In this experiment, the test section at the exit has dimensions 0.6m×0.6m, which is large enough for deflected flow as a result of yaw and wind veer to remain within the streamtube. The measurements are carried out at the free-stream velocity Uref=10.3 m s⁻¹. At this flow velocity, the Reynolds number based on disc diameter is approximately 68 000, and the turbulence in the undisturbed flow is of the order of 0.5 %. A rough schematic sketch in Fig.  shows the top view of the experimental setup employed in the current study to measure wake cross-section (Fig. a) and wake propagation in streamwise planes (Fig. b). Figure  shows a photograph of the experimental setup for cross-stream plane measurements, with key components labeled. It should be noted that in the laboratory setup, the disc is yawed with respect to the y axis of the coordinate system shown in Fig. 2, resulting in a physical wake deflection in the z direction. To measure streamwise wake evolution, the laser measurements are therefore performed in the xz plane of the laboratory coordinate system. For data analysis and presentation, the coordinate system is rotated in post-processing to follow the standard convention in the literature, where yaw occurs about the z axis and the wake deflects laterally in the y direction.

A uniform porous disc of diameter 10 cm and porosity 0.6 was 3D printed to replicate the effects of a wind turbine. Load measurements were performed to determine the thrust coefficient of the porous disc for the no-veer case and non-yawed conditions using a KD24S 2N (ME-Meßsysteme GmbH) force sensor with an accuracy of 0.1 % of the full-scale value (2N). Based on the load measurements, the thrust coefficient (CT) of the porous disc is approximately 0.69, which also aligns well with the general trend of the porosity vs. CT curve given in the literature .

The airfoil shape used in wind veering vanes is a NACA 0014 with a chord (c) of 0.2 m. This airfoil was selected based on its symmetric shape and relatively high angle of attack at which stall occurs (i.e., α ∼ 14°). Two wind veering vanes are employed in this study: one producing a wind veer of ∼10° and the other a veer of ∼20° across the porous disc. The variation in the angle of attack with height is shown in the left of Fig. . The vanes are designed so that at the center of the porous disc, the angle of attack is zero and increases symmetrically in positive and negative directions above and below the disc center. This is achieved by twisting the airfoil in opposite directions from the center outward, extending to the edges of the wind tunnel exit. For instance, in the 10° veer configuration, the vane twist varied linearly with a gradient of 1° cm⁻¹ until the edges of the wind tunnel, resulting in a total wind veer of 10° across the porous disc (+5° at one edge of the disc and -5° at the opposite edge of the disc). This design approach reflects common assumptions in stable boundary layer simulations, where the wind veer angle is set to zero at hub height and varies in the opposite direction above and below the hub height. The porous disc is positioned ≈20 cm downstream of the exit of the wind veering vanes. The disc was placed relatively close to the tunnel exit because the facility operates in an open-jet configuration, which limits the maximum downstream distance at which stable wake measurements can be obtained before flow quality deteriorates.

However, such a continuous transition in angles from the disc center to the edges of the wind tunnel exit for the wind veer model of 20° would result in excessively high angles of attack, which are undesirable. To address this, the vane is twisted smoothly from 0° at the center of the disc to ±10° at 5 cm away from the center. Beyond this point, the twist does not increase further – instead, the vane keeps a constant angle of ±10° all the way out to the edges of the wind tunnel exit (see orange line in the left Fig. ). This configuration ensures a smooth gradient of angle across the span of the guiding vane, as shown in Fig.  (left); the corresponding model installed at the tunnel exit is shown in Fig.  (right).

In the present study, stationary vanes were designed to produce wind veer; however, active grids represent another promising avenue for generating veered inflows . While these grids are commonly used for generating shear flows, gusts, and homogeneous and isotropic turbulence, they theoretically possess the capability to generate wind veer. By switching from standard counter-rotating shaft protocols – which are designed to neutralize deflection – to co-rotating adjacent shafts, the system could effectively steer the flow. Alternatively, Multi-fan wind tunnels (MFWTs) offer a state-of-the-art solution that overcomes the mechanical constraints of physical grids. MFWTs can achieve directional flow by employing multi-directional driving models, which selectively activate fan pairs oriented toward the desired flow angle .

The test matrix of the measured cases is summarized in Table . As it is of interest to investigate varying degrees of wind veer and its impact on wake recovery, two veered inflows are examined in addition to the reference uniform inflow (no-veer) for comparison. Initially, flow characterization is conducted in the absence of the porous disc for all three inflows (uniform flow, 10° veer, and 20° veer). Subsequently, wake measurements are performed in the presence of the disc at four cross-stream planes and three streamwise planes, as illustrated in Fig. , for different yaw angles of the disc. In this study, yaw misalignment is applied only in the direction of the inflow veer (positive yaw). This is motivated by the fact that spanwise velocity induced by negative yaw angles counteracts the direction of veer that can restrict the wake to rotor frontal projection , whereas positive yaw acts constructively with veer to enhance wake deflection . Other LES-based wind farm studies also show that negative yaw results in overall power losses in the Northern Hemisphere .

Due to the combined effect of turbine yaw and wind veer, all three velocity components of the flow become relevant. Consequently, stereoscopic particle image velocimetry (SPIV) was employed in the present study for wake flow measurements in the cross-stream and streamwise directions. In the past, SPIV has been commonly used for wake measurements of yawed HAWTs and also for VAWTs . In the present experiment, the seeding is done via a SAFEX smoke generator, which releases smoke in the form of water-glycol fluid particles of average diameter of 1 µm and a particle density of 103 kg m⁻³. The field of view (FOV) is illuminated by the Quantel Evergreen double-pulsed laser operating at a wavelength of 532 nm and delivering 200 mJ of energy per pulse. The thickness of the illuminated laser sheet is approximately 4 mm. Finally, images are captured using two LaVision sCMOS cameras (2560 × 2160 px, pixel pitch of 6.5 µm px⁻¹) at a frequency of 15 Hz positioned on the opposite side of the laser sheet, as depicted in Fig. . A Scheimpflug adapter was used to adjust the focus of the camera plane to the measurement plane.

SPIV works on the principle of stereoscopic imaging, where two cameras simultaneously record the same image plane at different angles over two very closely spaced time intervals. The two views allow the extraction of out-of-plane motion of particles, along with the in-plane displacement of the tracer particles. Once the images are captured, they are divided into small interrogation windows, and cross-correlation techniques determine particle displacement within each window . The resulting FOV is trapezoidal in shape, with a width and height of approximately 35.5 cm at a camera angle of 78.84°, using AF Micro-Nikkor lenses of 105 mm focal length set at an aperture of f#=8. Time steps of 45 and 225 µs between consecutive images are used for cross-stream and streamwise measurements, respectively. Table  summarizes the main parameters of SPIV employed in this study. A total of 100 vector fields were captured for each measurement plane and averaged. Vector calculation was performed using a multi-pass stereo cross-correlation approach with decreasing interrogation window sizes. The first pass employed a 128 × 128 px (18 × 18 mm) window size, followed by a second pass using 64 × 64 px window size (9 × 9 mm) with an overlap factor of 50 %, resulting in a final vector grid resolution of 32 pixels (4.5 mm).

The two cameras and laser are rigidly mounted on the traverse system with a spatial accuracy of 0.001 mm. The traverse system is capable of translational motion in both streamwise and spanwise directions. The spanwise measurements (i.e., wake cross-sections) at multiple downstream locations ranging from x/D=2 to x/D=8 are captured by moving the traverse system in the streamwise direction (shown in Fig. a). Meanwhile, streamwise wake measurements are taken across three different FOVs, with overlapping regions between consecutive imaging planes to ensure a smooth transition in wake propagation downstream, as illustrated in Fig. c. The streamwise FOVs are stitched together during the postprocessing of results by applying a Gaussian function in the overlapping region to ensure the gradients are smoothed out. It should be noted that the rotation stage that holds the porous disc is fixed at its location and is not mounted on the traverse system. Additionally, a new calibration was performed after the experimental setup was rotated by 90° to measure streamwise velocity fields.

Following the work of , the uncertainty in velocity components, along with the derived quantities, is discussed in this section. Uncertainty quantification is crucial in PIV, particularly for an experiment as complex as this. It provides an estimation of the range that likely contains the true value of the variable of interest, thereby enhancing the reliability and interpretability of the results. The uncertainty in the time-averaged streamwise velocity can be expressed as 1Uu‾=kσuN, where σu is the standard deviation of the streamwise velocity, k is the 95 % confidence interval, i.e., 1.96, and N is the total number of instantaneous images (N=100). Similarly, uncertainties are calculated for in-plane velocity components (v‾ and w‾). Moreover, the uncertainty in derived quantities, such as vorticity, can be expressed as follows: 2Uωx=Uv‾orw‾d1-ρ(2d), where Uv‾orw‾ is the uncertainty in mean in-plane velocity components, d is the grid spacing between the consecutive interrogation windows (here, d=4.54 mm), and the cross-correlation factor ρ(2D) is approximated to be 0.45 . In addition to the uncertainties in velocity components and vorticity, uncertainty in Reynolds normal stress and turbulent kinetic energy (TKE) is also computed as follows: 3UR,uu=σu22N-1,4UTKE=12UR,uu2+UR,vv2+UR,ww2, where σu is the standard deviation of streamwise velocity and TKE = 12(ui′ui′‾)=12(Ruu+Rvv+Rww).

Table  summarizes the maximum uncertainty values for various variables across the three inflow conditions. As expected, the uncertainty in the veer inflow cases is slightly higher compared to the no-veer case, due to additional complexity in the flow introduced by vane-induced veer. We also estimated the standard uncertainty associated with the turbulent kinetic energy Uk as described in : 5Uk=u′u′‾2+v′v′‾2+w′w′‾2⋅12N. Therefore, the error in TKE computation is of the order of 12N. For N=100, the error in TKE is around 7%. Please note that the uncertainty values reported correspond to the shear region of the wake (at x/D=5), where the flow is heavily separated.

For a complex system used to generate wind veer, such as the one employed in this study, it is essential to perform flow characterization in the absence of the porous disc. Ensuring a stable flow across all measurement planes is essential for obtaining reliable and consistent results in a wind tunnel experiment. Figure  shows the mean streamwise, spanwise, and vertical velocities, along with turbulence intensity and wind veer variation for all three inflow conditions. The reference velocity (Uref) used to normalize the velocity components is defined as the average velocity over the disc area located at x/D=0. For the uniform inflow case, Uref= 10.3 m s⁻¹. It is important to note that the placement of veering vanes at the tunnel exit induces a slight flow acceleration downstream of the vanes. As a result, for the veered inflow of 10°, Uref=11.43 m s⁻¹, while for the veered inflow of 20°, Uref=10.85 m s⁻¹. Uref is defined as the area-averaged velocity magnitude (U‾=u2+v2+w2) over the disc projection area in the absence of the porous disc.

The streamwise velocity profile for the clean (uniform inflow) case remains consistent across all downstream locations (see Fig. a). A minor spanwise velocity component of approximately 2 % is observed, which is attributed to the inhomogeneities within the wind tunnel. Additionally, the uniform inflow also has a persistent nonzero vertical velocity (see Appendix  for v‾ and w‾ velocity contours in the yz plane). The turbulence intensity within the disc region remains below 1 % at all streamwise planes, indicating highly stable flow conditions in the measurement domain.

For the case of the 10° wind veer model (Fig. b), the presence of vanes, however, results in a non-uniform streamwise velocity distribution across the disc due to wakes of the vanes. The minimum streamwise velocity typically deviates by less than 8 % from the mean reference velocity across all downstream locations. The veering vanes induce spanwise velocity in the flow that varies across the porous disc: generally positive above the centerline and negative below. The presence of vanes also results in nonzero vertical velocity, which can be attributed to the pressure difference induced by the vanes in the wake. The actual total veer across the porous disc location (x/D=0) is 8.91°. This value drops down to 6.97° at x/D=7. The veer configuration, however, increases turbulence intensity due to partial flow blockage induced by the vanes. Despite this, the turbulence intensity across the disc area remains below 2.5 % at all measured streamwise locations for both veered cases, indicating relatively low levels of added turbulence by the veering vanes. The effect of turbulence on wake recovery is isolated in the momentum budget analysis discussed in Sect. .

Lastly, for the 20° wind veer configuration, the actual veer across the porous disc is 14.3° at x/D=0, reducing to around 10.81° at x/D=7. The dashed magenta line in Fig. d shows the ideal wind veer profile. It is calculated based on the fact that wind veer is a linear function of the vertical coordinate, i.e., f(z)=A(z/zmax), where A represents the maximum veer amplitude at the domain boundaries z=±0.5. The root mean square error (RMSE) between the target and measured veer profiles lies within the range of 1.23 to 1.53° for both veer cases, within the vertical range of z/D∈[-0.5,0.5]. Although the reduction in wind veer downstream does not perfectly replicate real conditions, this deviation does not substantially affect the qualitative behavior of the wake evolution. As a result of producing wind veer using vanes, a localized deviation in the velocity profile is observed in the region z/D∈[-0.2,0] for both 10 and 20° wind veer cases (see 20° veer inflow in Fig. d). This non-uniformity stems from the wakes of the veer-generating vanes; the resulting local low-pressure regions induce a slight migration of the freestream flow, causing a localized distortion of the intended veer profile. An additional discussion of flow characterization in the cross-stream yz plane and the horizontal xy plane is given in Appendix .

The normalized streamwise velocity contours corresponding to all three inflows – no-veer, veer 10°, and veer 20° – under two different yaw angles of 0 and 30° are presented in Figs. , , and , respectively. Each figure presents the inflow velocity field to the disc and a three-dimensional representation of wake evolution, where cross-stream planes at x/D=3,5, and 7 are overlaid on the streamwise plane, for yaw angles of 0 and 30°. The figures also include planar views of streamwise velocity contours with in-plane velocity vectors at x/D=5 for both yaw angles. As the wakes are not axisymmetric for veered inflow, it is of interest to find the location of the wake center and how it moves downstream for both non-yawed and yawed cases. Here, the wake center is computed at each cross-stream plane (x/D=3, 5, and 7) using the center of mass method , given by the following equations: 6yc(x)=∫∫yΔu(x,y,z)dydz∫∫Δu(x,y,z)dydz,zc(x)=∫∫zΔu(x,y,z)dydz∫∫Δu(x,y,z)dydz.

In Eq. (), yc(x) and zc(x) are the wake centers in the spanwise and vertical directions, and the streamwise velocity deficit is defined as Δu(x,y,z)=U∞-Ux(x,y,z). The integration is performed at each cross-sectional plane. The wake center for different inflows and yaw angles of 0 and 30° is shown in Fig. .

For the no-veer and non-yawed disc case, the wake exhibits a nearly circular and symmetric profile in the region not influenced by the tower. Given the relatively large diameter of the tower in proportion to the disc diameter (with a disc-to-tower diameter ratio of 10), it induces a small vertical transport of momentum pointing downwards (also evident from the in-plane vectors in Fig. b behind the tower, where arrows point downwards), which makes the wake asymmetric and shifts the wake center downward toward the lower half of the disc (also see Fig. a). This was also observed in some other wind tunnel studies . As the wake advects downstream, it expands in both lateral and vertical directions due to flow entrainment from the free-stream. Notably, across all cross-stream planes in the non-yawed case under uniform inflow (see Fig. b), the wake consistently exhibits a lateral displacement toward the left, as further illustrated by the wake center trajectory in Fig. a. This is attributed to the inhomogeneity in wind tunnel inflow, which results in a systematic lateral deflection, particularly in the central region of the tunnel (see Appendix  for flow characterization contours). Supporting evidence for this is found in flow characterization measurements conducted without the porous disc (see Fig. a for reference), which also reveal a small spanwise velocity component of approximately 2 % of the free-stream velocity in the disc region. Furthermore, the relatively thick tower may contribute to this shift by acting as a bluff body that sheds vortices, thereby introducing asymmetries through complex interactions between the tower and disc wakes. As this lateral shift is systematic and consistent across all cases, it does not qualitatively or quantitatively impact the comparative analysis and conclusions presented in this study.

In the yawed case under no-veer inflow (Fig. c), the characteristic curled (or kidney-bean-shaped) wake was observed. This shape arises from the lateral force exerted by the disc on the incoming flow, which induces a significant spanwise velocity in the wake and displaces the wake laterally in the direction opposite to the yaw angle. As a result, two counter-rotating vortices form, originating from the top and bottom edges of the disc. Consequently, the yawed configuration breaks the wake's symmetry in both the spanwise and vertical planes. These findings are consistent with previous experimental works , where turbines or discs were immersed in uniform boundary-layer inflow. As is evident from the streamwise plane velocity contours, the wake deflects sideways and recovers faster when the turbine is yawed. This is primarily due to the reduced thrust coefficient of the porous disc under yaw, which results in a higher mean streamwise velocity in the wake (or smaller wake deficit). The wake width in the horizontal xy plane also appears thinner for the yawed disc because the mean spanwise velocity induced by yaw deflects and pushes the wake core sideways – a result that is also consistent with prior studies (see, for instance, , and ). Quantitatively, Fig. a demonstrates that this lateral displacement increases monotonically with downstream distance. By the farthest measurement location (x/D=7), the wake center deflects by approximately 0.45D in the yawed case, significantly exceeding the 0.2D deflection observed in the no-yaw case.

The wake topology in veered inflow cases is distinctly different from the uniform inflow, as seen by comparing the contours in Figs.  and . As a result of the veer in the inflow, the wake shape appears skewed in the lateral direction that extends moving along in the streamwise direction. This can be attributed to the variation in spanwise velocity with height – positive above the disc and negative below – as also illustrated by the spanwise velocity components in Fig. b, c. Consequently, the wake stretches in the +y direction above the disc and -y direction below it. The elliptical wake shape for the veered inflow qualitatively agrees well with the existing numerical simulation studies on stably stratified flows . When the disc is yawed, the curled shape due to yaw is superimposed on the elliptical wake shape due to veer, resulting in a complex wake structure that helps direct the flow away from the disc area, thereby exposing the downwind turbine to a higher free-stream velocity. Under veered inflow conditions, the tower wake in Fig. b shows that the velocity deficit caused by tower blockage can be displaced by up to 1D distance to the right, due to the lower-half veer pointing in that direction. At farther downstream locations, it merges with the wind turbine wake due to turbulent mixing. As pointed out in earlier studies , the tower wake is important as it influences wake dynamics and interacts with surface fluxes. It should be noted that the periodic structures shed by the upstream veering vanes have a slight influence on the wake characteristics locally. A quantitative analysis confirms that their effect is secondary to the dominant forcing of the porous disc. For instance, the maximum velocity deficit due to the porous disc (Δuwake) at x/D=3 for the WV0Y0 case is 0.65⋅Uref, whereas the deficit due to the vane (Δuvane) at the same location in the absence of the porous disc is only 0.05⋅Uref. Since the disc-induced deficit is over an order of magnitude larger than the wake of the vanes, these periodic structures act as a passive background turbulence source rather than a driver of the mean wake shape.

As the magnitude of wind veer in the inflow increases (see Fig. ), the wake skews even further due to higher spanwise velocity, characterized by an extended major axis and a reduced minor axis. This means that the velocity deficit is concentrated more in a narrow band along the minor axis direction. As hypothesized in , this effect can accelerate wake recovery as the free-stream flow now has to travel a shorter distance to reach the wake core. A higher veer in the inflow results in wake thinning, which leaves more undisturbed free-stream wind speed for downwind turbines, potentially leading to more power available for them. An interesting observation from these results is that as the magnitude of wind veer increases, it exerts dominant control on the wake shape, reducing the relative influence of yaw, which is also consistent with the findings of . Figure b and c show the wake center evolution for no-yaw and yawed cases under veer 10° and 20° inflows, respectively. It is evident that as the inflow veer magnitude increases, the wake center progressively moves toward the -y direction relative to the reference uniform inflow case. It should be noted that there is a vertical bias in wake centers (i.e., zc/D<0), which is attributable to the tower wake. An additional discussion on the effect of yaw and veer on spanwise and vertical velocity fields in the streamwise xy plane is presented in Appendix .

To better quantify wake recovery under veered inflow, we adopt an integral analysis approach where we integrate the quantity of interest within the wake region defined as the boundary where u‾/Uref=0.9. Figure  shows the integrated wake deficit quantity within the wake region (Mx=∬Δu‾dA/UrefAw) as a function of streamwise position, normalized by the reference velocity of the inflow (Uref) and wake area Aw. A lower value of this term indicates that the wake has recovered more. It can be observed that the wake recovery is slowest in the reference uniform inflow case (WV0Y0) and is progressively accelerated under veered inflow. Quantitatively, at x/D=5, the wake deficit reduces by approximately 22 % when transitioning from the reference uniform inflow case and no yaw (WV0Y0) to the 10° veer inflow case (WV10Y0). This accelerated decay confirms that wind veer enhances wake recovery by entraining fresh momentum from the freestream into the wake, effectively re-energizing the wake core more rapidly than in the uniform inflow case. Notably, veer does not merely displace the wake laterally but actively promotes wake recovery (see Appendix , where we compare a simple advection model with the measurements). Furthermore, increasing the yaw angle also has a positive effect in reducing wake deficit, as is evident by the dashed lines for yaw cases, which consistently lie below their solid counterparts (non-yawed cases), indicating that the combination of veer and yaw further enhances wake recovery. However, the benefit diminishes with downstream distance, as the curves of no-yaw and yaw converge.

For all the measured cases in this study, Fig.  presents the heat map of the integrated wake deficit in the wake region across various cross-sectional planes for different yaw angles and all inflow cases. In the no-veer reference case, the effect of yaw misalignment is negligible at low yaw angles (≈10°). This is expected because the thrust reduction scales as CT(γ)∝cos2(γ), yielding a 3 % reduction at γ=10°; the corresponding power penalty, governed by the cosine law PγP0=cosp(γ), with p=2, is equally marginal (see works of , and ). The gains of yaw misalignment become appreciable only at higher yaw angles (γ≥20°), where the peak deficit decreases notably, indicating accelerated wake recovery under yawed conditions. Under veered inflows, several interesting insights can be gathered. The presence of wind veer results in lower values of integrated wake deficit compared to the no-veer reference case at all measurement planes. At x/D=3 with no yaw, introducing 10° of veer reduces the deficit from 0.37 to 0.34 (∼8%), whereas 20° of yaw with no veer only reduces it to 0.35. Therefore, veer is a stronger driver of wake recovery than yaw alone. When yaw is coupled with veer, the recovery is enhanced further. At 0° veer, increasing the yaw angle from 0 to 30° reduces the deficit by 11% at x/D=3. At the same location, on the other hand, a veer of 20° results in a reduction of around 24%. This is because wake stretching due to veer and yaw-induced lateral deflection exposes more of the wake boundary to the freestream, which increases the entrainment. Comparing increasing yaw angles for the veer inflow of 10 and 20° at x/D=3, there appears to be a synergistic interaction between yaw and veer: the benefits of veer are amplified when combined with yaw. At far downstream distances, yaw provides essentially no additional benefits under veered inflows, as evidenced by the same value of integrated wake deficit at x/D=7 for the veer 10° case; the range is only 0.16-0.17 for the veer 20° case. This implies that the wake has already recovered to a level where yaw only adds marginal benefits. This highlights a key finding: if wind veer is high in the atmosphere under stable conditions, as is typically the case and also reported in multiple field experiments (for instance, see and ), yawing the upstream wind turbine may not be advantageous for wake steering purposes. Since wake recovery is already enhanced by wind veer, there appears to be limited value in applying yaw control solely for the purpose of maximizing downstream wind turbine power production. This will be further discussed in Sect. .

The measured time-averaged streamwise vorticity contours for the three inflow cases at cross-stream location x/D=5 for the yaw angle of 30° are shown in Fig. a. For the reference uniform inflow case, the curled wake shape observed in velocity contours in Fig. c is directly attributable to the formation of a counter-rotating vortex pair (CVP) that results in spanwise induction of flow and is responsible for the lateral displacement of the wake. A slight asymmetry in the vorticity distribution of top and bottom vortices for this case can be attributed to the relatively thicker tower that sheds its own vortex, which is not negligible. This tower-induced vortex merges with the bottom vortex of the disc as the wake propagates downstream.

In contrast, veered inflow leads to increased stretching and distortion of streamwise vorticity contours, with the effect intensifying under stronger wind veer. It can also be noticed from Fig. a that the vorticity strength in veered inflow is significantly affected by background veer vorticity: the strength of the top vortex decreases compared to the corresponding top vortex in the uniform inflow case with no-veer. These observations are consistent with the numerical results of and further supported by the variations in maximum streamwise vorticity and the circulation of the top vortex as shown in Fig. c and b. Circulation in the top vortex was computed by spatially integrating the positive vorticity field over that region (Γ=∫∫ωxdydz) following the approach highlighted in . In general, with increasing downstream distance, the peak vorticity strength for the yawed cases under veered inflow is lower than that observed under uniform inflow. This can be explained as follows: because the inflow already contains nonzero streamwise vorticity with opposite orientation to the upper vortex, the background veer reduces its strength. Consequently, higher veer cases accelerate the decay of circulation in the CVP (Fig. b), with lower values than the yaw case under uniform inflow. As expected, due to the symmetric pressure distribution, circulation remains relatively constant downstream for the non-yawed disc under uniform inflow.

Conversely, lateral shear induced by veer in the inflow results in a non-symmetric pressure distribution around the disc, leading to a decrease in circulation with downstream distance. Interestingly, when the disc is yawed in the reference uniform inflow case, circulation slightly increases from x/D=2 to x/D=3, before continuing its downward descent, as also seen in . In contrast, the circulation for the veer inflow of 10° and yawed disc is nearly constant from x/D=2 to 3, before decaying further downstream, whereas the circulation in the veer inflow of 20° decreases monotonically. This again highlights the dominant role of background veer in reducing circulation strength, which increasingly overshadows yaw effects as veer magnitude grows.

Additionally, the vertical vorticity contours ωz=∂v‾∂x-∂u‾∂y shown in Fig.  indicate that veered inflow accelerates vortex sheet breakdown compared to the no-veer case. This is evident from the golden contour lines marking regions where ωzD/U∞=±1. For the no-veer case, the peak vorticity persists beyond a downstream distance of 6D, whereas in the veered inflow it disappears within 5D. This faster decay is attributed to stronger lateral (cross-flow) velocity components in the wake introduced by veer. The decay of vorticity is slightly faster when yaw is superimposed on veer, which results in additional spanwise velocity in the wake. For the yawed disc immersed in the wind veer of 20°, the counterclockwise (CCW) vortex bifurcates into two, visually consistent with the wake topology discussed in Sect. . Overall, these results suggest that increasing veer intensity leads to faster CVP decay and enhances turbulent mixing in the wake, thereby promoting faster wake recovery.

In this section, dominant mechanisms driving wind turbine wake recovery as a result of yaw and veered inflow conditions are analyzed via the momentum budget analysis of the Reynolds-averaged Navier–Stokes (RANS) equation in the streamwise direction. This analysis helps to explain how momentum is redistributed in the wake. The time-averaged RANS equation in the streamwise direction can be written as 7u‾∂u‾∂x=-v‾∂u‾∂y︸I-w‾∂u‾∂z︸II-1ρ∂p∂x-∂(u′u′‾)∂x-∂(u′v′‾)∂y︸III-∂(u′w′‾)∂z︸IV. In the above equation, u‾, v‾, and w‾ denote the mean streamwise, spanwise, and vertical velocity components, respectively. The overbar indicates the time-averaged values of these components, while the primes represent instantaneous velocity fluctuations. Term I in Eq. () denotes cross-stream advection that represents the transport of streamwise momentum by the mean spanwise velocity (v‾). Term II represents the transport of streamwise momentum in the vertical direction by the mean vertical velocity (w‾). Lastly, terms III and IV are the divergence of the cross-stream Reynolds stress (u′v′‾) and the vertical Reynolds stress (u′w′‾), which represent turbulent mixing of momentum laterally and vertically, respectively. It should be noted that the variation in streamwise Reynolds stress gradient on the RHS is not computed because of the relatively large spacing between the spanwise planes. Similarly, the pressure gradient term is also neglected as it is not measured in the experiments. Viscous terms are also neglected due to the high Reynolds number of the flow. Figure  represents the contributions of the remaining terms in Eq. () at X/D=5 for six different cases. The wake edge is represented by a dotted line. The positive (red) region indicates favorable contributions to wake recovery, and vice versa.

The contours in the first vertical column in Fig.  show the mean lateral advection of streamwise momentum. It can be seen that for the no-veer inflow, the positive values are concentrated along the right edge of the wake, and vice versa. For the yawed disc in the no-veer case, due to significant spanwise velocity induction into the wake, the positive values are much higher than in the non-yawed cases. Interestingly, it can be seen visually that the positive values for the yawed disc are slightly lower for the veered cases compared to the uniform inflow case. This could be attributed to the spanwise component of background veer counteracting the spanwise flow induced by yaw. In the no-veer case, the low-momentum wake is ejected out laterally from the center to the left side of the wake edges (blue region). In contrast, for the veered cases, this ejection takes place from both the left and right edges of the wake due to the varying spanwise velocity with height, whose sign also changes. Moreover, comparing the relative contributions of all terms in the second row of Fig. , it is obvious that the advective terms play a key role in redistributing momentum for the yawed case compared to the divergence of shear stress terms.

The contours in the second column show the mean vertical advection of streamwise momentum. The contribution of term II to wake recovery is small for the uniform inflow case relative to the veered inflow case. In the latter, a large red region is present along the upper edges of the wake. This can be attributed to the fact that, due to the skewed wake shape, the gradients in the shear layer are sharper, making the injection of free-stream flow into the wake easier.

The contours in the third column show the Reynolds stress term distribution in the lateral direction. A visual comparison of contours of term III for all cases reveals that its contribution remains similar across all cases.

Lastly, the contours in the fourth column show the Reynolds stress term distribution in the vertical direction. It can be seen visually that, compared to the reference uniform inflow case, the contribution of the streamwise-vertical Reynolds stress term -∂u′w′∂z to the overall budget is higher in the veered case compared to the uniform inflow case, where both its magnitude and area are relatively small. The positive and negative values are concentrated in the wake core and wake edges, respectively, which means more momentum is entrained from the top and bottom in the veered inflow scenario. This can be attributed to the fact that velocity gradients are sharper due to the elliptical shape of the wake.

To evaluate the overall impact of the momentum budget terms on wake recovery across different cross-stream planes, we present their individual contributions to the recovery process at each plane. Equation () can be rewritten as 8∂u‾∂x=1u‾-v‾∂u‾∂y-w‾∂u‾∂z-1ρ∂p‾∂x-∂u′u′‾∂x-∂u′v′‾∂y-∂u′w′‾∂z, where ∂u‾∂x is also called the streamwise wake recovery rate. Figure  shows the bar plot of each term's net contribution to wake recovery, which is integrated within the wake region at three different downstream locations of x/D=3, 5, and 7 for three different inflows and two yaw angles of 0 and 30°. At x/D=3 for the no-veer case, the lateral advection of streamwise momentum (term I) helps in wake recovery more than the other terms. This term is slightly higher for the yawed cases because of the induction of spanwise flow in the wake, which contributes positively to the wake recovery. At further downstream locations, the contribution of this term reduces in the overall wake recovery. For the veered inflow and no-yaw case, the vertical advection of streamwise momentum (term II) plays a dominant role in redistributing momentum in the wake, as is evident from the sharp peaks of term II at x/D=3. Term II for the 20° wind veer case is approximately 5 times higher than the corresponding term in the reference uniform inflow case. However, when the turbine is yawed, term II decreases with a corresponding increase in term I. This means that more entrainment of fluid occurs into the wake from the vertical direction than from the spanwise direction. Term IV is generally higher than term III for all cases, indicating greater turbulent transport of streamwise momentum in the z direction via u′w′‾ (i.e., more turbulent mixing vertically). Generally, veered cases exhibit higher values of term IV compared to the uniform inflow case.

Turbulence kinetic energy (TKE) reflects the energy content of turbulence in the wake and how it is spatially distributed. It is an important parameter influencing wind turbine performance and blade fatigue loads. The wake-added TKE is computed as follows: 9ΔTKE=TKEwithdisc-TKEwithoutdisc, where TKE =12(ui′ui′)‾. In this equation, ui′ is the fluctuating velocity component in the ith direction (where i=1,2,3 corresponds to u,v,w) and ui′ui′‾ is the time-averaged Reynolds stress tensor components.

Figure  shows the streamwise and cross-section planes of wake-added TKE for non-yawed and yawed discs for the three inflows. In the near-wake (x/D=3) for the non-yawed cases, the TKE appears enhanced in the lower half of the disc region due to the strong influence of vortices shed by the tower and their interaction with the porous disc wake. TKE distribution is ring-shaped in the near-wake, becoming more uniform at farther downstream distances due to flow entrainment and wake mixing. The wake-added TKE is highest for the veered inflow cases compared to the uniform inflow (no-veer) case, which is also consistent with the findings of . Yaw introduces asymmetry in the TKE field, with more TKE addition from one side of the wake, as is evident from the streamwise plane of TKE in the bottom panels in Fig. . Similar to the velocity wake contours, the wind veer has an effect of stretching the TKE laterally. A larger TKE in the wake also implies a higher degree of wake mixing, thus aiding in wake recovery. The TKE distribution in the horizontal xy plane under veered inflow and non-yawed conditions is markedly different from that in the reference uniform inflow case, with the TKE expanding more in the lateral direction for the veered inflow.

To better understand the TKE distribution within the wake, a budget analysis of the TKE transport equation is performed in the yz plane. The time-averaged TKE budget equation can be written as : 10u‾j∂k∂xj︸Advection(Ak)=-ui′uj′‾∂u‾i∂xj︸Production(Pk)-∂∂xj12ui′ui′uj′‾︸TurbulentTransport(Tk)+ν∂2k∂xj2︸ViscousDiffusion(Vk)-ν∂ui′∂xj∂ui′∂xj‾︸Dissipation(εk). We evaluate the terms of Eq. () on yz planes at x/D=5. Given the sufficiently high Reynolds number, the viscous diffusion term (Vk) is omitted, whereas the dissipation term (εk) cannot be measured by PIV and is also omitted in our analysis. Therefore, we only consider the in-plane contributions from advection (Ak), production (Pk), and turbulent transport (Tk) terms. Figure  shows the distributions of Ak,Pk, and Tk at x/D=5 for different yaw and veer cases. The red and blue regions in the Pk term indicate the source and sink of TKE, while the red/blue regions in Ak and Tk indicate gain/loss of TKE. The advective term (Ak) represents the transport of TKE by the mean flow. Comparing no-veer cases at yaw 0 and 30°, the yawed case shows stronger TKE advection along the lateral wake edges, driven by the induction of spanwise velocity due to yaw misalignment. In the presence of veer and no-yaw conditions, TKE is advected into the wake predominantly along the top and bottom edges of the wake boundary (red regions), while it is carried away from the lateral sides.

The production term (Pk) represents the transfer of kinetic energy from the mean flow to turbulence. The enhanced wake-added TKE for the veered cases in Fig.  is attributed to the enhanced production term, which is much higher compared to the no-veer cases. The high Pk region is concentrated along the wake edges. This is due to higher lateral and vertical shear in the former case than in the latter. Comparing the top and bottom panels in Pk, yaw primarily alters the spatial distribution, whereas wind veer is the stronger driver of increased production magnitude. It should be noted that in the wake core, the TKE production is almost negligible, primarily because the mean velocity gradients vanish in this region.

Lastly, the turbulent transport term (Tk) represents the redistribution of TKE by the turbulent fluctuations. In the veered cases, TKE generated by the shear production along the wake edges is transported into the wake core by turbulent fluctuations, as indicated by the red regions within the wake core and blue regions along the outer wake edges where production is the highest. This redistribution is notably stronger in the veered cases compared to their no-veer counterparts, consistent with the stronger production observed in these cases.

The analysis so far has focused on the mean wake deficit and turbulence quantities of porous disc wake under uniform and veered inflows, and the dominant terms in the wake recovery mechanism. To further highlight the role of wind veer inflow in replenishing kinetic energy in the wake, an analysis of available power (AP) is performed at selected downstream locations. Rather than comparing velocities at a single point, the AP metric integrates the cube of the streamwise velocity over the frontal projection of a hypothetical downstream rotor and then slides this integration window laterally across the wake. This area-integrated, sliding-window approach captures the combined effects of wake shape, lateral displacement, and any spatial inhomogeneity in the inflow. Figure  shows the resulting lateral variation in available power for different cases at two downstream locations x/D=5 and 7. The AP is calculated using the relation given in and : 11fAP(xT,yT,zT)=∬GU(xT,y,z)3dydz∬GUin(y,z)3dydz, where fAP(xT,yT,zT) is the normalized available power at the downstream turbine location, U(xT,y,z) is the wind speed in the wake, and Uin(z) is the inflow wind speed. The integration is performed over the region G, defined as 12(y-yT)2+(z-zT)2≤R2, which represents a circular area of radius R centered at hub height zT and lateral position yT. In this study, the disc center is positioned at the origin of the (y,z) coordinate system, such that (yT,zT)=(0,0). The window is then traversed in the lateral direction by varying yT, yielding the profiles shown in Fig. .

For the no-veer inflow and non-yawed disc condition (solid blue line in Fig. ), fAP is lowest among all other cases, increasing from 32 % at x/D=5 to 47 % at x/D=7 for an inline configuration with the downstream turbine positioned at y/D=0. When the disc is yawed (dashed blue line), the wake center shifts to the left, and the overall available power increases. The lowest value of fAP for the yawed case is 33 % higher than that for the non-yawed case. In the presence of veer in the inflow and a non-yawed disc (solid red and green lines), a stronger veer (20°) results in more available power than a weaker veer (10°). This is expected, as veer results in a stretched and elongated wake structure, exposing the downstream turbine to higher free-stream velocity inflow and thus leading to higher AP. To quantify this, if a hypothetical turbine is placed at x/D=5, the AP for it under veered inflow of 20° is 45 % more than when the turbine is operating in no-veer inflow. When yaw is introduced in the presence of veered inflow (dashed red and green lines), wake recovery is enhanced even further because yaw deflects the wake farther away in the positive y direction and fAP increases to 77 %. Similar trends are also observed at the downstream location of x/D=7. At this location, the highest gains in available power up to 85 % can be observed for the veered inflow of 20° and a positive yaw steering of 30°.

To investigate optimal downwind turbine placements with lateral offsets, Fig.  presents contour maps of the available power coefficient for all cases, computed using a sliding integration window across the spanwise plane. The dashed and solid lines on the contours denote locations where the coefficient of available power, fAP, is 0.50 and 0.75, respectively. A similar analysis for vertical-axis wind turbines has been conducted by .

In the uniform inflow case with no-veer, the available power deficit exhibits a lateral shift to the left with increasing yaw angle. As discussed previously, yawed turbines generate reduced CT, resulting in greater fAP values further downstream. However, a downwind turbine positioned directly inline – even at x/D=7 from the upstream turbine – would still experience a partial wake, potentially exacerbating structural loading. In such scenarios, a more favorable placement would involve a slight negative lateral offset to avoid the lower wake regions of the upstream turbine.

In contrast, veer in the inflow leads to a more laterally uniform distribution of the available power deficit, particularly at x/D=5 and x/D=7. This is evident from the absence of the fAP=0.5 contours (dashed lines) at x/D=7, indicating that fAP exceeds 0.5 across the entire span. A stronger veer of 20° further enhances this uniformity, with the dashed lines disappearing even at x/D=5. Under this condition, the fAP contours for yaw angles of 0° and 30° appear nearly identical.

This observation is particularly noteworthy as wake steering strategies typically perform best under low-turbulence conditions, which are characteristic of stable atmospheric boundary layers. Wind veering is more prevalent under such stratified conditions. Therefore, even without active yaw control, the wake may have sufficiently recovered by the time it reaches a downwind turbine located at x/D=7. This recovery could be beneficial for dense wind farm configurations, where turbine spacing often falls within the range of x/D=5to7.