Numerical investigation of rotor asymmetry to promote wake recovery

Tip vortices in the wakes of wind turbines are known to have detrimental effects on downstream turbines such as reduced performance and increased fatigue loading. Rotor asymmetry is investigated as a passive method for mitigating these effects by triggering the helical vortex pairing instability. The study is conducted using MIRAS, a multi-fidelity vortex solver, to compare the wakes of the standard NREL 5MW turbine and a modified asymmetric version where one blade is extended radially relative to the other two. The asymmetric rotor is shown to successfully trigger the vortex instability, increasing the wake average velocity by a maximum of 3.5% and the power available to a downstream turbine by up to 11%. The turbulence in the wake of the asymmetric rotor is also modified, exhibiting enhanced mixing. Using the available power gains from the simulations and operational data from the Lillgrund wind farm, the total impact of rotor asymmetry on wind farm efficiency is estimated, showing increases > 2% under certain wind conditions. The findings of this study indicate that rotor asymmetry has strong potential as a wake control method and would benefit from further investigation to understand the effects of inflow turbulence and the impacts on rotor loading.


Introduction
The near wake of a wind turbine is bounded by helical vortices shed from the tips of the rotor blades.These coherent vortices can cause increased fatigue loading on downstream turbines within a wind farm [1].In addition, previous studies have shown that these tip vortices delay wake recovery by blocking mixing between the low-speed flow inside the wake and the freestream flow outside [2,3].This delayed wake recovery limits the minimum allowable spacing between turbines within a wind farm, reducing the amount of power available for a given area.In existing wind farms, wakes can lead to power losses of up to 40% [4].Inducing tip vortex breakdown could help mitigate these detrimental wake effects by enhancing mixing between the wake and the freestream and by reducing the amount of coherent vortical structures in the flow.
Tip vortices are subject to various instabilities due to their helical geometry, which can cause them to break down.These instabilities fall into two categories: short-wave and longwave instabilities [5].Short-wave instabilities are characterized by perturbations within the vortex core, whereas long-wave instabilities involve displacement of the entire vortex.Longwave instabilities, which are the focus of the current investigation, lead to pairing between adjacent vortex loops and leapfrogging, where the upstream loop rolls up around and passes in front of the downstream loop.This pairing has been shown to play a significant role in tip vortex breakdown [3,6].
Previous studies have employed various methods to trigger tip vortex pairing with the goal of inducing breakdown.Odemark and Fransson [7] used pulsed jets on the nacelle of a threebladed rotor to experimentally investigate the effect of perturbation amplitude and frequency on vortex strength and energy.Lignarolo et al. [3] characterized the impact of pairing and vortex breakdown on energy fluxes into the wake, triggering the instability by installing the two rotor blades at slightly different pitch angles.Using a one-bladed rotor, Quaranta et al. [8] sinusoidally perturbed the rotation speed and compared the resulting instability growth rate to the theoretical prediction calculated by Gupta and Loewy [9].They also explored the effect of perturbation amplitude on the downstream distance of vortex pairing.Quaranta et al. [10] extended these findings to a two-bladed rotor, investigating both local pairing, induced through rotor speed modulation, and global pairing, triggered by offsetting the rotor from the center of rotation such that one blade was slightly longer than the other.Huang et al. [11] and Marten et al. [12] numerically modeled rotors with flaps, observing the pairing instability in the wake even when small-amplitude unsteady perturbations were applied.The most recent studies have focused on practical implementations, with Brown et al. [13] using blade pitch and rotor rotation speed to trigger the instability, Hodgkin et al. [14] investigating the effects of atmospheric shear and thermal stratification, and Kleine et al. [15] exploring perturbations induced by wave motion for floating turbines.
The current study examines the use of passive rotor asymmetry to trigger the tip vortex instability in the wake of a three-bladed rotor by numerically modeling a wind turbine with one blade that is slightly longer than the other two.This study builds on a previous comparison between different types of rotor asymmetry conducted using a simplified point vortex model [16].In the current investigation, the effects of this asymmetry on the velocity and turbulence intensity in the wake are analyzed, and the impact of such a modification on an entire wind farm is estimated.The methodology, including descriptions of the numerical solver, wind turbine model, and wind farm data, is presented in Section 2. The findings of the investigation are presented and discussed in Section 3. Finally, a summary and discussion are provided in Section 4.

Methods
Simulations were conducted using the multi-fidelity vortex solver MIRAS, which employs a hybrid filament-mesh vortex method to model the wind turbine wake flow [17].This method models vorticity shed from the turbine blades as filaments in the extreme near wake, then the vorticity is interpolated onto a mesh as it is advected downstream.By using this hybrid method, a balance between accuracy and computational efficiency is achieved.MIRAS was previously validated against detailed measurements conducted in the wake of an asymmetric two-bladed model rotor in [18].The National Renewable Energy Laboratory (NREL) 5 MW turbine model, which has a rotor diameter of D = 126 m and a hub height of H hub = 90 m, was chosen for the current investigation.Lifting lines with 120 points along the radial span were used to represent the rotor blades, which were modeled with infinite stiffness.The tower and nacelle were not included in the model to avoid interference with the tip vortex dynamics.A uniform inflow (0% turbulence intensity) with U 0 = 8 m/s was imposed to ensure the turbine was operating below the rated wind speed of 11.4 m/s, where wake effects are strongest.At this wind speed, the rotational speed of the rotor was Ω = 0.95 rad/s, corresponding to a tip speed ratio of λ = ΩD/(2U 0 ) = 7.5.
Two different cases were simulated.The first was the baseline symmetric case, where all three blades were the same.In the second, asymmetric case, one blade was modified to be 2% longer than the other two (62.732m versus 61.5 m for the baseline blade -note that the hub has a radius of 1.5 m).For the long-wave helical vortex instability, the relevant geometric scaling parameter is h, the spacing between adjacent vortex loops, which is determined by λ.A modification of 2% of the rotor radius corresponds to ∼ 0.1h for λ = 7.5.This modification was chosen to enable comparison with the experiments presented in [19] for an asymmetric rotor in a water channel, which explores asymmetries between 0.05 and 0.15h.In the current investigation, the radial extension was implemented by slightly reducing the taper of the outer 7% of the blade, at the location of a change in taper angle of the baseline blade, so that the tip chord remained the same (figure 1).For the modified blade, the twist of the extension was held constant at 0 • , corresponding to the twist angle at the tip of the baseline blade.
The extent of the simulation domain was −D ≤ x ≤ 8.5D in the streamwise direction, −D ≤ y ≤ D in the spanwise direction, and −H hub ≤ z ≤ D in the vertical direction, with (x, y, z) = (0, 0, 0) corresponding to the center of the rotor hub.This domain was selected to ensure sufficient downstream distance to capture the effects of the wake on a virtual downstream turbine situated at a typical distance of ∼ 7D [20].Free boundary conditions were applied to all domain boundaries except the bottom surface of z = −H hub , which had a slip boundary condition.The spatial resolution of the mesh was 0.46 m (equivalent to ∼ D/274), yeilding a total of 1.4 × 10 8 cells.Such a high resolution was required to resolve the tip vortices, which drive the dynamics of interest in the current study, and to avoid unphysical vortex merging.With a timestep of 0.015 s, the Courant-Friedrichs-Lewy (CFL) number was 1.95.A CFL value less than 2 is acceptable for vortex methods, which have looser CFL conditions than Eulerian solvers [21,22].The simulations ran for 30000 time steps to ensure sufficient convergence of the mean flow fields.For the first 15000 time steps, the wake was still developing so the data were not saved.During the second 15000 time steps, the x−z plane at y = 0 and y −z planes at every 0.5D between 0 ≤ x ≤ 7D were saved every 12 time steps.This total duration corresponds to 225 s or 34 rotor rotations.
A database covering 10 years of operation of the Lillgrund offshore wind farm, provided as Supporting Information for Sebastiani et al. [23], was used estimate the impact of rotor asymmetry on the overall efficiency of a wind farm.This analysis was conducted by comparing the wake effects on power output recorded at Lillgrund to those estimated from the MIRAS simulations of asymmetric rotors under a distribution of wind conditions.The Lillgrund wind farm consists of 48 Siemens SWT-2.3-93turbines with rotor diameters of 92.6 m, hub heights of 65 m, and rated powers of 2.3 MW.The available database includes 10-minute average power output values for each turbine in the wind farm, binned by wind speed intervals of 2 m/s and wind direction intervals of 1 • , and normalized by the power of the most-producing turbine for the given wind conditions.The probability of each bin is also provided.Because the power outputs of all turbines are normalized by different values for each wind condition, the total wind farm power cannot be computed.Instead, the efficiency of the array for each wind condition, η array , is used to investigate the overall wake effects.This array efficiency is defined by Sebastiani et al. [23] as where N t is the number of turbines in the farm, P ref is the power of the most-producing turbine used for normalization, and P i is the power of each turbine.The database also includes the geographic coordinates of each turbine.

Results
The instantaneous vorticity field in the wake of the asymmetric rotor clearly reveals the occurrence of vortex pairing, which is not observed in the wake of the symmetric rotor (figure 2).This pairing occurs within 1D downstream of the rotor, showing that even such a small modification causes the instability to develop quickly.The impact of this instability on the wake velocity field is explored below.

Wake velocity
Figure 3 compares the velocities in the wake of the symmetric and asymmetric rotors at 7D downstream of the rotor plane, corresponding to the typical spacing between turbines in a wind farm [20].A circle with a diameter D is included for reference, representing a virtual downstream rotor.These velocity fields show that the vortex interactions generated by the rotor asymmetry cause the wake to spread more than that of the symmetric rotor.This spreading leads to a lower velocity on the edges of the wake and a higher velocity in the center, characteristic of increased mixing between the freestream and wake flows.
To quantify the effect of rotor asymmetry on power available to a downstream turbine located within the wake, the mean velocity within the virtual rotor disk positioned at different streamwise and spanwise locations is calculated.The gain in asymmetric wake velocity relative to the Figure 3: Mean velocity fields in the wakes of the symmetric and asymmetric rotors at x = 7D, as well as the relative difference between the two cases.The dashed circle represents a virtual rotor with diameter D located directly downstream of the simulated turbine.symmetric wake velocity, calculated as where ⟨⟩ represents the spatial average over the virtual disk (y 4(a) for all computed streamwise and spanwise positions of the disk center, (x c , y c , H hub ).This plot shows that the maximum gain of 3.5% occurs when the downstream turbine is located in the center of the wake of the upstream turbine where the velocity deficit is strongest.As the downstream turbine moves towards the edge of the wake and a smaller portion of the rotor is immersed, the difference between the two cases decreases.A turbine in the wake of the asymmetric rotor with a spanwise offset of y c /D > 0.3 actually experiences a reduced velocity relative the symmetric rotor wake.This decrease is due to the lower velocities at the edges of the asymmetric wake caused by the enhanced mixing with the freestream flow.The distribution of the mean velocity gain is asymmetric about y c /D = 0 due to the rotation of the wake and the symmetry-breaking boundary condition on the bottom surface.In the streamwise direction, the gain increases until x c /D = 6, at which point it begins to decrease again.The increase is caused by the time required for the tip vortices to break down and the high-speed flow outside the wake to become fully entrained.After x c /D = 6, the gain decreases because the wake of the symmetric turbine is also beginning to recover at that point, so the benefit of rotor asymmetry diminishes.Note that locations with x c /D < 3 are not included in the figure, as a downstream turbine would not be positioned within this region.Within x c /D < 3, the difference between the symmetric and asymmetric cases generally continues to decrease with decreasing streamwise distance from the rotor.The velocity field is then used to compute the available power inside a disk in the rotor wake based on the definition provided by Vollmer et al. [24] and van der Hoek et al. [25]: The gain in available power in the asymmetric case relative to the symmetric case, defined as is presented in figure 4(b).Because f AP is directly dependent on the velocity, the spanwise and streamwise trends are the same as those described above.The maximum gain in available power, however, is more than 11% since f AP scales with the cube of the velocity.Figure 5: Turbulence intensity fields in the wakes of the symmetric and asymmetric rotors at x = 7D, as well as the difference between the two cases.The dashed circle represents a virtual rotor with diameter D located directly downstream of the simulated turbine.

Turbulence intensity
In addition to increasing the available power, the pairing instability induced by rotor asymmetry has the potential to reduce the structural loading on downstream turbines due to the destruction of the tip vortices.Turbulence intensity (TI) in the wake is used to investigate this effect, as it has been shown to significantly impact fatigue loading [26].TI fields at x = 7D are presented for the symmetric and asymmetric case, alongside their difference, in figure 5.The middle of the asymmetric rotor wake exhibits lower TI than the symmetric case, while the TI at the edges is increased.This trend is consistent with a wake that is more spread out and well-mixed.Figure 6 shows the percent increase in TI within the virtual disk centered at different streamwise and spanwise locations in the wake of the asymmetric rotor relative to the symmetric rotor, defined as (⟨T I a ⟩ − ⟨T I s ⟩)/⟨T I s ⟩.
For x/D ≤ 4.5, the TI is always higher in the wake of the asymmetric rotor than the symmetric rotor, indicating that the tip vortex pairing instability enhances mixing in the wake.Farther Figure 6: Chart showing the increase in mean turbulence intensity within a virtual rotor disk in the wake of the asymmetric rotor relative to the symmetric rotor centered at different streamwise (x c ) and spanwise (y c ) locations, as defined in equation ( 5).
downstream, the TI is still higher within a virtual rotor disk that is offset from the wake center in the −y direction.When the downstream rotor is aligned with the center of the asymmetric rotor wake, it experiences a reduction in TI relative to the symmetric rotor wake, consistent with the results presented in figure 5.These increases and decreases in TI may result in corresponding changes in structural loading on downstream turbines, though other factors such as the characteristics of the ambient flow turbulence (e.g., atmospheric stability) and the properties of the wake-added turbulence also influence turbine response.For example, work is currently underway to investigate the effects of asymmetry on the size of turbulent structures in the wake, which have been shown to play an important role in the structural response of a downstream turbine [27].

Total wind farm effects
The efficiencies of the turbines in the Lillgrund wind farm are used to explore the total impact of rotor asymmetry on wind farm performance.Note that throughout the following analysis, the Lillgrund turbines are not simulated directly.Rather, the percentage changes in the wake velocity computed for the NREL 5MW turbine are applied to the Lillgrund farm, based on the assumption that the effects of rotor asymmetry on the wake scale with turbine size.Because of the tightly packed layout of Lillgrund (figure 7a), many turbines operate in the wakes of their neighbors under multiple different wind conditions.These periods with strong wake effects appear as regions of low efficiency in the top plot of figure 7(b) because the power production values of most turbines are less than the unwaked reference power (equation 1).To estimate the effect of adding asymmetric rotors to the wind farm, the following procedure is applied.First, for each wind direction, the region of influence of the wake of every turbine is determined by extending lines from the edges of the rotors along the direction of the wind.If a downstream turbine is within this region of influence, it is determined to be in the wake of the upstream turbine.Then the streamwise and spanwise distance from the upstream turbine is recorded for all waked turbines.If a turbine is in the wake of multiple turbines, only the nearest upstream turbine (i.e., the one with the shortest streamwise distance) is considered to be influencing it.In the current investigation, rotor asymmetry is only expected to modify the wakes of turbines that are experiencing clean inflow, as the flow encountered by waked turbines will be too disturbed to allow for persistent coherent tip vortices [28].Therefore, only a turbine that is in the wake of an unwaked turbine, termed a "second row turbine", can benefit from the reduced wake effects caused by the asymmetry of the turbine in front of it.Furthermore, rotor asymmetry is expected to reduce wake effects only when both the upstream and waked turbines are operating below the rated wind speed (12 m/s for the Lillgrund turbines) where the wake is still strong and the turbine is not yet producing its maximum amount of power.For the second row turbines where these conditions are met, their efficiencies are increased relative to the case where they are in the wake of a symmetric turbine (i.e., the efficiency values recorded in the database).The magnitude of this increase is determined using the chart in figure 4(b), based on the streamwise and spanwise distance from the upstream turbine.This increase is not allowed to push the efficiency above 1.These modified efficiencies, along with the unmodified efficiencies of the turbines that are not in the second row, are all averaged together for each wind condition, yielding the plot shown in the middle panel of figure 7(b).Though the differences between the top two panels are not immediately apparent, the bottom panel of figure 7(b) shows the gain in efficiency of the entire wind farm due to the introduction of rotor asymmetry for each wind condition.This total gain is greater than 2% for multiple different wind directions, particularly those where the initial recorded efficiency is low because of wake effects.The largest gains occur when the wind is in the east-west direction, aligned diagonally with the turbine rows, not when the wind is directly aligned with a row.This result is due to the trends in available power gain presented in figure 4(b), which show that the benefits of asymmetry increase as the wake propagates downstream.Therefore, the larger spacing between turbines in the east-west direction allows waked turbines to benefit more from the upstream rotor asymmetry than they would under rowaligned wind conditions.Under some wind conditions, the total farm efficiency is reduced, but these bands are narrower and their magnitudes are lower (< 1%) than the periods of efficiency increases.All gains are then weighted by the probabilities of each corresponding wind condition in order to estimate the total impact of rotor asymmetry throughout the lifetime of the farm.This weighted average yields an overall efficiency increase of 0.1%, as many of the wind conditions experienced by the farm do not exhibit a significant efficiency change (bottom panel of figure 7b).

Summary and discussion
The current investigation explores the impacts of asymmetry, in the form of the radial extension of one blade, on the wake of a wind turbine rotor.By modeling the NREL 5 MW turbine using the multi-fidelity vortex solver MIRAS, such asymmetry is shown to perturb the tip vortex system, triggering the pairing instability.Comparison of the velocity fields in the wake of the symmetric and asymmetric rotors generally exhibits an increase in velocity in the center of the asymmetric rotor wake, which grows with downstream distance until 6D.The velocity field is used to calculate available power, showing that a downstream turbine in the wake of the asymmetric rotor experiences up to 11% more available power than one in the wake of the symmetric rotor.Turbulence intensity is also compared in order to gain insights into the effects of rotor asymmetry on the fatigue loading of a downstream turbine.Increases in turbulence intensity on the edges of the asymmetric rotor wake are consistent with enhanced mixing with the freestream flow, while reductions are observed in the wake center after 4.5D downstream.Finally, the Lillgrund wind farm is used as a test case to examine the total potential impact of rotor asymmetry on wind farm efficiency.Under certain wind conditions, efficiency gains > 2% are obtained.
The magnitudes of these available power gains are comparable to other wake control methods such as wake steering (total power increases of 6% [29] and 5% reduction of wake losses [30] for two in-line turbines in turbulence levels of 5-6%), dynamic induction control (total power gains of 20% for two in-line turbines in uniform flow [31]), and dynamic individual pitch control (total power production increases up to 7.5% for two in-line turbines in 5% turbulence intensity [32]).The additional advantage of rotor asymmetry compared to other wake control methods is that it is passive, so the turbine controls do not need to be modified for its implementation.Furthermore, adding an asymmetry such as the radial blade extension investigated in the current study can increase the power generation of the upstream turbine, whereas other wake control strategies tend to reduce it.When extending the wake control investigation to an entire wind farm rather than an in-line turbine pair, the total benefit of rotor asymmetry reduces substantially.However, this reduction would be observed for any wake control method, as wind farms are typically designed to minimize the amount of time the turbines spend in waked conditions.For example, Howland et al. [33] report energy gains of 3% over an entire wind farm for a range of wind speeds where the turbines are operating below their rated power and the wind direction is less favorable.When averaged over all wind conditions, the total gain is only 0.4%, even considering that the wind direction is in the unfavorable range about 50% of the time based on the provided wind rose.Still, considering wind generated 1870 TWh of electricity globally in 2021 [34], even a small percentage increase applied to all wind farms would represent several additional TWh of renewable energy.The key question then becomes whether the costs of such wake control technologies outweigh the benefits.
One important aspect of cost is the loading on the turbine structure and components, which will be investigated in more detail for asymmetric rotors in future studies.While increased turbulence intensity in some parts of the asymmetric rotor wake could lead to increased fatigue loading on downstream turbines, the reduction of the velocity deficit would decrease the forces and moments experienced by a partially-waked turbine [35].Furthermore, the scales of the flow structures in the wake of an asymmetric rotor must be compared to those generated by a symmetric rotor, as these scales have important implications for structural loading [27].The asymmetry of the rotor is also expected to greatly increase the fatigue loading on the upstream turbine components due to the imbalance between the blades, unless a counterweight is used to offset the added forces generated by the longer blade.These considerations must be explored thoroughly so the loading experienced by each turbine component can be understood.
Finally, the impact of inflow turbulence on the efficacy of rotor asymmetry at accelerating wake recovery must be quantified under flow conditions experienced by turbines operating in the atmosphere.Based on previous investigations of other wake control methods, the benefits are expected to decrease, though not disappear, with the introduction of inflow turbulence [31].As the turbulence intensity of the inflow is increased, the benefits are expected to reduce further [31,30].With a certain level of turbulence in the inflow, the tip vortices will likely be destroyed before asymmetry-induced pairing can occur, erasing the benefit of such a wake control method.Further investigation is underway to identify such a threshold.Because turbulence will reduce the impact of vortex pairing on the wake, the technology is expected to be most suitable for offshore wind farms where turbulence levels are relatively low and wake effects persist for several diameters downstream.If the benefits of rotor asymmetry are found to outweigh the costs under atmospheric flow conditions, blade add-ons could be introduced to existing wind farms to mitigate detrimental wake effects.Alternatively, new farms could be designed with asymmetric rotors in the outer rows to reduce the necessary spacing between turbines and increase the energy density of the site.

Figure 1 :
Figure 1: Outlines of the baseline NREL 5MW blade (black) and the blade modified with a radial extension (red), used to replace one of the blades on the asymmetric rotor.The inset on the right shows a comparison of the baseline and modified blade tips.

Figure 2 :
Figure 2: Instantaneous vorticity field in the y = 0 plane of the (top) symmetric and (bottom) asymmetric rotor simulations.The inset highlights the occurrence of vortex pairing in the wake of the asymmetric rotor.

Figure 4 :
Figure 4: (a) Chart showing the gain in mean velocity within a virtual rotor disk in the wake of the asymmetric rotor relative to the symmetric rotor centered at different streamwise (x c ) and spanwise (y c ) locations, as defined in equation (2).(b) Chart showing the gain in available power within a virtual rotor disk in the wake of the asymmetric rotor relative to the symmetric rotor centered at different streamwise and spanwise locations, as defined in equation (4).

Figure 7 :
Figure 7: (a) Lillgrund wind farm layout based on the data provided by Sebastiani et al. [23].Each circle represents a turbine, and the filled circle is the reference turbine from which the distances are measured.(b) Total efficiency of the wind farm, as defined by equation (1) for each turbine, binned by wind speed and direction.The top panel shows the baseline case, the middle shows the case with the benefits of rotor asymmetry applied to the second row turbines (see main text), and the bottom shows the percent change between the two cases.White regions represent wind conditions where insufficient data is available.