ABSTRACT

INTRODUCTION
Wind turbines operate in large-scale farms where the wake of turbines interact with each other.  A turbine's wake has a velocity deficit and impacts the turbulence intensity, there- fore turbines which sit within a wake experience reductions in power generation reported to be in the order of 10-25% and increases in fatigue loads of a similar magnitude (Barthelmie et al. , 2010; Hansen et al. , 2012).  The interaction between atmospheric turbulence and turbine wakes is complex so de- composing flow phenomena to understand the impact of dif- ferent components of the atmospheric boundary layer (ABL) is useful.  In particular, wind turbines which exist high up in the ABL can be subjected to strong shear and thermal strati- fication which vary in strength throughout the day.  Thermal stratification impacts the shear profile and the intensity and structure of turbulence. The thermal stratification gradient can be used to classify the ABL into neutral, stable and convec- tive. When the thermal stratification is positive, the boundary layer is considered to be stable. In the stable boundary layer the wake has been found to be the longest, with the highest reductions in power output, so is of particular interest (Abkar & Porté-Agel, 2015). Hodgkin et al. (2022) conduct thorough analysis on wakes in a comprehensive range of shear and ther- mal stratification strengths.  They find that shear has a large impact on the wake length and shape and breakdown loca- tions, whereas thermal stratification only has a limited effect on the wake shape. Additionally, due to Coriolis forces caused by the rotation of the Earth, wind turbines can be subjected to veer, where the direction of the wind changes with height. This can be especially significant in stable conditions, causing the wind turbine wake to skew (Churchfield & Sirnivas, 2018). By studying the mean kinetic energy (MKE) flux across the wake, Lignarolo et al. (2015) find that the helical tip-vortices of the wake inhibit turbulent mixing and prevent transition to turbulence. Therefore, the wake will not break down without an additional disturbance. The stable wake length is a key parameter in wind farm optimisation.  Knowledge of the mechanisms that cause wake breakdown can improve farm efficiency and design. The sta- ble wake length can be defined as the length between the tur- bine rotor plane and the location where the wake's tip-vortex structure begins to break up into turbulence.  The near-wake field transitions into the far-wake field as the tip-vortices break down into smaller turbulent structures which mix with the am- bient fluid as the wake recovers (Vermeer et al. , 2003).  Wid- nall (1972) states that the mutual inductance instability is one of three main instabilities that leads the vortex breakdown. It occurs when out-of-phase perturbations causes neighbouring vortex filaments to disturb each other leading to vortex pairing and breakdown. Control methods which aim to shorten the stable wake length by manipulating a turbine's induction factor, known as dynamic induction control, are growing in popularity.  Using periodic fluctuations in torque or blade pitch, the thrust coeffi- cient of a wind turbine can be altered in a way to enhance wake mixing. Munters & Meyers (2018 a ) find the optimal frequency for the perturbation has a Strouhal number of 0.25.  Frederik et al. (2020) test the periodic scheme of Munters & Meyers (2018 a ) using pitch control in wind tunnel and large-eddy sim- ulation (LES) experiments and confirm their findings, showing a potential power increase of up to 4% (mostly coming from the first downstream turbine). Brown et al. (2021) identify two optimal forcing strategies, one of which is to force the mutual induction instability, with a frequency equal to one and a half 1 times the rotation rate.  The optimal frequency for dynamic induction control is expected to depend on the turbine and at- mospheric conditions, and the physical mechanisms which are responsible for the wake's breakdown need to be investigated in depth (Houck, 2022). In this work, the breakup mechanisms of wind turbine tip-vortices in flow with shear, thermal stratification and veer are investigated using high-fidelity turbulence resolving sim- ulations, with a focus on the near-wake field.  We study im- pact of dynamic pitch control on the wake length and shape in these conditions. Data-driven analysis methods such as proper orthogonal decomposition and mean kinetic energy flux are utilised to further understand the role coherent structures have on the wake’s breakdown.
METHOD
The high-fidelity turbulence resolving simulations in this work are carried out with the wind farm simulator WInc3D (Deskos et al. , 2020), which is part of the high-order, finite-difference, framework of flow solvers Xcompact3d (Bartholomew et al. , 2020). The simulations are based on the incompressible Navier-Stokes and temperature scalar transport equations, coupled through a gravitational term,
where u is the velocity vector field, p is the pressure field, θ is the potential temperature and F i accounts for additional forcing (with i = 1,2,3 corresponding to the streamwise ( x ), vertical (y) and spanwise ( z ) directions, respectively), Re = U 0 R/v is the Reynolds number based on the reference free-stream ve- locity ( U 0 ), the turbine radius ( R ) and the kinematic viscos- ity ( v ), and Sc is the Schmidt number.  These equations are solved with sixth-order finite-difference schemes on a Carte- sian mesh.  More details of the code implementation can be found in Laizet & Lamballais (2009); Laizet & Li (2011). Sub- grid stresses are handled with an implicit LES scheme which uses the hyper-viscous momentum diffusion term ( β ) or tem- perature diffusion term ( Q ) to introduce the required dissipa- tion; more details can be found in Dairay et al. (2017).  A conventional actuator line method is used to model the turbine via a forcing term added to the momentum equation. The simulations setup is based on a scale-model three- bladed turbine (with radius R = 10.24 m) from experiments conducted in the wind-tunnel of NTNU (Krogstad & Eriksen, 2013). The computational domain, 10R × 10R × 10R , is dis- cretised with a uniform mesh of 513 × 513 × 513 nodes. The turbine is placed at mid-height, 2R downstream of the inlet. The Reynolds number is equal to Re = 30,000 with a tip speed ratio, λ = ω ⁄ R ⁄ U 0 = 6, where ω ⁄ is the angular velocity of the turbine. The effect of shear and thermal stratification on the flow is studied by imposing vertical positive gradients of stream- wise velocity and temperature. The velocity gradient is main- tained with free-slip boundaries in the spanwise and vertical direction with inflow/outflow boundaries in the streamwise di- rection. The temperature gradient is maintained with vertical wall fixed-temperature boundary conditions. The strength of these gradients is defined as a percentage change in the quan- tity from the centre-line of the turbine ( y = 0) to one vertical radius ( y = R ). We test shear strengths of 10%, 20% and 30% and thermal stratification of 1%.  To study veer we alter the wind-direction by a constant 4° change in flow angle per verti- cal radius and maintain this using periodic boundary condition in the spanwise direction. The method used to trigger tip-vortex instabilities is dy- namic induction control, practically applied by altering the pitch of the blades. The pitch of the blades are defined in time with a sinusoidal function, θ b = A sin (ω, + ω, ) t , where θ b is the pitch of blade b , ωε is the excitation angular frequency, and A is the amplitude.  This work follows previous recom- mendations (Munters & Meyers, 2018b ; Frederik et al. , 2020) to have a low excitation frequency of ως ≈ ω, /10 and with an amplitude of 2.5°. Proper orthogonal decomposition is an analysis technique that finds coherent structures of a flow field by finding orthog- onal basis vectors which can be reconstructed to return to the complete flow field using
where " indicates the averaged value over N snapshots, index k corresponds to the k th mode velocity field, φ k is the velocity of mode k in direction i and a k is the corresponding temporal coefficient (amplitude).  The modes are ordered in terms of their total energy contribution to the flow, and appear in pairs with the same frequency, shape and energy content. The mean kinetic energy (MKE) flux can be used to quan- tify the impact that coherent structures have on the wake break- down.  A positive MKE flux indicates that the flow helps the wake break down, as this refers to turbulent energy being brought into the control volume from the ambient, whereas a negative MKE flux is the opposite, and hence suggests it is delaying breakdown. The MKE term is extracted from the transport equation for the MKE:
with ιℓιd , = ụụ– ụj ụ̄j . To relate this to a specific POD mode the following approximation is used: where λ k is the eigenvalue of mode k , which represents the strength of the k th mode.  Combining equation 3 and 4 and integrating this term over a control volume (V c ) enclosed by a surface ( S c ) finds the turbulent MKE flux per unit area for a single mode:
The MKE flux of a mode pair is simply the addition of the MKE flux for each mode in that pair. This method is explained in more depth in Hodgkin et al. (2022).
RESULTS
Uniform Flow
The wake from a turbine in uniform flow with dynamic pitch control can be seen in Figure 1. The stable wake length is about 1.5 R and multiple 'overlap' locations can be seen where the vortex filaments have rolled over each other and interacted. Breakdown ultimately occurs when neighbouring vortex fil- aments, which oscillate partially out-of-phase, interact caus- ing their perturbations to grow. The vortex interaction pattern caused by the selected excitation frequency can be seen more clearly in Figure 2.  This plot shows the vortex filaments if they are cut along the top and flattened out so that 0° corre- sponds to the top of the domain and 180° to the bottom. The interaction sees vertical 'overlap' locations which are spaced approximately 0.7R apart and become more turbulent as they progress downstream.
Shear
Shear acts to lengthen the wake along its top and shorten it along its bottom.  This can be visualised by looking at the vorticity contours, in Figure 3, which show the wake as it expands downstream of the turbine for 10%, 20% and 30% shear.  With shear, the spacing between neighbouring vortex filaments is increased along the top and reduced along the bot- tom, which consequently causes the wake to break down first at the bottom.  Increasing shear increases the lengthening ef- fect at the top and the shortening effect at the bottom. Turbu- lent structures are seen at the bottom of the domain once the wake has started to break down. We can look at the planar vor- tex lines to observe the vortex filament pattern more clearly. Figure 4 shows these for the 20% shear only case (as well as for 20% shear with and without thermal stratification and veer, discussed subsequently) . It can be seen that, although shear causes the vortex filaments to bend, there remains 'overlap' locations where the wake is breaking down. The first overlap is located at x = 1.9R for 0° and at x = 1.2R for 180°. Shear increases the power extracted from the wind. The power co- efficient compared with uniform flow increases by 0.22% for 10% shear, 0.78% for 20% shear and 1.56% for 30% shear.
Thermal Stratification
Thermal stratification is added to the 20% shear case to mimic the shear and thermal stratification levels in a sta- ble boundary layer (Abkar & Porté-Agel, 2015).  In a stable boundary layer strong shear is expected with a positive tem- perature gradient.  Hodgkin et al. (2022) shows that the ef- fect of thermal stratification is amplified with strong shear so this set up is the most interesting. There is little visible differ- ence between the flow with and without thermal stratification. Thermal stratification alone does not change the location of the vortex breakdown. There is a slight advance in the breakup lo- cation with thermal stratification but this is negligible, shown in Figure 4. Thermal stratification reduces the power extracted from the wind. Adding 1% thermal stratification to 20% shear reduces the power coefficient compared with uniform flow to -1.01%, a 1.79% reduction from the 20% shear only case.
Veer
For interest, we add a constant veer gradient to the inflow with shear and thermal stratification.  Veer is caused by the Earth's rotation and may have an impact on the wake length and shape.  As seen in Figure 4 veer does not change the lo- cation of the vortex breakup or the stable wake length signif- icantly.  The power coefficient compared with uniform flow is reduced to -1.13%, a -0.12% reduction compared with the 20% shear + 1% thermal stratification case. This is small com- pared with the changes caused by adding thermal stratification or shear.
Wake Area
The wake area can be impacted by the simulation condi- tions. Figure 5 shows the boundary of the wake, calculated as the convex hull of the points which have a vorticity above a minimum average vorticity. The wake boundary is provided for 3 downstream locations, spaced 1 R apart. For all cases the wake expands as it moves downstream. The uniform flow case has an equal radius for all angles about the turbine, of 1.26R at 3R downstream. Adding shear reduces the radius at 0° (top), at 3R this reduction is 3% for 10% shear, 5% for 20% shear and 12% for 30% shear. Shear increases the radius at the bottom, at 3R this increase is 4% for 10% shear, 9% for 20% shear and 19% for 30% shear.  The change due to shear at the bottom is larger than at the top.  Thermal stratification mainly has a impact on the radius at the sides, most clearly seen at 3R. The radius at the sides is increased on average by 4% compared with the 20% shear only case.  This change is comparable to the top and bottom effect caused by 10% shear. The most dra- matic change to the wake boundary is observed when veer is present. Veer causes the wake boundary to skew, with maxima found at 45° and 225°. The location of the maximum radius for 3R (at 225°) is 8% bigger than the maximum radius found for the same case without veer (at 180°). The wake area could have a significant follow on impact to the far-wake field and dynamics.
Coherent Structures
We use POD to extract coherent structures from the flow field. The first mode of the four most dominant mode pairs for uniform flow are shown in Figure 6. The first three mode pairs are related to the control scheme, as these modes are not present when there is no control.  The dominant frequency of mode pair 1 is the same as the pitch control frequency, 1.1 w t , and distinct rings can be seen.  Mode pair 4 is related to the helical spiral and is dominant in the first half of the domain when the wake in laminar or transitioning. The four dominant POD mode shapes for 20% shear are shown in Figure 7.  The mode frequencies and structure are the same as for uniform flow, but the modes are slanted to due to the effects of shear. In uniform flow, mode pair 1 extends through the whole domain, whereas in shear this mode maps onto the laminar part of the wake only, ending by 1.5R at the bottom of the domain.  The shape of mode pair 2 also maps onto the laminar wake only, with its main structure ending at about 2.2R in uniform flow and 1.5R (bottom) and 3.5R (top) in sheared flow. Mode pair 4, which represents the helical spi- ral is dominant in the same parts of the domain as mode pair 2 for both inflows. Mode pair 3 on the other hand, begins once the wake starts to break down to turbulence, becoming strong by about 1.5R in uniform flow and 1 R (bottom) and 2.5 R (top) in shear flow.  We note that mode pair 1 can be characterised as an anomaly in that it maps slightly differently for uniform and shear flow.  The POD modes for the cases with thermal stratification and veer have the same structure as with uniform flow and shear only flow, and only their area and length shapes change slightly to map the overall shape of the wake.
Mean Kinetic Energy Flux
We calculate the total MKE flux for the four most domi- nant POD mode pairs (ֲֲֲֲֲֲֲֲ( α̣λ̣α̣α̣α̣), as well as for the com- plete flow ( − σ x ) ( ū i l i i )). The control volume for the calcu- lation is a cylinder, centred on the turbine, of radius 1.2R and length spanning from x = 0.5R to x = 3.5R.  The MKE flux values are normalised by the total MKE flux for the complete flow in uniform inflow.  It is important to note that the wake area for each setup is different (Figure 5) so the control vol- ume will be crossing the wake boundary at different points which can affect the results. However, we fix the boundary ra- dius at 1.2R to account for some expansion of the wake. The data are shown in Figure 8.  10% shear and 20% shear cases both have a larger complete flow MKE flux than the uniform case, with the largest MKE flux at 20% shear (101% of uni- form), however for 30% shear the MKE flux is about 90% of the uniform value. The wake radius at the bottom of the do- main for 30% shear is much larger so this could be responsible for the reduction in flux, at least in part. Adding thermal strati- fication to 20% shear reduces the total MKE flux to just below 100%. The lowest MKE flux for complete flow (86% of uni- form) is found for the case with thermal stratification and veer. However, as shown in Figure 4, veer does not reduce the stable wake length which indicates this difference could be related to the wake area rather than a true reduction in MKE flux. Changing the inflow conditions has an impact on the total MKE flux in the four studied mode pairs.  For all cases mode pair 4 relates to the helical spiral and has a negative MKE flux indicating that this mode pair delays wake recovery.  This is expected as without any added disturbance the helical spiral vortex itself will not breakdown, as shown by Lignarolo et al. (2015).  The magnitude of the MKE flux for mode pair 4 is largest for uniform flow, this suggests that the other inflow set ups reduce the shielding effect of the helical spiral.  Shear, thermal stratification and veer all further reduce the magnitude of the MKE flux of this mode pair. Mode pairs 1-3, which are related to the control strategy, all have a positive MKE flux, hence they accelerate break- down by providing the needed turbulent energy to disturb the wake's helical spiral structure.  Mode pair 1 has the smallest flux for uniform flow and shear increases the flux of this mode, whereas, mode pairs 2 and 3 see the largest flux at uniform flow and shear decreases the flux for these modes. This means that mode pair 1 has the largest flux at 30% shear and it con- tributes the most to wake breakdown for this setup (out of the four studied modes). Thermal stratification slightly increases the flux in mode pair 1 and 3 but decreases the flux in mode pair 2. Adding veer decreases the flux in all 3 of these mode pairs.  Veer dramatically changes the wake area which could be influencing this result. For all cases, the MKE flux in the first four modes is in the order of a third of the complete flow MKE flux, suggest- ing that small scale turbulent structures (lower energy POD modes) contribute significantly to wake breakdown.
CONCLUSIONS
In this paper we have presented selected results from a set of simulations of a wind turbine with dynamic pitch control in flow with shear, thermal stratification and veer. The aim was to further understand how the dynamic pitch control is impacting the wake development in these flow conditions. We found that with dynamic pitch control, the stable wake length was short- ened as vortex filaments were disturbed and interacted with each other by about 1.5 R downstream of the turbine.  With shear, the wake length is increased along the top and short- ened along the bottom.  This effect is more important when shear is increased. Thermal stratification alone does not im- pact the wake length significantly, but does reduce the power output from the turbine. For all cases, the wake expands as it travels downstream. In uniform flow the wake area is radially the same. The addition of shear decreases the radius at the top and increases it at the bottom.  Thermal stratification causes an increase in the radius at the sides. The most noticeable im- pact from veer is a skew in the wake boundary shape, with a maximum effect found at 45° and 225°. We found three dominant POD mode pairs relating to the pitch control, and the fourth most dominant was related to the helical spiral vortex.  The four most dominant POD modes have the same structure for all inflow conditions, but their mode shape and area change to map onto the wake shape and area.  Mode pairs 1-3 provide necessary MKE flux to cause wake breakdown and mode pair 4, which relates to the helical spiral vortex, provides a negative MKE flux. The magnitude of these fluxes is impacted by the inflow. Future work will expand the analysis of these simula- tions using other decomposition methods based on frequency. We would then like to conduct simulations in a full turbulent boundary layer to see if these key conclusions hold in a more realistic configuration.