The blade geometries of the DTU 10 MW RWT with a rotor diameter of 178.3 m and a hub height of 119 m are used . Tower, nacelle, and hub are not included in the simulation.

The turbine is controlled by the k-ω2 controller in region 2 of the power curve. It relies on the generator's rotational speed, ω, as input. For simplicity, we assume the turbine is a direct-drive turbine, meaning the generator and rotor speeds are equal when losses are neglected. In order to understand the underlying assumptions of k-ω2 control, the control law will be derived in the following. We start with the turbine power, which for a given wind speed URef can be calculated as 1P=12ρR2πCP(λ,θ,X)URef3, with air density ρ; rotor radius R; and power coefficient CP, which depends on the tip-speed ratio λ=ωR/URef, the blade pitch θ, and the location X, as shown by the aforementioned research . In the following, these dependencies will not be explicitly mentioned. The definition of the reference wind speed URef is provided in Sect. . The rotor torque is related to the power as 2τRot=Pω=12ρR2πCPURef3ω. To design a controller that tracks optimal performance, it is desired to keep the local inflow angles at the blade constant under the conditions where the maximum power coefficient CP,max is reached. This means that the tip-speed ratio must remain constant under optimal conditions. By substituting URef=ωR/λopt, one obtains 3τRot,opt=12ρR5πCP,maxλopt3︸kω2, where the torque constant k is introduced, representing optimal operational conditions in the given environment. This equation can now be used to set the generator torque τGen as a function of the rotor speed to ensure optimal operation: 4τGen=kω2. The substitution of URef to maintain constant local inflow angles at the blade is essential for understanding how the controller works from an aerodynamic perspective. Of course, the controller does not measure the inflow angles. But maintaining τRot∝ω2 is, from an aerodynamic perspective, only possible when aerodynamic forces also scale with the rotor speed squared, which is only the case when local inflow angles at the blades are kept constant. This will become apparent in the course of this paper. In steady state, conservation of angular momentum between the rotor and the generator yields 5τGen=τRot6⇔kω2=12ρR5πCPλ3ω27⇔CP,maxλopt3=CPλ3. Thus, the turbine will always track the curve defined by Eq. (), irrespective of the surrounding flow.

In this work, the k-ω2 controller is implemented as part of a five-region generator torque controller as described by . In every iteration i of the simulation, ω is updated until convergence based on the conservation of angular momentum of the rotor, which can be written as 8ωi+1=ωi+ΔtJτRot,i-τGen,i, with the time step Δt; the rotor moment of inertia J; and the rotor torque τRot,i, which is extracted from the flow simulation. A low-pass filter for the rotational speed is used in Eq. (). In steady state, aerodynamic torque and generator torque are balanced, ensuring that in region 2, the turbine tracks maximum power performance in flat terrain. Since steady-state simulations are performed, the dynamic response of the controller is not of interest here, and Δt/J in Eq. () is solely tuned to improve solution convergence. The blade pitch is controlled by a proportional-integral (PI) controller as also described by , which modifies the blade pitch above rated conditions to maintain rated rotor speed and consequently power. By design, the PI controller saturates below rated conditions at θopt.

The optimal blade pitch θopt, tip-speed ratio λopt, and power coefficient CP,max can be obtained from aeroelastic simulations of the turbine without the flow field being resolved by CFD, as has also been done for the DTU 10 MW RWT by . The optimum identified in this way is not necessarily the same optimum as can be found when resolving the flow field with CFD. In this work, we chose to tune the control constants based on our RANS simulation setup, as shown later in Sect. . The resulting control constants are listed in Table .

We introduce the following decomposition of the mean flow: 9u=U+u′, where u is the flow field including the turbine interacting with it, U is the undisturbed flow field, and u′ quantifies the disturbance by the turbine. The reference wind speed employed throughout this work is the undisturbed rotor-equivalent wind speed, which we calculated as an average not only over height as originally proposed by , but also over the lateral dimension as 10URef=1AR∫ARUR3dA3, with UR being the undisturbed rotor-normal velocity component at the location of the rotor. This approach takes the variation in available kinetic energy over the rotor plane into account and allows for a more accurate assessment of the efficiency of a turbine. However, it does not consider the streamwise development of the undisturbed flow field.

At every turbine location, URef is obtained separately. Therefore, quantities like λ, CP, and the axial induction a are normalized by the respective local URef evaluated when the turbine is turned off and not by an upstream velocity. By defining the reference wind speed like this, any bias in the performance results due to a change in kinetic energy of the flow induced by the topography between wind speed measurement position and turbine position is avoided. It is also more consistent defining the reference wind speed like this compared to choosing an arbitrary position upstream, where the relation between the wind speed at this position and the turbine position depends on the terrain.

In Sect. , the case of a turbine placed on top of a hill (location C) is used to analyze the combined effect of the increased kinetic energy at the turbine position and the flow deceleration behind the hill.

The shape of the quasi-two-dimensional Gaussian hill is described by 11h=h0exp⁡-x22σ2, and has, in its standard configuration, a height h0 of 1 D and a standard width σ of 1.5 D, with D denoting the turbine diameter. When the turbine is located at the foot of the hill (B), it is 4.5 D away from the top of the hill. The surface roughness is 0.001 m, corresponding to a snowy surface . Due to this low roughness, the flow does not separate behind the crest, leading to a strong deceleration and noticeable impact on the induction . The configuration as a whole is chosen because it modifies the flow in the proximity of the AD at a length scale comparable to the rotor size. When characteristic lengths of the hill, such as height or width, are significantly smaller than the turbine, its impact on performance vanishes, and the same holds when the hill is significantly bigger than the turbine . Effectively, the turbine is subject to a flow comparable to flat terrain in such cases. The setup is similar in dimensions to other studies but purposely idealized to investigate the isolated effect of flow acceleration and deceleration. To allow for more general conclusions, three additional sets of simulations of the turbine on top of the hill are performed. In the first set (C2), the surface roughness is changed to 0.1 m. In the second set (C3.X), the hill height is varied, while the width is kept constant, and in the last set (C4.X), both hill height and width are scaled with the same scaling factor, thus keeping the ratio between height and width constant.

An overview of all simulations is listed in Table . The simulations are performed in two steps. First, the domain is simulated without the turbine to extract the undisturbed URef at the turbine position. Second, after convergence of the empty domain, the turbine is placed into the domain, and the simulation is run until convergence again to extract the relevant turbine and flow metrics.

In all cases, the computational domain is a curvilinear grid with a size of 45 × 18 × 34 D3 in the x, y, and z directions, respectively, which correspond to the streamwise, lateral, and vertical dimensions, respectively. The hill is generated by deforming the bottom surface of a flat domain. In total, the grid has a size of 256 × 192 × 192 = 9.44 × 10⁶ cells. In the turbine region, the mesh is refined with nearly cubic cells with a side length of approximately D/32.

The turbine is simulated as an AD, which is represented in the flow domain by a polar grid, and forces are projected onto the computational grid by the actuator shape approach . The tip-loss correction by Glauert is applied, and the disk grid has 17 radial points and 64 azimuthal points.

The simulations are performed as RANS simulations using the k-ε-fp model as a closure model. The standard model coefficients are left unaltered, and the inflow is described by the analytical log-law solutions for the velocity U, the turbulence kinetic energy k, and the dissipation ε : 12U=u*κlog⁡z+z0z0,k=u*2Cμ,ε=u*3κ(z+z0), with the friction velocity u* and the von Kármán constant κ. It is important to mention that the level of turbulence intensity is independent of the friction velocity, which is varied in order to change the hub height velocity.

The described inflow is set as a boundary condition at the inlet and at the top of the domain in order to maintain the logarithmic profile in the absence of obstacles. At the outlet, a zero-velocity gradient condition is imposed, and at the bottom, a rough wall boundary condition as described by is used. The lateral boundaries are periodic.

The steady RANS equations are solved using the incompressible finite-volume solver EllipSys3D using a modified version of the SIMPLEC algorithm to enforce pressure–velocity coupling .

The sensitivity of the simulation setup to the cell size is analyzed. We evaluate the variation in the reference velocity URef and in the power coefficient CP because they are the most important variables characterizing power performance. The grid is structured; therefore, the cell size is varied by combining several cells into a single cell or, in other words, by running at a lower grid resolution. In Fig. , the results are reported as changes in URef and CP relative to the solution obtained on the finest grid, which is also the grid we use in this work. URef varies by less than 0.01 % when changing the resolution from the finest grid to one level coarser. Increasing the cell side length by a factor of 8 affects the reference velocity in all cases by less than 0.4 %. When the turbine is operating with a controller, the power coefficient varies by less than 2 % between the two finest grid levels. At the coarsest investigated grid, the power coefficient is up to 15 % higher.

The impact of the domain size on the results has been studied in a previous work. For a turbine located on a hill, increasing the cross-sectional area of the computational domain by a factor of 8 results in a change in the disk-averaged velocity of less than 0.1 % for a fixed CT .

In Fig. , the undisturbed velocity profiles and turbulence intensity profiles at all considered turbine positions are shown. For all cases, the inflow profiles are identical to the profiles in flat terrain (A0/A1), corresponding to a neutral inflow at z0=0.001 m as described by Eq. (). Only for C2 are the inflow profiles different due to the higher surface roughness of z0=0.1 m.

In flat terrain, the rotor-averaged turbulence intensity is 6.76 %. At the foot of the hill, the shear of the velocity profile is slightly higher than in flat terrain, and the turbulence intensity is also higher at 7.54 %. On top of the hill (C0/C1), negative shear due to the hill-induced speed-up close to the surface occurs, and the turbulence intensity is only 5.90 %. The cases C3.X–C4.X are mostly enclosed by the flat-terrain profiles and the highest-hill (C0/C1) profiles. With decreasing height, the profiles generally converge towards flat terrain. The hills of C4.X are steeper than the hills of C3.X, leading to a stronger speed-up close to the ground and higher levels of turbulence intensity. The case C2 shows very little shear over the rotor area and the highest levels of turbulence intensity of 9.25 %. The different shear profiles result in different distributions of kinetic energy across the rotor area, but this bias is accounted for by calculating the rotor-equivalent wind speed (Eq. ) as a reference wind speed. Since turbulence intensity also affects turbine power performance, we assess the bias in the simulation results introduced by the different levels of turbulence intensity at the different turbine positions. The impact of turbulence intensity I on power can be estimated as 13P(I)=PI=01+3I2. Based on the rotor-average of the turbulence intensity, it is expected that when neglecting any other effects, such as streamwise flow development, the power and also the power coefficient at the foot of the hill are approximately 0.33 % higher than in flat terrain at the same undisturbed wind speed and approximately 0.32 % lower on top of the hill for the lowest case of turbulence intensity. In the case of C2, we expect an impact of 1.18 %.

In Fig. , the undisturbed flow fields for representative cases at low roughness and high roughness are visualized. Behind the hill with high roughness, a separation region is clearly visible, whereas at low roughness, the flow remains attached to the surface. Notably, as mentioned before, the inflow fields are also slightly different.

Figure  shows CP as a function of the blade pitch angle and the tip-speed ratio for cases A0, B0, and C0. The surfaces are obtained on a coarse grid with Δλ=1 and Δθ= 1°, and the maxima are found by a nested grid search, where, based on the discrete maximum on the coarse grid, a finer region is defined. In this finer region, a surface with spacing Δλ= 0.1 and Δθ= 0.1° is simulated, and based on the obtained maximum within this region, an even finder grid with Δλ= 0.02 and Δθ= 0.02 is simulated. The final maxima are, in all cases, within the finest regions and not on their boundaries, ensuring that the optimum is actually in this region. On top of the hill, a sufficiently converged flow solution could not be reached for cases of λ = 10 and θ=-5, -4° due to the high loading on the flow. The combination of a high disk-based thrust coefficient (CT>1) and a deceleration of the flow behind the ridge leads to oscillations in the solution, suggesting that it may be unsteady and not solvable with a steady-state RANS model.

In flat terrain, the optimal CP is 0.533 at a blade pitch angle of θ=-1.94° and a tip-speed ratio of 8.03. These values differ from the ones specified for the DTU 10 MW RWT (CP≈0.474, θ=0.0, λ=7.5). The DTU 10 MW RWT was designed using an aeroelastic optimization tool with a blade element momentum model to represent the flow . In AD-CFD simulations, the induction is commonly underpredicted , leading to a higher performance compared to solely aeroelastic simulations, as shown by for the DTU 10 MW RWT. In combination with the region around maximum performance being comparably flat and the optimum being sensitive to modeling choices, this provides a likely explanation for the different optimal performance point we observe.

When the turbine is located at the foot of the hill, the maximum power performance increases by 2.46 %, accompanied by a decrease in optimal pitch (-2.16°) and an increase in the tip-speed ratio (8.16). At the top of the hill, the opposite is true. The maximum power coefficient decreases by 18.25 % together with an increase in pitch (-0.24°) and a decrease in the tip-speed ratio (7.17). Both on top of the hill and at its foot, the change in CP is significantly greater than what one would expect due to the different levels of turbulence intensity at the position of the turbine, as discussed in the previous section.

The undisturbed velocity contours in Fig.  show that the flow accelerates behind the turbine located at the foot of the hill, while it decelerates behind the turbine at the hill top. In line with previous research , deceleration results in a decrease in power performance, while acceleration results in an increase. The presented CP surfaces indicate that there is no possibility of operating at the same optimal power coefficient as in the flat case when the turbine is operating on the hill, regardless of pitch and tip-speed ratio, because the power coefficient is limited by the flow development. However, this does not necessarily mean that placing a turbine on top of a hill leads to worse power performance in absolute numbers because wind speeds on top of hills are often higher. The actual benefit one can expect by placing a wind turbine on a hill is discussed in Sect. .

The pitch and torque controllers are tuned based on the steady-state optimal performance in flat terrain (A0), and the resulting control constants are listed in Table . Simulation results with the controller are presented in two steps. First, quantities characterizing performance, such as power, CP, and λ, are presented; second, the resulting flow fields around the turbine are shown.

With the calibrated controller, the power curves are obtained by varying the friction velocity u* in Eq. (). The results for flat terrain, ahead of the hill, and on top of the hill are shown in Fig. , with the power shown in panel (a). The turbine on top of the hill produces significantly less energy below rated wind speed than the one in flat terrain for the same undisturbed wind speed at the rotor, while the one at the hill's foot produces slightly more energy for the same undisturbed rotor wind speed. Note that although the wind speed changes, Reynolds similarity leads to the development of similar flow features independent of the wind speed , which is why the induction below rated wind speed is unaltered by the inflow velocity. Above rated conditions, all turbines produce the same power as a consequence of the pitch controller aiming to maintain a certain rotor speed independently of the maximum available power in the flow. The corresponding rotor speed, shown in Fig. b, exhibits the same behavior as the power, being slightly higher ahead of the hill and significantly lower at the hilltop. Inspection of the blade pitch in Fig. c shows that, below rated conditions, the pitch is identical for all cases, settling at -1.94°. When reaching rated conditions, the controller of the turbine ahead of the hill starts to pitch at lower wind speeds than in flat terrain, whereas the opposite is true for the turbine on the hilltop. This can be attributed to the different rotor speeds, which serve as an input signal for the pitch controller. Overall, the reported trends are in agreement with the results by .

Turning towards the non-dimensional quantities characterizing turbine operation in region 2, Fig. a shows the CP-λ curves of A0, B0, and C0 at θ=-1.94°, including the points where the torque controller settles in region 2 during the power curve calculation for all considered simulations. Regardless of the turbine position, the controller settles at the intersections of the control curve given by Eq. () and the respective performance curves as shown for cases A0/A1, B0/B1, and C0/C1. Specifically for case C1, it becomes apparent that this intersection does not necessarily represent the point of maximum power performance. In numbers, at the foot of the hill, the CP at which the controller settles is 2.40 % higher than in flat terrain, while it is 20.63 % lower on top of the hill. If the torque constant were tuned to match the optimal tip-speed ratio in both cases, the gain ahead of the hill would be around 2.45 %, while on top of the hill, the loss would reduce to 18.67 %. Adjusting also the blade pitch would further improve the results, as shown in Fig. .

Figure b shows the axial induction as a function of λ. At the foot of the hill, it is always lower than in flat terrain, while on top of the hill, it is always higher, corresponding to a higher and lower CP, respectively. At the top of the hill, the optimal tip-speed ratio would be approximately 6.7, resulting in a lower induction relative to the flat case. However, the controller settles at a tip speed ratio of 7.4 with an increased induction relative to the flat case. There is a general trend that for the controlled turbine, the induction decreases with an increasing tip-speed ratio.

In Fig. c, the azimuthally averaged tangential induction a′=uR,t/(ωr) evaluated at 0.75 R is presented. The a′-λ curves show an opposite trend to the a-λ curves. The tangential induction generally decreases with an increasing tip-speed ratio; however, for a given tip-speed ratio, it is always lower at the top of the hill and higher ahead of the hill, which is opposite to the trend followed by the axial induction. In comparison to the axial induction, the tangential induction of the controlled cases seems to stay rather constant during operation at the three different locations. Only a small trend can be observed: ahead of the hill, the tangential induction increases, while it decreases on top of the hill. An outlier is case C2, the one with the higher surface roughness compared to case C1, showing a higher tangential induction than all other cases. In Sect. , it is discussed whether tangential induction is expected to stay constant during torque control. There, the radial distributions of axial and tangential induction are also shown.

After presenting performance quantities only, we will now extend the results by also presenting the respective flow fields for all cases. Figure  shows velocity contours in the x–z plane through the rotor center. Velocities are normalized by the undisturbed velocity at the turbine position. Note that in all cases, the x coordinate is relative to the position of the rotor.

Case C1 in Fig.  shows that the flow speed is lower everywhere around the turbine, while for case B1, one can see that the hill behind the turbine accelerates the flow around the turbine. Comparing cases C1 and C2, the velocity deficit in the wake is lower in the latter case. It has already been noted by that there is a strong negative correlation between the maximum velocity deficit in the wake and the respective power performance, although there are exceptions specifically for cases where flow separation occurs. A comparison of the second and third rows of Fig.  shows that at a given hill height, the power performance is higher for the wider hill or the lower maximum slope, respectively, which is in agreement with and . Comparing the panel of C3.3 with the panel of A1 already indicates how an increasing widening of the hill leads to a flow field around the turbine converging towards a field close to flat conditions.

Figure  shows the undisturbed and disturbed velocities at constant height above ground (hub height). The figure reveals several noteworthy characteristics. First, it is observed that the undisturbed flow does not accelerate at the turbine position ahead of the hill but rather slightly behind it. This indicates that predicting power performance bias solely by evaluating acceleration at the turbine position is insufficient and that the flow field downstream also needs to be considered, as previously mentioned . A comparison of cases C1 (low roughness) and C2 (high roughness) shows a nearly identical development of the undisturbed hub height velocity ahead of the hill, whereas the deficit behind the hill is markedly deeper in C2. When the turbine is operating, the opposite behavior is observed. This indicates a non-linear interaction between turbine, flow, and terrain. Comparing cases with the same hill height but different widths (C3.1–C3.3 and C4.1–C4.3) shows that, at a given hill height, the flow field differs only near the turbine, with stronger acceleration and deceleration for the narrower hills. In all cases, there is a speed-up directly at the turbine position when the turbine is operating (also visible in Fig. ). This is a consequence of the missing nacelle, which leads to a channeling of the flow through the rotor center.

Figure a shows that the torque controller follows its prescribed control curve as expected, even when the surrounding flow field changes to conditions the controller was not calibrated for. On the one side, this is not surprising because, as shown in Eq. (7), the torque controller effectively enforces 14CP,maxλopt3=CPλ3 for any operational state, with the left-hand side being constant and calibrated, for example, for flat terrain in our case. On the other hand, this relation does not directly yield insights into how a and the local a′ change due to complex terrain, which is analyzed in the following.

The pitch controller tracks the difference between the actual and rated rotor speed. In region 3 of the power curve, this leads to the observed behavior that even in a non-uniform background flow, the same rated power is reached in the different cases. In region 2, the pitch controller remains inactive, although Fig.  suggests that performance could be increased by adjusting the pitch. The reason for this is the implementation of the pitch controller as a PI controller. Below the rated wind speed, or rather the rated rotor speed, the difference between actual and rated rotor speed is negative, resulting in a negative pitch signal saturating at the minimum pitch independently of the flow state. Similarly to the torque controller, one could imagine an algorithm that modifies the minimum pitch seeking maximum power performance in region 2 of the power curve. However, in practice, it might again be difficult because of the small differences between optimal and actual pitch.

A common way to account for the effect of complex terrain is to use speed-up factors. Because power scales with the wind speed cubed in the flat-terrain case, the effect of terrain on the power of a turbine is usually estimated as 25Phill≈Pflat(1+ΔU)3, with ΔU=(URef,hill-URef,flat)/URef,flat being a non-dimensional velocity speed-up factor. This estimate is only accurate as long as the actual power coefficient of the turbine is the same as that in the flat-terrain case. However, as shown above, placing the turbine on a hill reduces the power coefficient due to the streamwise evolution of the downstream flow. This raises the question of what is actually the maximum performance that can be reached by placing wind turbines on elevated spots, such as hills.

To address this question, we analyze all simulations with the turbine located on top of the hill using the model by . The model can predict the maximum power coefficient CP,max as a function of the speed-up, ΔU, by assuming that the maximum deceleration behind the hill is comparable to the speed-up ahead of it and that the speed-up region is smaller than the region in which the wake pressure equalizes with the surrounding pressure. This is outlined in Appendix  in more detail. Because this model yields identical results to momentum theory for ΔU=0, the scaling of the power on a hill can be estimated to be 26Phill=PflatCP,max(ΔU)CP,max(ΔU=0)(1+ΔU)3,27=PflatCP,max(ΔU)16/27(1+ΔU)3,28≈Pflat(1+ΔU)1.5, with CP,max calculated from Eqs. () and () and the third line being an approximation based on visual inspection of the resulting curves. This estimate requires that the flow recovers to the flat-terrain state directly behind the turbine and can therefore be interpreted as a lower bound to the possible maximum power performance. To investigate the power increase on top of the hill in the presented simulations, Fig.  compares the simulated power increase with the traditional cubic trend (Eq. ), the lower bound from Eq. () and its approximation Eq. (), as well as a quadratic scaling for reference. It is important to keep in mind that the simulation results also include a controller; thus, the actual maximum available power is expected to be around 1 % higher, as shown in Fig. a.

As expected, the actual power increase on top of the hill does not follow a cubic trend but is lower in all cases. When the flow separates behind the hill, as is the case for z0=0.1 m (C2), and the velocity immediately behind the turbine does not decelerate so strongly, the power is closer to the cubic relationship. The same conclusions can be made comparing simulations C3.1–C3.3 with C4.1–C4.4. When the width does not change, the deceleration behind the turbine is weaker, and the respective power performance is higher. Reducing both width and height results in a more local speed-up followed by a strong deceleration. None of the presented cases seems to scale according to Eq. () but rather mostly show a scaling close to quadratic. Only for very small speed-ups does the scaling seem to approach our theoretical predictions. As outlined previously, this can be explained through the assumptions made for deriving Eq. (), which requires a very local speed-up with immediate wake recovery. These observations suggest that if terrain and flow features in the vicinity of the turbine are more similar to flat-terrain features, such as a very long hill or a separation bubble that delays deceleration, the maximum power coefficients are also similar, leading to a more cubic scaling of power. If this is not the case, and flow and terrain features vary on similar length scales to those of the turbine, a stronger influence on the power coefficient can be expected.

The above analysis shows that, although higher wind speeds at a hilltop lead to increased power output, flow deceleration reduces the achievable gain compared to that predicted by the classical cubic scaling. In cases where turbines are located in smaller local freestream speed-up regions, the scaling is closer to quadratic and may theoretically be even lower.

Although this work deals with complex terrain, the studied case of an AD on a quasi-two-dimensional Gaussian hill remains a significant simplification. The undisturbed flow field is quasi-two-dimensional and varies only in the vertical and streamwise direction. Furthermore, the effect of atmospheric stability was not included. It remains a subject for future studies how a three-dimensional, unsteady flow would interact with the wind turbine and to what extent the result that optimal induction decreases in a decelerating flow and increases in an accelerating flow also holds there. In this light, it would also be important to apply a more sophisticated turbine model, such as an actuator line model or a fully resolved turbine model, which leads to a more accurate representation of the local blade flow. This study focused on k-ω2 control only; however other approaches to maximizing power performance in region 2, such as tip-speed ratio tracking , exist as well, and it would be of interest how these would perform in complex-terrain conditions.