Wake-induced alteration of C p - λ characteristics and operating equilibrium in downstream wind turbines

Abstract

Wake interaction in wind farms modifies the inflow conditions for downstream turbines. However, their influence on the aerodynamic performance of individual turbines remains unclear. This study investigates how wake-added turbulence (WAT) and the disturbed inflow shift the normalised power coefficient ( $C_{p}$ ) and tip-speed ratio ( $λ$ ) using Fast.Farm. Four aligned configurations of NREL 5 MW turbines were simulated under steady inflows of 4 m/s to 25 m/s, with WAT enabled, for two inter-turbine spacings (7.5D and 10D). The results show that turbines operating within strong and wake superimposed regions exhibit higher $C_{p}$ and $λ$ values, with T3 showing a maximum $C_{p}$ of 0.596 at 7.5D spacing, despite reduced power compared with the upstream turbine. Increasing spacing to 10D reduced the wake-induced shift in aerodynamic operating behaviour and improved downstream power recovery, especially within the 8-12 m/s wind speed range. These findings highlight WAT’s influence on the operating equilibrium of downstream turbines.

Keywords

wind farm wake effects downstream turbines aerodynamic performance power coefficient tip-speed ratio

Introduction

Concerns about energy crises, pollution, and climate change motivate the world to rely on cleaner energy sources. Among these, wind energy is one of the most widely tapped renewable energy sources and could serve as a reliable, scalable solution to meet global energy demand while reducing dependence on fossil fuels. For these reasons, the installation of wind farms has increased globally over the past decades (Jung, 2024; Tumse et al., 2024). There are two main types of wind turbines based on the orientation of their hub axis: horizontal-axis wind turbines (HAWT) and vertical-axis wind turbines (VAWT). The HAWT, which has a comparatively higher efficiency due to its operation in the lift effect, has been widely used (Manwell et al., 2010); among these, the three-bladed HAWT is the most common. The turbine’s capacity, hub height, rotor diameter, and all other parameters depend on the site’s potential. Hence, the numerous turbines will be installed to utilise a site’s full potential.

The major problem that a wind farm faces is the wake effect. For a lower power density, it is always recommended to occupy a smaller area by the wind farm (Denholm et al., 2000). The major problem with keeping the turbines closer is the wake effect. When an upstream turbine rotates in one direction, it extracts the energy from the inflow wind, generating a wake region behind it with a deficit velocity (Sanderse et al., 2011). This effect not only affects the power production of downstream machines, but also their fatigue life (Karlina-Barber et al., 2016; Thomsen and Sørensen, 1999) by inducing higher turbulence in the inflow, depending on the position of the downstream rotors (Meng et al., 2018; Rinker et al., 2021).

While analysing the performance of a single turbine, the power coefficient ( $C_{p}$ ) is the key factor that determines a turbine’s aerodynamic performance. This factor is dependent on the tip-speed ratio (TSR) and blade pitch (Manwell et al., 2010). The TSR is the rotational speed of the rotor normalised by the incoming wind speed, which is highly dependent on the blade shape and its number (Çetin et al., 2005). Studies have been conducted to find out the optimum TSR with reduced rotor speed for better energy production (Hosseini et al., 2022). It is always desired to define the optimal operating point using $C_{p} - λ$ curve to analyse the aerodynamic performance of a rotor disc. According to actuator disc momentum theory, the theoretical maximum power coefficient is limited to 16/27, known as the Betz limit (Manwell et al., 2010). But this ideal behaviour was observed under steady wind conditions. However, this typical analysis becomes questionable when turbines operate under a wake zone or in a turbulent inflow because the local speed experienced by the downstream turbine will not be the same as the ambient condition as it passes the upstream turbine. This inflow for downstream machines will exhibit spatial non-uniformity and temporal fluctuations due to wake-added turbulence (WAT). Studies have highlighted that scenarios where tidal turbines exceed the Betz limit due to their arrangements and channel conditions (Vennell, 2013). A few studies have also shown that the exceedance of the Betz limit is common even in vertical-axis wind turbines (Debnath et al., 2024; Thönnißen et al., 2016).

It is important to consider the WAT, which accounts for the additional turbulent kinetic energy generated by rotor and vortex breakdowns (Lignarolo et al., 2015). Considering WAT to analyse wake-induced effects on dynamic loading, power production, and altered performance of downstream turbines will help accurately understand the downstream turbines (Göçmen et al., 2016). In addition, wake superimposed models have been developed to account for multiple overlapping wakes in wind farm arrangements (Jonkman and Shaler, 2021). Even though high-fidelity computational fluid dynamics (CFD) methods, such as Reynolds-averaged Navier-Stokes (RANS) and large-eddy simulations (LES), have been utilised to accurately study wake mixing and wake characteristics (Sanz Rodrigo et al., 2017), their high computational costs and duration are major drawbacks for large-scale studies. Hence, mid-fidelity engineering tools have gained popularity for their lower computational expense and their ability to capture physical realism. Several experimental and numerical studies utilising high-fidelity tools have addressed wake characteristics (Cao et al., 2023; Dörenkämper et al., 2015; Li et al., 2016; Zhang et al., 2023). Studies have shown that mid-fidelity tools can also account for significant effects on power loss, rotor effects, and wake recovery, compared with high-fidelity tools (Jonkman et al., 2018; Shaler et al., 2020). Most engineering applications of these wake models have focused solely on understanding farm-level power production and structural loads (Li et al., 2025; Liu et al., 2020; Wu et al., 2020; Xu et al., 2024). Few studies have been conducted to understand how wake-altered inflow will affect the aerodynamic equilibrium of each downstream turbine, particularly in terms of $C_{p} - λ$ behaviour and its optimal operating condition.

Despite this great progress in wake modelling and wind farm layout optimisation (Cao et al., 2022; He et al., 2024; Sinner and Fleming, 2024; Zhao et al., 2022) knowledge gap remains in analysing the characterisation of the wake-induced shift in aerodynamic operating equilibrium. Specifically, it is unclear how downstream rotors adjust their TSR and power coefficient in response to wake-added turbulence compared with steady ambient conditions. Studies had quantified the power loss compared to upstream turbines, but rarely analysed the modification in $C_{p} - λ$ behaviour or identify changes in optimal operating regions caused by the wake effect (Gao et al., 2024; Tang et al., 2019).

The present study addresses this research gap by investigating wake-induced changes in aerodynamic performance in an aligned configuration of multiple turbine arrangements using a validated mid-fidelity engineering tool. Unlike other wake studies, which mainly focus on farm-level power production and dynamic responses, this study focuses on the altered aerodynamic equilibrium induced by wake-affected inflow for each downstream turbine. By systematically varying inter-turbine spacing and evaluating downstream turbine performance under controlled steady ambient conditions with WAT enabled to quantify shifts in the optimal power coefficient ( $C_{p}$ ) and corresponding tip-speed ratio ( $λ$ ). Furthermore, insights into altered aerodynamic operating regions may lead to the development of adapting rotor speed and pitch settings to wake-influenced inflow conditions. As wind farms grow, addressing these shifted operating states is essential to increasing wind farm efficiency and ensuring the economic feasibility of the wind energy system.

Numerical modelling

The present study was conducted utilising Fast.Farm, which is a Multiphysics mid-fidelity tool for predicting the power and structural load in a wind farm (Jonkman et al., 2017). It uses various OpenFast modules to complete the analysis. In addition, it solves complex wake features, such as wake advection, deflection, and merging. To study the wake interaction effect on the performance and aerodynamic efficiency of the downstream turbines, four turbines are placed back-to-back to capture the complete wake effect on downstream turbines (Jonkman et al., 2018) where the upstream turbine is named T1, the downstream machines are named T2 to T4. After the fourth row, the variation in the effect of wake is negligible (Li et al., 2025). To generate a Cp–λ curve for the downstream turbines and compare it with the upstream turbine, a series of steady-state ambient winds is generated from 4 m/s to 25 m/s at 1 m/s intervals, enabling wake-added turbulence, which is already validated by Fast.Farm (Branlard et al., 2024).

This introduces turbulence into the system, which in turn facilitates the breakdown of the vortices. So, it is easy to compare how the downstream turbine aerodynamic efficiency changes as turbulence or WAT are encountered. The effect of wake changes with wind speed as well as the inter-turbine spacing, or the distance up to which the wake is travelled. To account for this, these four turbines are arranged in two different inter-turbine spacing with 7.5D and 10D, where D represents the rotor diameter. The schematic representation of farm layouts and turbine position is shown in Figure 1. For this study, the NREL 5 MW baseline turbine is used, and its details are presented in Table 1.

Figure 1.

Schematic representation of wind farm layout; (a) ITS 7.5D; (b) ITS 10D.

Table 1.

5 MW baseline description.

Parameters	Values
Maximum rated power	5 MW
Hub height	87.6 m
Rotor diameter	126 m
Height above ground (MSL)	87.6 m
Hub inertia	115,926 kg m²
Hub diameter	3 m
Rotor Orientation, Configuration	Upwind, 3 blades
Cut-in, rated, cut-out wind velocity	3 m/s, 11.4 m/s, 25 m/s

The domain width and height to accommodate the growing wake are selected as per (Jonkman et al., 2018) and the domain length is 6 km. For each wind speed, the spatial and temporal discretisation changes; hence, these values can vary from simulation to simulation, as this study considers several wind velocities. The discretisation values for both low and high-resolution domains are selected less than recommended by (Jonkman and Shaler, 2021) for every velocity and for higher accuracy.

The power coefficient $C_{p}$ and the tip-speed ratio λ are non-dimensional parameter for assessing turbines primary performance (Manwell et al., 2010). Where $C_{p}$ can be simply expressed as

C_{p} = \frac{P}{\frac{1}{2} ρ A U^{3}} = \frac{R o t o r P o w e r}{P o w e r i n t h e w i n d}

(1)

Where $ρ$ is the air density, $A$ is the rotor swept area and $U$ is the free stream velocity.

This represents the rotor’s efficiency to extract power from the wind. The generated power is not the same as the rotor power; it should also account for gearbox, generator, and other mechanical efficiencies. So, the optimum $C_{p}$ always represents the maximum power the rotor can provide for power production. Considering $C_{p}$ , it is a function of both λ as well as blade pitch β. λ defines the ratio of blade tip speed to the wind velocity.

λ = \frac{Ω R}{U}

(2)

whereas β is the angle of the blades relative to the inflow wind to control rotational speed and output power. So, in practice, a turbine’s control system always tries to operate in this

λ

to get optimum

C_{p}

From ideal momentum theory, the optimum $C_{p}$ was found to be 16/27. As per Actuator Disc momentum theory, this approximation, the rotor disc imposes an axial velocity induction factor a. Thus, conservation of mass and momentum gives a classic relation between Thrust coefficient $C_{T}$ and $C_{p}$ leading to

C_{p} = 4 a {(1 - a)}^{2}

(3)

Resulting in an optimum $a$ of 1/3 and $C_{p}$ of 16/27. But these are derived for ideal steady wind conditions, which differ from real-life scenarios.

Wake deficit model

Recent models discovered that the wake deficit has a Gaussian profile in radius R (Jonkman and Shaler, 2021) such as

\frac{Δ U (r, x)}{U_{\infty}} = (1 - \sqrt{1 - C_{T}}) \exp [- \frac{r^{2}}{2 σ {(x)}^{2}}]

(4)

where r is the radial distance from the wake centre, x is the downstream distance, and

σ (x)

is the wake spreading parameter with width increasing approximately linearly with downstream. In Fast.Farm the wake dynamic model is axisymmetric, which produces deficits approximately in Gaussian on far wake zones. When multiple turbines are arranged so that the wakes are superimposed, the ambient wind and array effects module of Fast.Farm merges wakes by considering the axial deficits as a scalar quantity whose local kinetic energy could be combined using the Root-Sum-square method. For each point, the combined axial deficit is calculated using equation (5)

| Δ U_{t o t} | = \sqrt{\sum_{w a k e s} {(Δ U_{i})}^{2}}

(5)

and the radial velocity components are combined by vector summation (Katic et al., 1987).

The major contribution of wake is that it alters the turbulence. Fast.Farm explicitly add small wake-added turbulence of each turbine. This WAT model assumes that this extra turbulence is proportional to the local velocity deficit and its gradient. Hence, a scaling factor represented in equation (6) is calculated for each wake plane in its meandering frame.

k (x, y, z) = \frac{k_{def}^{W A T}}{\bar{U}} | V_{x}^{W a k e} (x, y, z) | + \frac{k_{g r a d}^{W A T} D}{2 \bar{U}} [| \frac{\partial V_{x}^{W a k e} (x, y, z)}{\partial r} | + | \frac{1}{r} \frac{\partial V_{x}^{W a k e} (x, y, z)}{\partial θ} |]

(6)

where D is the reference diameter of the wake plane, and the

\bar{U}

is the mean velocity normal to the rotor disk considered in the turbine location. x, y, z and

θ

are coordinates taken in the meandering frame of the wake plane reference. The parameters

k_{def}^{W A T}

and

k_{g r a d}^{W A T}

are the two tuning parameters account for the wake deficit and its gradient.

Fast.Farm modelling framework

Fast.Farm couples these wake models with other turbine dynamics. It utilises multiple OpenFast modules for each turbine, where it calculates the thrust from the incoming disturbed flow. These individual turbines solve the blade-element momentum equations, also with tip-loss and yaw corrections. At the same time wake dynamic (WD) module uses each turbine’s $C_{T}$ and yaw to find out the wake deficit and meandering, utilising the thin shear layer approximation of the Reynolds-averaged Navier-Stokes equation, which only assumes the velocity gradient is greater in the radial direction than in the axial direction (Jonkman and Shaler, 2021). At last, the ambient wind and array effects (AWAE) module identifies where wakes overlap and applies RSS or vector sum, as described above. The output is each turbine’s inflow at every time step, ensuring that the upstream turbine’s wake consistently updates the inflow to downstream rotors, changing their local $C_{p}$ and operating conditions.

Numerical validation

To show the credibility of this study and the author’s ability to correctly utilise the tool, the upstream turbine shown in Figure 2 is considered for validation. As mentioned earlier, a series of simulations was done with wind velocity ranging from 4 m/s to 25 m/s, with steady-state increments of 1 m/s, enabling wake-added turbulence, which will alter the inflow to downstream turbines. So, the upstream turbine experiencing steady wind will give each point on the $C_{p} - λ$ plot. Instead of using a legacy controller, which uses a simple algorithm, in the present study, we utilised an updated controller with industry-standard features, which can function very well with different turbulent flows (Abbas et al., 2020). From the NREL published report, which utilises a legacy controller, the optimum $C_{p}$ was found to be 0.482 at a $λ$ of 7.55 and $β$ of ${0.0}^{0}$ (Jonkman et al., 2009).

Figure 2.

Fast.Farm setup for validation.

Figure 3 compares the normalised $C_{p} - λ$ values of the upstream turbine (T1) obtained from the present study with NREL published data. The predicted optimum normalised $C_{p}$ and corresponding $λ$ values show good agreement with the reference data, as mentioned earlier. This confirms the right application of the adopted turbine model, controller implementation, and simulation methodology for the present study.

Figure 3.

$C_{p} (λ, β)$ plot for 5 MW baseline (validation of present study).

Numerical methodology

Unlike a steady-state condition, the outputs from a turbulent operating turbine will differ. Figure 4 represents when T1 operates at a steady wind speed of 12 m/s how the WAT is altering the $C_{p}$ and $λ$ for T2. It can be described as a collection of transient portions, so dealing with these results requires careful handling. Instead of simply averaging the entire value over time, a different methodology is used that accounts for the dynamic nature of both values. Each case ran for three trials of 1500 seconds to account for the statistical convergence of the turbulent nature. During post-processing, the first 150 seconds were ignored to ensure that no transient portion was included in the results.

Figure 4.

(a) $C_{p}$ vs time; (b) $λ$ vs time; (Left) T1 and (Right) T2.

In the present study, the aerodynamic coefficients of each downstream turbine were obtained from the AeroDyn outputs of individual turbines. This approach considers rotor-disc averaged wind velocity normal to the rotor plane and instantaneous rotor swept area for the calculation of the non-dimensional parameter (Cp). Hence, the normalisation quantities change with time under wake-influenced turbulent inflow conditions. This consideration helps in understanding the wake-affected behaviour of each individual downstream turbine rather than at the farm scale. Therefore, the reported values of $C_{p}$ differ from the power coefficients defined using fixed free stream reference conditions. The downstream $C_{p}$ values reported in this study should be considered due to the altered wake disturbed inflow conditions and their resulting aerodynamic operating behaviour, rather than considering it as a measure of power extraction efficiency.

The aerodynamic performance data were extracted from simulations for each turbine in the farm layout as shown in Figure 2. $C_{p}$ , $λ$ , and $β$ were obtained from the output list. The $C_{p}$ used is time-averaged over a sufficiently long averaging period to average out the dynamic inertial effect. To obtain a statistically consistent plot, the $(λ, β)$ domain was divided into uniform bin with fixed bin widths. For each bin $(λ_{i}, β_{j})$ , the mean power coefficient was computed using equation (7).

\bar{C_{p}} (λ_{i}, β_{j}) = \frac{1}{N_{i j}} \sum_{k = 1}^{N_{i j}} C_{p, k}

(7)

where,

N_{i j}

is the number of data samples within the bin. This averaging technique reduces the influence of turbulence-induced spikes and transient overshoots, such that the contour will represent the steady aerodynamic behaviour rather than instantaneous variations. From the resulting plot, the maximum

C_{p}

value (

C_{p}^{\max})

was identified.

Results and discussion

Aerodynamic performance analysis

After processing the outputs as per Section 4, the resulting plots are shown in Figure 5. This contour plot shows how the aerodynamic performance and its optimum conditions are altered when the turbine interacts with wake inflow. While analysing Figure 5, the optimum value of $C_{p}$ and its corresponding $λ$ was modified for the downstream turbines when compared with T1 operating in steady state (Figure 3). For every turbine operating in the wake inflow $C_{p}^{\max}$ has increased to a higher side with a higher $λ$ . But the pitch angle remains almost the same and low as T1, which confirms that these maximum values were observed in the below-rated region. After the turbine enters the rated region, β increases with increasing wind speed. This shift in optimal $λ$ indicates that wake-induced inflow modifies the rotor operating equilibrium.

Figure 5.

Bin-averaged $C_{p} (λ, θ)$ contour maps for T2–T4 under WAT inflow conditions; (a)T2, (b) T3 and (c) T4; (Left) ITS 7.5D and (Right) ITS 10D.

From the plot for T2, which is operating under the wake of T1 with considerable turbulence, the $C_{p}^{\max}$ value was found to be 0.5611 with a TSR of 9.2 at an ITS of 7.5D. When the distance between turbines is increased to 10D, these values are reduced to 0.527 and 8.6, respectively.

While observing T3, which is operating in a wake superimposed zone, that is having a higher turbulent flow than T2, the optimum $C_{p}$ value changes from 0.596 to 0.531, with respective TSRs of 9.2 and 8.8, when the spacing from T2 is increased from 7.5D to 10D. For the last turbine operating in a weak superimposed wake from T2 and T3, the optimum values observed were 0.564 and 0.502, respectively, for closer and farther turbine spacing. Since the calculation of non-dimensional coefficients to assess aerodynamic performance is based on instantaneous rotor disc averaged inflow velocity, the reduction in local inflow velocity can increase normalised $C_{p}$ values. Therefore, the reported downstream $C_{p}$ may be considered a wake-induced shift in the aerodynamic operating equilibrium rather than an increase in energy extraction.

Every downstream turbine’s optimum $C_{p}$ and corresponding $λ$ value increased while operating at a closer ITS, with T3 showing the greatest increase in power coefficient in a strong wake superimposed zone. The WAT is the induced standard deviation of the wind speed from its mean, resulting in gusts and lulls. This positive bias in the power coefficient arises because the rotor extracts more power from gusts than it loses in lulls, given the cubic relationship between power and wind speed.

The tip-speed ratio $(λ = \frac{Ω R}{U})$ increased downstream, when the local wind available for the rotor ( $U$ ) is decreased due to the momentum extracted by the upstream turbine or due to the fluctuations implied by the turbulence. But the rotor speed $Ω$ does not reduce accordingly, hence this scenario is causing an increased shift in TSR. Furthermore, the rotor and the drive train together have a large inertia $J$ . So, a change in the net torque $Δ T$ could change the rotor speed according to the relation $J \dot{Ω} = T_{g e n} - T_{a e r o}$ , where $\dot{Ω}$ is the angular acceleration, $T_{a e r o}$ is the aerodynamic torque and $T_{g e n}$ is the generator torque. With a fixed $J$ , $Ω$ could only change slowly; it does not instantly follow a decrease in windspeed.

For these reasons, T3, which has a higher turbulent inflow and a requirement for a rotor for rapid adjustment of its speed to the highly fluctuating wind inflow, exhibits higher optimal $C_{p}$ and $λ$ values. When ITS is increased from 7.5D to 10D, a reduction of nearly 11% for $C_{p}^{\max}$ values were observed for T3 and T4, showing this complete alteration is from wake and turbine position relative to it. Also, it can be inferred that when wake is normalising to its ambient condition (as ITS increases), these overshoots in aerodynamic parameters are also moving towards that of T1 (Figure 3). The change observed in the contour line of $C_{p}$ in Figure 5 also agrees with this statement. Even though the $C_{p}$ value for the downstream turbines was found to be higher than that of T1, it does not imply that the generated power is greater than T1 for downstream turbines. It was reduced due to the wake effects and reduced coefficients that take into account for the final power due to the fluctuating wind. The generated power for each turbine at each ambient wind for all turbines at two ITS is shown in Figure 6.

Figure 6.

Generator power vs wind velocity; (a) ITS 7.5D; (b) ITS 10D.

From Figure 6, it is clear how increasing ITS from 7.5D to 10D results in an improvement of the generator power of downstream turbines. This improvement with ITS was dominant in the sub-rated and near-rated region (8 m/s to 12 m/s), where power is very sensitive to inflow velocity. When turbines are closely spaced, WAT and velocity deficit together result in power reduction in T2 to T4, even though they found better $C_{p}$ than T1. By increasing the spacing, partial recovery of wake occurs before reaching every downstream turbine, showing a noticeable restoration of performance. With increasing ITS, it is found that downstream turbines reached rated operation at a lower ambient velocity than at closer spacing. These results highlight the importance of sufficient turbine spacing in aligned wind farm layouts to reduce wake-induced power losses and improve overall farm efficiency.

To show how the wake influences downstream turbines in other aspects, a separate study is carried out at one of the velocities (12 m/s) in the next section.

Wake effects on different inflow angles (12 m/s ambient)

Instead of just assessing the effects of wake on velocity deficit, TI, and rotor thrust on the downstream turbine, a secondary study was also conducted to show how these effects vary with different inflow angles for the above turbine arrangements. As per the visualisation shown in Figure 7, for a downstream turbine, the effect of the upstream wake was fully escaped when the inflow angle is at $15^{0}$ . This effect was agreed by a schematic diagram plotted using a drawing tool utilising the Jensen wake model (Jensen, 1983; Shakoor et al., 2016) as shown in Figure 8.

Figure 7.

Fast.Farm visualisation for different wind inflow angles; (a) $5^{0}$ , (b) $10^{0} and$ (c) $15^{0}$ (at 12 m/s inflow, near-rated case).

Figure 8.

Schematic representation of wake growth using the Jensen Model (wake tail considered at 8D position);(a) $0^{0}$ inflow and (b) $15^{0}$ inflow.

The Equation used is given below as follows:

R_{w} = d (1 + 2 \propto x) / 2

(8)

Where d is the rotor diameter, x is the distance at which the radius of the wake is calculated, and $\propto$ is the wake decay constant, which is 0.075 for onshore and 0.04 for offshore.

For this section, only the 12 m/s ambient condition is considered, which is near the rated speed (11.4 m/s) of the NREL 5-MW reference turbine. As per equation (8), the wake geometry (for a given induction setting) is qualitatively the same at other speeds. However, in practice, the $C_{T}$ does vary. Above-rated speeds, which trigger blade pitching, will result in reduced thrust, yielding a smaller wake deficit. Thus, the 12 m/s ambient condition represents a near maximum deficit condition.

The rotor behaviour for different inflow angle from Figure 9 infers that, when the inflow angle is $15^{0}$ the wake effect was deviated away from every downstream turbine. When the inflow angle is $0^{0}$ the wake effect was perfectly encountered to downstream turbines where T3 and T4 experienced more deficit velocity with higher TI than T2.

Figure 9.

Effect of inflow angle on downstream wake characteristics and turbine performance (12 m/s): (a) wind velocity, (b) turbulence intensity, (c) rotor thrust, and (d) generator power.

For every angle, this trend remains the same, with a gradual progression to the ambient condition when the angle progresses to $15^{0}$ . The key observation is that even though the downstream turbines showed a higher $C_{p}$ than T1, its rotor thrust decreased as it approached downstream turbines, indicating that the machine was capturing less energy from the wind. T3 and T4, operating in a wake superimposed zone, showed the lowest rotor thrust and generator power compared with T2 when the turbines are fully aligned with the inflow. This reduction was normalising to T1 as the inflow angle progressed and the wake deviated.

Conclusion

The present study shows how the aerodynamic performance of downstream turbines changes under the influence of wake-added turbulence, compared with common steady-state performance analysis. By comparing aligned and yawed inflow cases, the present study examines how wake deficit and turbulence modify both rotor efficiency and overall farm performance. Providing useful guidance for wind farm layout design and wake mitigation strategy. Below are a few key takeaways from the present study:

1. The downstream rotors operating under WAT exhibited higher peak power coefficients and larger optimal tip-speed ratios than the upstream steady case. The $C_{p}^{\max}$ value for T2 increased to 0.5611 (TSR = 9.2) at ITS 7.5D and reduced to 0.527 (TSR = 8.6) at ITS of 10D proving that turbulence and velocity fluctuations bias $C_{p}$ through gust-dominant power production. Although $C_{p}^{\max}$ increases at closer ITS, especially for T3 (59.6%) operating in the wake superimposed zone, the actual generator power is lower due to reduced mean inflow and higher turbulence. Increasing ITS to 10D partially restores the mean inflow and increases generator output, mainly in ambient conditions of 8 m/s-12 m/s.

2. The present study observed a nearly 11% reduction in $C_{p}^{\max}$ value for turbines operating in wake superimposed zone (T3 and T4) after increasing ITS from 7.5D to 10D, showing higher $C_{p}$ values at closer spacing are induced by stronger turbulence, while larger spacing between turbines aids more stable aerodynamic performance closer to the upstream condition. These observations lead to the idea that the power coefficient is no longer solely a function of turbine design, but also of the turbulent inflow it operates in and the separation distance between turbines.

3. For a fixed farm layout, changing the inflow angle effectively navigates the wake away from downstream turbines. With sufficient angular offset, the downstream machines experience lesser wake interference, resulting in significant progress in thrust and power output. Planning wind farm layout at angles to the dominant wind direction will help reduce wake interference.

In conclusion, this study highlights the importance of understanding wake behaviour when planning wind farm layouts, accounting for performance losses. Understanding $C_{p} - λ$ behaviour of downstream machines is especially useful, as it provides insight into how WAT alters optimal operating points and affects aerodynamic efficiency.

Limitations and future scope

The present study utilises steady ambient wind conditions to facilitate comparison with ideal aerodynamic characteristics. Implying a realistic turbulent wind field could possibly capture realistic features of the wake and $C_{p} - λ$ shifts. The closest turbine spacing considered in this study was 7.5D; reducing ITS will enable it to capture features and changes induced by a more vigorous wake.

Analysing the shift in the operating equilibrium under different atmospheric stability conditions may provide more insight into realistic conditions. In addition, Fast.Farm is a mid-fidelity engineering framework which does not resolve all turbulent structures that are achievable using a high-fidelity framework.

Future work may extend the present study by incorporating realistic atmospheric and ambient turbulent conditions, varying atmospheric stability, and conducting detailed validation against experimental or high-fidelity LES datasets. Further studies may also investigate blade pitch control strategies to utilise the increase in normalised $C_{p}$ values for higher power production for downstream turbines.

Footnotes

Acknowledgements

The authors would like to convey their gratitude to the Department of Civil Engineering, NIT Calicut, for providing the facilities for carrying out this present study.

ORCID iD

R. Prethiv Kumar

Author contributions

Roshan Thomas: Writing original draft, Formal analysis, Investigation, Software implementation, Validation, Conceptualisation.

Satheesh Jothinathan: Conceptualisation, Methodology, and supervision.

R. Prethiv Kumar: Conceptualisation, Methodology, Investigation, Review & Editing and supervision.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

All relevant data are available from the authors upon reasonable request.*

Appendix

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