Improved Lift for Thick Flatback Airfoils in the Inboard Blades of Large Wind Turbines

Abstract

Thick airfoils are often used in the inboard sections of blades in commercial wind turbines. The main reason for this is to give the blade greater structural strength, but it is well known that thick airfoils degrade aerodynamic performance by stalling at relatively small angles of attack. The adoption of flatback airfoils instead of sharp trailing edges allows high lift coefficient to be maintained in thick airfoils. In this paper, we propose a novel airfoil design based on a passive flap to further improve the lift coefficient. This new design was tested by numerical simulation on several airfoils with different maximum thickness and different TE thickness. The improved design for flatback airfoils yields a higher lift coefficient, while the drag behaviour is strictly related to the baseline airfoil shape: some airfoils show a decrease in drag at certain angles of attack, while others exhibit a drag increase. In conclusion, the practical implications of the flap’s utilisation on a state-of-the-art blade designed for a 5 MW wind turbine are analysed. The findings demonstrate that, due to the enhanced lift coefficient, it is feasible to shorten the chord while maintaining the power output, thereby reducing material costs.

1. Introduction

Wind energy is a crucial component of clean energy production and reducing greenhouse gas emissions. To achieve the climate targets, the European Union, for example, aims to cover more than one third of its electricity demand with wind power by 2030 and over 40% by 2050. The trend in recent years has been for wind turbines to increase in size. In fact, to be economically viable, wind turbines should increase their power output, which is proportional to the swept area of the rotor. This requires turbines with increasingly longer blades. At the same time, the mass of the blade increases approximately as the cube of the blade length, resulting in greater gravitational loads, particularly torsional, edgewise bending and flapwise bending loads (as explained in [1]). The latter is the most significant load, and to prevent the turbine from structural problems, it is necessary to adopt thick airfoils in the area of the blade close to the root. A thick airfoil (i.e., airfoil with $t/c\ge 0.25$ [2], where t is the maximum thickness and c is the airfoil chord) can be utilised to increase the resistance of the blade to load due to the enhanced moment of inertia. The downside is that the increase in thickness reduces the aerodynamic performance of the sharp trailing edge (TE) airfoil, since it stalls at small angles of attack, thereby preventing the achievement of a high lift coefficient $C_L$. As a compromise between structural requirements and aerodynamic performance, blunt trailing edge or flatback (FB) airfoils offer a good solution: they increase the maximum lift coefficient and the lift coefficient slope [3]. This is due to the different pressure distribution along the airfoil compared to the sharp trailing edge airfoil, resulting in a lower adverse pressure gradient on the suction side, which prevents premature boundary layer separation. In addition, flatback airfoils offer some transport advantages [4,5]. They are used in the inboard area where the chord is at its highest, so they allow for smaller dimensions.

On the other hand, flatback airfoils show a higher drag coefficient ( $C_D$) compared to sharp TE and a wake development characterised by vortex shedding. However, when looking at the turbine as a whole, one of the most important parameters is the torque coefficient. Since the inboard region of the blade has a lower moment arm and lower relative velocity, the contribution to the torque generated by the entire blade is relatively small. Moreover, the kinetic energy available in the wind in a ring of thickness $\delta r$ is proportional to the area of the ring, thus exhibiting a decrease towards the root. However, increasing the aerodynamic performance of the inboard region could be beneficial.

Considering Figure 1, a generic blade element located at a radial position r is shown. The infinitesimal lift $\delta L$ and drag $\delta D$ forces are highlighted as well as the relative velocity to the blade W. The infinitesimal torque and thrust acting on the blade element can be obtained considering the total force along the tangential (t) and normal (n) directions, respectively:

$$\delta Q=(\delta L·sin\left(\varphi \right)-\delta D·cos\left(\varphi \right))r={\displaystyle \frac{1}{2}}\rho c{W}^2{C}_{Q}r\delta r$$

(1)

$$\delta T=\delta L·cos\left(\varphi \right)+\delta D·sin\left(\varphi \right)={\displaystyle \frac{1}{2}}\rho c{W}^2{C}_T\delta r$$

(2)

where $\varphi$ is the inflow angle, the sum of the angle of attack $\alpha$ and the twist angle $\beta$, and $C_Q$ and $C_T$ are the torque and thrust coefficients, respectively, defined as follows:

$$C_Q=C_L·sin\left(\varphi \right)-C_D·cos\left(\varphi \right)$$

(3)

$$C_T=C_L·cos\left(\varphi \right)+C_D·sin\left(\varphi \right)$$

(4)

The main contribution to the torque in the inboard region of the blade is due to the lift coefficient $C_L$: near the root, the inflow angle $\varphi$ is large, so the dominant term in the torque coefficient $C_Q$ shown in Equation (3) is that which depends on $C_L$ (as observed in [2,6,7]). A high $C_D$ is therefore acceptable when combined with a high $C_L$, hence the adoption of flatback airfoils even when they have a high $C_D$.

The aim of this paper is to provide a solution to improve the lift coefficient of a flatback airfoil while keeping the drag coefficient almost the same. The latter, as mentioned, has little influence on the torque at the root of the blade but has a negative effect on the thrust coefficient $C_T$ Equation (4). Consequently, it is necessary to identify novel geometries that will not substantially enhance the drag.

A review of the literature reveals numerous studies that have sought to reduce the drag coefficient of flatback airfoils. The primary objective of these studies was to reduce the drag coefficient and subsequently minimise noise emission due to vortex shedding. Various passive devices, including splitters and cavities (as shown in [8,9,10]), reveal a common trend in the aerodynamic performance resulting in a reduction in drag at relatively low angles of attack, accompanied by a concomitant reduction in the maximum lift coefficient achieved. Other works adopt a wavy TE such as in [11], where the drag reduction is accompanied by a lift loss. Another interesting solution is proposed in [4], where the swallow tail concept is studied. The trend shown in these works is that it is possible to attain a drag reduction paying the cost of a lift decrease. In the aforementioned works, it is not clear whether the lift losses are fully compensated for by the drag reduction or whether an increase in chord length may be necessary to maintain the same power output.

Other studies instead aim at increasing the lift coefficient; one example is given in [12] where the Gurney flap and vortex generators are analysed. The Gurney flap is a device that is typically set at a right angle at the TE on the pressure side of the airfoil. Its function is to enhance the camber of the airfoil, which results in a higher maximum lift coefficient and a reduction in the zero-lift angle. Vortex generators, on the other hand, delay the separation of the flow on the suction side of the airfoil, allowing for a higher maximum lift coefficient. These devices are beneficial for the lift, but a significant increase in drag is often linked to the increase in lift. The present paper has a similar objective: to enhance the lift coefficient of thick flatback airfoils. This may result in a slight increase in the drag coefficient, but given our focus on large-scale turbines, specifically those used in offshore applications, we will disregard the noise implications. The goal is to develop more efficient airfoils to achieve higher power production or to attain the same power output with a shorter chord, reducing plant costs. The techniques studied in the literature and listed above are not specifically designed for FB airfoils (except for the swallow tail), resulting in a poor overall performance (considering both the lift and drag coefficient). The present study proposes a flap designed specifically for FB airfoils. It consist of a passive device, which represents an economical solution to control the flow. Furthermore, the device is fixed to minimise maintenance costs.

The paper is structured as follows: Section 2 shows the methodology adopted and describes the geometry of the proposed flap. Section 3 describes the validation process, which encompasses the verification of calculation grids and the setup. To assess the robustness of the methodology, validation was conducted on multiple flatback airfoils, varying the airfoil under analysis. Subsequently, the novel geometry was implemented in a number of airfoils, as detailed in Section 4. In Section 5, a case study is presented for analysis. This case study involves the inboard blade of a turbine from the literature, which was analysed using conventional flatback airfoils. The results of this analysis are then compared with the new geometry proposed in this work. The paper concludes with a summary of its principal findings in Section 6.

2. Methodology

The objective of this study is to assess the potential for enhancing the aerodynamic performance of flatback airfoils. In particular, the use of passive devices has been explored as a means of increasing the $C_L$ while minimising the impact of these devices on the $C_D$. Given the pivotal role of the $C_L$ in determining the torque generated at the root of the blade, small differences and increments in $C_D$ are deemed acceptable. The method used to enhance the lift is a flap of a specific design. The latter is depicted in grey in Figure 2. This kind of device can be added to any flatback airfoil; the slope of the flap is given by the line passing through points 1 and 2 (red in the figure), with point 1 located on the pressure side of the airfoil at a distance of $0.2\%c$ from the TE, while point 2 is on the TE. The extent of the flap is 5% of the chord length in the x direction, and the thickness is 2% of the chord length.

In order to conduct this study, 2D fluid dynamic simulations were employed. All computational grids were generated with the ANSYS ICEM v19.3 software and were structured multi-block grids. Such grids are a combination of the O-grid and the C-grid concept. This approach allows the advantages inherent to both grid types to be exploited. As documented in the literature [13], C-grids are well-suited for sharp TE airfoils, but they are less effective for finite thickness TE airfoils. Conversely, O-grids are adept at capturing the behaviour of flatback airfoils, but they tend to give poor predictions in the development of the wake and consequently in the $C_D$ coefficient. Indeed, O-grid results in an excessively coarse grid in the wake region. Figure 3 shows a generic grid used in this work for flatback airfoils: a C-grid is visible with an O-grid surrounding the airfoil with a thickness of $2\%c$, which is enough to encompass the boundary layer thickness (the grid details in the red and violet boxes). The number of grid points on both the suction and pressure sides is 350, and there are 100 points on the TE. The wall distance from the first layer of cells is set to $1\times {10}^{-5}c$; this ensures that $y^{+}$ values are largely less than 1 throughout the airfoil wall. The computational domain extends $33c$ in the flow direction and $28c$ in the normal direction: the distance from the inlet to the airfoil nose is $14c$ as shown in Figure 3 on the left side. Figure 4 illustrates the grid employed in this study for the analysis of all FB airfoils with the flap. It differs from the grid utilised for flatback airfoils (as shown in Figure 3) in that an additional C-grid is incorporated at the TE in order to reproduce more complex geometry at the TE. The entire dominion extents are the same as those of Figure 3 on the left side. The boundary conditions used are a velocity inlet where the x velocity component is specified at the inlet and a pressure outlet at the exit where a free-stream pressure is set. The inlet has a C shape and also encompasses the top and bottom side; therefore, in order to set the free-stream velocity, we have to specify the velocity components: the x-component was set to $U_{\infty x}$, which is the free-stream velocity, while the y-component was zero. The numerical simulations were carried out with ANSYS Fluent v19.3 using URANS: a pressure-based approach was used with the SIMPLE scheme and default settings (also default controls). The turbulence model is $k-\omega$ SST, and the convergence criteria were $1\times {10}^{-4}$ for all residuals. A total of 8000 time steps were used for each simulation, and the simulation time was such that the entire domain was traversed twice from the air flow. Therefore, the time step size was defined as $\Delta t=2L_x/\left(U_{\infty x}8000\right)$, where $L_x$ is the domain length in the x direction.

3. Validation

Since the new flap proposed in this work is intended to be added to any flatback airfoil, the validation of the computational grid and the computational setup was carried out using three different airfoils to ensure a wide range of validity. The first airfoil is the one used in [14] called DU97FB (a flatback airfoil developed by modifying the DU97-W-300 airfoil); the second is called DU97ST and it is the airfoil used in [14] with the “swallow tail” (ST) concept (first appearance in [4]); while the third is the flatback airfoil proposed in [8], called FB-3500-1750. In Figure 5, the x-y coordinates of the profile are shown. The three airfoils are very different: DU97FB and DU97ST have a maximum thickness of $30\%c$ and a TE thickness of $10\%c$, while the FB-3500-1750 airfoil has a maximum thickness of $35\%c$ and a TE thickness of $17.5\%c$. Additional information about these validation cases is listed in

Table 1. Validation setup.

	DU97FB	DU97ST	FB-3500-1750
chord c [m]	0.65	0.65	0.66
$U_{\infty x}$ [m/s]	25	25	15
$Re$	$1\times {10}^6$	$1\times {10}^6$	$6.66\times {10}^5$
$Ma$	0.07	0.07	0.04
transition	free	free	free

. As previously stated, the simple flatback airfoils were evaluated using a grid type comparable to that illustrated in Figure 3, whereas FB airfoils with modifications at the TE (DU97ST in this case) were examined using a grid type analogous to that depicted in Figure 4.

Figure 6 illustrates the lift and drag coefficients for all the airfoils analysed in the validation task. Comparisons are made between the 2D simulations of the present study and the corresponding experimental measurements. With regard to airfoils DU97FB and DU97ST, the experimental data are available in [14], while for FB-3500-1750, the free-transition measurements available in reference [15] were considered. The general trend of 2D URANS simulations is an overestimation of both the lift and drag coefficient (as also observed in [16]). Furthermore, it is evident that 2D results have difficulty in accurately capturing the occurrence of stall. A similar trend for FB-3500-1750 airfoil is shown in [17]. In particular, the linear trend is accurately represented with regard to the $C_L$, but the simulations overestimate both the maximum $C_L$ and the alpha at which it occurs, resulting in a delayed stall. Therefore, when using 2D URANS it is important to operate at angles of attack far enough from the stall condition. For $C_D$, the simulated data consistently exceed the experimental data. These behaviours are consistent with those observed in other 2D numerical studies, as in [11,18,19]. Moreover, 2D URANS simulations have exhibit weakness in drag prediction as shown in [20], where a significant spanwise change in drag was measured.

4. Results

The flap geometry described in Figure 2 was applied to the test airfoils DU97FB and FB-3500-1750. Figure 7 illustrates the lift and drag coefficients obtained with and without the flap. It can be observed that the use of the flap allows for a higher lift coefficient across all angles of attack for both airfoils under consideration. With regard to the drag coefficient, the DU97FB airfoil exhibits a deterioration, manifested as an increase in drag across all considered angles of attack, with the only exception being for $\alpha =0°$. Conversely, the FB-3500-1750 airfoil benefits from the use of the flap at low angles of attack (up to 12°). Figure 8 illustrates the pressure coefficient $C_p$ for both airfoils with and without the flap, at two angles of attack, $\alpha =0°$ and $\alpha =12°$. The pressure coefficient is defined as follows:

$$C_p={\displaystyle \frac{p-p_{\infty }}{\frac{1}{2}\rho U_{\infty }^2}}$$

(5)

where p is the static pressure on the blade, $p_{\infty }$ is the undisturbed pressure and $\rho$ is the air density. It is evident that the incorporation of the flap results in a higher overpressure on the pressure side of the airfoil, accompanied by a reduction in pressure on the suction side. The enhanced pressure difference across the airfoil is responsible for the observed lift enhancement. Conversely, the drag reduction observed at certain angles of attack can be attributed to the interference with the vortex shedding caused by the flap. The pathlines of the static pressure of DU97FB and FB-3500-1750 are plotted in Figure 9 and Figure 10, respectively. The figures illustrate the situation with and without the flap at two different angles of attack, $\alpha =0°$ and $\alpha =12°$, and due to the oscillating trend of the $C_L$ (and $C_D$) caused by the vortex shedding, the figures represent the instantaneous snapshot at the $C_L$ peak. It is evident that the presence of the flap consistently confers an advantage to FB-3500-1750 across all angles of attack. The reduction in flow disturbance in the wake due to vortex shedding is a notable benefit of the flap. In the case of DU97FB, this remains true at $\alpha =0°$, while the situation with the flap at $\alpha =12°$ exhibits a slight deterioration. This is accompanied by an increase in drag, as illustrated in Figure 7. A similar pattern is evident in Figure 11 and Figure 12, where the root-mean-square error (RMSE) of the mean velocity is plotted. It can be observed that the presence of the flap generally reduces velocity fluctuation in the wake, with the exception of DU97FB at $\alpha =12°$. The preceding renderings were obtained through the utilisation of the data-sampling option in Fluent, given that vortex shedding is a periodic phenomenon, exhibiting varying frequencies across different scenarios, i.e., different airfoils under consideration. Hence, to ensure a fair comparison, the pressure coefficient of Figure 8 and the velocity fields of Figure 11 and Figure 12 were calculated as the mean value of further 4000 time steps of simulation.

In order to facilitate a comparative analysis between the novel flap geometry proposed in the present work and a widely utilised flap geometry, namely the Gurney flap (GF), the GF was applied to the simple DU97FB flatback airfoil. In this instance, the Gurney flap was scaled to the same dimensions as the proposed flap, with a thickness of 2%c and a vertical length of 5%c. Figure 7(top) presents a comparison of the lift and drag coefficients obtained with the addition of the Gurney flap and with our flap. The GF generates a notable increase in the drag coefficient at all alpha values, while the lift coefficient is comparable to that of our flap up to $\alpha =8°$. For larger alpha values, the lift coefficient is higher when using the GF, with the exception of very high alpha values (e.g., $\alpha =20°$). Figure 13 compares the streamlines colouring by static pressure at $\alpha =10°$ for the simple flatback DU97FB airfoil, the airfoil with the flap and the airfoil with the GF. It is evident that the flap is capable of extending the overpressure region on the pressure side of the airfoil while maintaining relatively low drag values. The wake development in the case of the simple DU97FB (Figure 13a) and in the case with the flap (Figure 13b) are observed to be quite similar. In the case of the GF (Figure 13c), it is evident that the GF enhances the overpressure on the pressure side of the airfoil, but simultaneously increases the drag. The wake displays more pronounced depression regions. Therefore, the flap geometry proposed in this work, in contrast to the GF, allows for improvements in the lift coefficient while limiting the drag coefficient increment in comparison to the simple flatback airfoil case.

The domain extent in the y direction was maintained at $28c$ for all simulations presented in this paper. To ensure that no significant boundary effects can occur with such a domain width, we conducted additional simulations, doubling the y extent. As illustrated in Figure 14, the new domain extends $47c$ in the flow direction and $56c$ in the normal direction. In particular, the new domain contains the old $28c$ domain, as illustrated in Figure 14, where the blue line encompasses the $28c$ domain. Consequently, the grid refinement in the calculation domain delimited by the blue line remains unchanged. The consideration of a larger domain has resulted in a slight increase in the number of grid cells, from 180 k to 200 k. In order to evaluate the influence of the domain y extent, the DU97FB airfoil was analysed at several angles of attack. The results demonstrate that enlarging the domain has no significant impact on the lift and drag coefficients, as illustrated in Figure 15. This is probably due to the fact that the grids used in this work have high refinement near the walls, so the grid surrounding the airfoil is very fine. Consequently, local effects are of greater consequence than far-field effects.

The flap geometry was then applied to two further airfoils: the DU40FB and the DU35FB airfoils. These airfoils are derived from the sharp trailing edge airfoils called DU40 and DU35, which are described in detail in [21,22] (coordinates available in [23] and in [24]). The DU40 and DU35 airfoils are obtained in turn from DU 99-W3-450 and DU 99-W3-350, respectively. All of the aforementioned airfoils are sharp TE airfoils, and the flatback airfoils were optimised in [25], where the x,y coordinates are provided for analysis. As would be expected, these airfoils have a maximum thickness of 40% and 35%, respectively, and a TE thickness of approximately 11% and 9%. The simulations for both airfoils were performed at $Re$ of $7\times {10}^6$, with an undisturbed flow velocity of 34.4 m/s and a chord length of 3 m. All other settings are the same as the previous simulations. Figure 16 illustrates the lift and drag coefficients. For these airfoils, the flap exerts a beneficial influence on lift at all angles of attack, whereas the drag exhibits disparate behaviour. The DU40FB airfoil with the flap demonstrates a general increase in drag. In contrast, the DU35FB airfoil with the flap exhibits a drag reduction up to $\alpha =10°$.

A uniform conclusion regarding the flap’s influence on drag cannot be drawn from the analysis of the results presented in Figure 7 and Figure 16. The flap confers an advantage on some airfoils at relatively low angles of attack, while this is not the case for others. This behaviour appears to be independent of the thickness of the TE or the ratio between the maximum thickness of the airfoil and the TE thickness. Therefore, the limited number of airfoils tested precludes the formulation of a general rule for leveraging the flap solution in terms of drag reduction. However, the flap proposed in this work appears to offer a promising solution when compared to other devices. For instance, as discussed in reference [12], the adoption of the Gurney flap or vortex generators has been shown to significantly enhance lift, particularly in the case of Gurney flaps. However, this increase in lift is often accompanied by a significant increase in drag. In contrast, the flap proposed in this work has been observed to result in a reduction in drag at certain angles of attack.

Given the inherent limitations of 2D URANS analysis, namely the overestimation of drag and the delay in stall prediction, it is recommended that the proposed flap geometry be subjected to further analysis using a more sophisticated tool, such as 3D simulations or wind tunnel tests. However, without considering quantitative data, it is possible to make a qualitative assessment of the flap geometry proposed in this paper, which suggests that it has the potential to enhance the performance of the airfoil, particularly by increasing the lift coefficient.

In other words, a comprehensive campaign of 2D URANS fast simulations could serve as the initial phase of an airfoil optimisation methodology. However, the final verification should be conducted with more reliable CFD techniques, despite the increased computational cost.

5. Case Study

The DU40FB and DU35FB airfoils were selected for the purpose of analysing a portion of a realistic blade. The turbine in question is the NREL 5MW, for which the distribution of airfoils, chord and twist are available for reference in [21,26]. The focus of this analysis was on the portion of the blade from $\mu =0.18$ to $\mu =0.38$ (where $\mu$ is the dimensionless radius along the blade defined as $\mu =r/R$ and r is the local radius along the blade, while R is the turbine radius that in this case is 63 m). It should be noted that the original blade was constructed with sharp trailing edge airfoils; however, for the purposes of this analysis, these were substituted with FB airfoils. As previously stated, the DU40FB and DU35FB airfoils were optimised in [25], beginning with the sharp TE.

The objective of this section is to undertake a comparative analysis of the same portion of blade utilising both the flatback airfoils and the modified geometry with flaps. The analysis of the blade portion was based on blade element/momentum (BEM) theory. In particular, in order to calculate the torque Q and thrust T generated by the blade, it is necessary to calculate the relative velocity to the blade W (see Figure 1) as follows:

$$W=\sqrt{U_{\infty }^2{(1-a)}^2+r^2{\mathrm{\Omega }}^2{(1+a^{\prime })}^2}$$

(6)

$$a={\displaystyle \frac{1}{3}}$$

(7)

$$a^{\prime }={\displaystyle \frac{a(1-a)}{{\lambda }^2{\mu }^2}}$$

(8)

$$\lambda =\frac{R\mathrm{\Omega }}{U_{\infty }}$$

(9)

where a is the axial induction factor, $a^{\prime }$ is the tangential induction factor, $\lambda$ is the Tip Speed Ratio (TSR) and $\Omega$ is the rotation speed of the turbine. Equation (7) and consequently the $a^{\prime }$ value represent the condition of maximum power extraction. The angle of inflow $\varphi$ and the angle of attack $\alpha$ are calculated as follows:

$$\varphi =arctan\left({\displaystyle \frac{1-a}{\lambda \mu (1+a^{\prime })}}\right)$$

(10)

$$\alpha =\varphi -\beta$$

(11)

In order to evaluate and compare the torque and thrust of the “blade FB” and the “blade flap”, where “blade FB” denotes the blade obtained using simple FB airfoils, while “blade flap” denotes the blade using FB airfoils with the flap, it is first necessary to ascertain the chord distribution. Given the analytical nature of this analysis, it is essential to pay close attention to certain aspects. To ensure a fair comparison in the torque and thrust calculation, it is necessary to use a different chord distribution when changing the airfoil under consideration. The theoretical blade design procedure proposed in [27] is based on the assumption of optimised operating conditions, which are represented by Equations (7) and (8). In this analysis, the fundamental parameter is the geometry parameter, defined as follows:

$${\displaystyle \frac{Bc}{2\pi R}}\lambda C_L={\displaystyle \frac{4{\lambda }^2{\mu }^2{a}^{\prime }}{\sqrt{{(1-a)}^2+{\left(\lambda \mu (1+a^{\prime })\right)}^2}}}$$

(12)

Equation (12) demonstrates that the chord distribution and the lift coefficient employed for the design are inversely proportional. Consequently, an airfoil with a higher lift coefficient will result in a shorter chord at each blade section. In light of the aforementioned concepts, modifying the airfoil entails modifying the chord distribution. In Equation (12), the chord distribution is a function of solely the $C_L$. The $C_D$ is not considered, which is an acceptable simplification given that in [27] it was demonstrated that an almost equal chord distribution would have been obtained considering also the $C_D$. The drag effect will be evident only on the turbine performance, in particular, on the torque and thrust generated. If we assume that the value of $C_L$ remains constant from the tip to the root, and thus the $\alpha$, then the blade will exhibit a strongly non-linear tapering for $\mu \le 0.6$. This is because the chord will result in very high values towards the root. Moreover, if a constant value of $C_L$ is assumed, the blade will exhibit a high degree of twist, with a significant difference in the $\beta$ value between the root and the tip. From an economic standpoint, the design procedure entails a linear tapering from the tip to the root, whereby the chord at the root is reduced and the loss is compensated for by an increase in the value of $C_L$ and consequently in $\alpha$. The left member of Equation (12) ( ${\displaystyle \frac{Bc}{2\pi R}}\lambda C_L$) necessitates a specific product of $c·C_L$ to achieve a given power extraction. Consequently, it is possible to reduce the chord when increasing the value of $C_L$, and vice versa. For the present analysis, we take R, the $\beta$ distribution, $\lambda$ and the rated wind speed from the real turbine proposed in [26]. Therefore, using equations from (6) to (11), it is possible to find the $\alpha$ distribution along the blade. The procedure is as follows:

• Given R, $\lambda$, the $\beta$-distribution and the $U_{\infty }$ of the NREL 5MW turbine ( $R=63$ m, $\lambda =7$, and $U_{\infty }=11.4$), we calculate $\alpha$;
• With the found $\alpha$-distribution, we obtain the $C_L$-distribution;
• With the $C_L$-distribution, we obtain the new chord distribution through Equation (12);

The $\varphi$-distribution is shown in Figure 17 (bottom right side) along with the $\alpha$ and $\beta$ distributions. Using Equation (12), we found the chord distribution shown on the right side of Figure 17 (in the red box). The discontinuity at $\mu \approx 0.25$ is a consequence of the rapid change in the airfoil. For $\mu \le 0.25$, the airfoil employed is DU40FB (or DU40FB with the flap), whereas otherwise the airfoil used is DU35FB (or DU35FB with the flap). In the case of a real blade, it would be advisable to incorporate a fitting to soften the impact of the airfoil change (see, for example, the airfoil distribution suggested in [28]). Figure 17 on the left illustrates the optimal chord distribution when utilising the DU35FB airfoil up to the tip of the blade. However, the present study focuses exclusively on the section of the blade between $\mu$ 0.18 and $\mu$ 0.38, as delineated by the red box.

The alpha value exhibits a notable decline as the $\mu$ value rises, which is responsible for the observed increase in the chord for $\mu \ge 0.25$.

Using Equations (3) and (4), it is possible to assess the torque and thrust coefficients obtained using the new designed blade section. Figure 18 illustrates the torque and thrust coefficients. As anticipated, the flap’s deployment results in an elevated thrust coefficient, attributable to the enhanced lift coefficient and the concomitant increase in drag coefficient at specific angles of attack. With respect to the torque coefficient, the flap enables an increased value across all radial positions, substantiating the aforementioned assertion that at the inboard blade, the dominant influence on the torque coefficient is the lift coefficient, despite the rise in drag coefficient.

By integrating Equations (1) and (2) along the blade section and multiplying by the number of blades, it is now feasible to obtain the torque and thrust, which are shown in Figure 19. The observed trend is consistent with the theoretical prediction, exhibiting a growth pattern from the root to the tip. The integration of the Q and T along the blade section yields an increase in total torque by 4% for the “blade flap” compared to the “blade FB” and at the same time a decrease in total thrust by 1%. It should be noted that all BEM calculations used to determine the Q and T values are based on the airfoil lift and drag coefficients presented in Figure 16, which were calculated at $Re=7\times {10}^6$. The Reynolds number along the blade sections varies approximately between $6\times {10}^6$ and $10\times {10}^6$. Despite the simplifications employed in the analysis of this case study, the qualitative benefits of flap adoption are evident. It may be possible to construct the blade section with a shorter chord, reducing material requirements and costs while maintaining power generation and thrust on the structure. This can be achieved compatibly with structural requirements at the root.

6. Conclusions

In this study, we propose a novel flap geometry with the objective of enhancing the lift coefficient of flatback airfoils. The latter are employed at the inboard region of the blade in order to satisfy both structural and aerodynamic requirements. The flap can be added to any FB airfoil, and it typically results in an increase in the lift coefficient. With regard to the drag coefficient, no uniform trend is observed, as it is strictly dependent on the baseline airfoil shape. At specific angles of attack, some airfoils exhibit a reduction in drag when the flap is adopted, while other airfoils show a general increase in drag. The flap was evaluated on a number of airfoils, namely, DU97FB, FB-3500-1750, DU40FB, and DU35FB. However, due to the limited number of airfoils tested, no general conclusions can be drawn. Further analyses are required to assess the flap proposed in this work, as the selected airfoils exhibit significant variation in maximum thickness, thickness of the TE and camber.

The observed behaviour of the flap results in an increase in lift coefficient for all the airfoils, which coincides with the objective of this study. In the inboard region of the blade, where the torque coefficient is mainly influenced by the lift coefficient, enhancing the lift coefficient of the airfoil chosen for the blade design is of paramount importance. As a case study, we have attempted to evaluate the potential of adopting the flap in the inboard region of a realistic blade turbine. The method adopted is analytical, specifically based on BEM evaluations. The blade under consideration is the NREL 5MW turbine. The original blade features a sharp TE; however, a blade with FB airfoils (“blade FB”) and a blade with the same airfoils but with the addition of the flap (“blade flap”) were compared. In order to ensure a fair comparison through analytical evaluations, the chord distribution used for the two blades was modified. In particular, the airfoils with a higher $C_L$, i.e., airfoils with the flap, exhibit a shorter chord. The utilisation of the flap enables the attainment of a higher torque coefficient whilst concurrently a higher thrust coefficient. The total torque and thrust (Q and T) on the inboard region of the blade were evaluated, resulting in a +4% and −1% change, respectively. This is due to the fact that the BEM analysis enables the redesign of the blade, in particular, the chord distribution, as a result of the change in airfoil performance. The chord length exerts an influence on both Q and T; consequently, despite the fact that the thrust coefficient of the “blade flap” case is higher, the total thrust is lower due to the shorter chord. Neglecting quantitative results, the important thing in this case study was to evaluate the potential of adopting the flap in the inboard region of a realistic blade. What we can conclude is that it is possible to obtain approximately the same power output using fewer materials because of a shorter chord.

Future development of this work could be to further analyse the flap behaviour with more sophisticated tools, for instance, LES simulations or testing a couple of airfoil geometries with the flap addition in a wind tunnel. For a realistic blade inboard region, it will be useful to simulate an entire blade with 3D fluid dynamic simulations in order to assess the chord reduction. With 3D simulations, neglecting the BEM analytical approach, it will be possible also to analyse and compare airfoils with and without the flap using the same chord distribution, evaluating whether the flap could lead to higher power output.