1. Introduction
The reduction of pollutant emissions from fossil combustion and the need for a valid alternative to fossil fuels, due to an increasing demand of coal and natural gas resources, have motivated the European Union (EU) to invest into research on renewable energy. EU has established that renewable energy contribution will be 20% higher than the current amount and at least one-third of this production could be possibly derived from wind energy within the 2020s [
1,
2]. The technological progress achieved in the renewable energies field has been relevant and, in particular, wind power has become the main renewable energy source in many countries.
Wind turbines represent the main technology able to convert kinetic energy extracted from the wind into electric energy [
3,
4]. Wind turbines are exposed to various environmental conditions during their operating life (i.e., wind/dust, rain, hail, snow, ice formation, ultraviolet exposure). These operating conditions combined with blade tip speeds result in damage to the leading edge due to leading edge erosion (LEE) effect [
5]. There are several studies in the literature about the mechanisms, modelling, and possibilities of preventing the surface erosion of wind turbine blades [
6,
7].
Wind turbines are classified as horizontal-axis wind turbine (HAWT) and vertical-axis wind turbine (VAWT), according to the spatial orientation of the rotating axis. In the last years, an increasing interest in vertical-axis wind turbine has been experienced because this configuration presents some relevant advantages compared to the horizontal-axis layout. Specifically, it is able to work with an omnidirectional asymptotic velocity without a yaw mechanism that generates power loss. This configuration is also more compact and less noisy [
8]. Moreover, with the prospective of urban application, the micro-VAWT can represent a valid alternative to photovoltaic devices.
On the other hand, VAWT is less efficient compared to HAWT, due to the presence of unsteady aerodynamic phenomena, e.g., dynamic stall, deep stall, and blade–wake interaction. The steep variation in time of the angle of attack is among the main causes of the dynamic stall phenomenon, which increases the noise and the fatigue life of the turbine components [
9,
10]. The delay of the dynamic stall phenomenon has been the subject of study in recent years, and different strategies have been considered in literature. Wang et al. [
11] proposed a new technology based on flexible blades. According to Huang’s research [
12], blade sections are able to change automatically their camber angle to prevent flow separation. These effects were noticed by Bishop [
13] and by Swartz et al. [
14] when studying flying squirrels and bats, which are able to fly at high angles of attack by changing the shape of their wings during the flapping flight. Furthermore, Ismail and Vijayaraghavan [
15] proposed a combination of semicircular inward dimple and a Gurney flap at the surface of a Darrieus turbine. They observed that the adoption of an optimized combination of a Gurney flap and a dimple produced an increase of the tangential component of the aerodynamic force. Another strategy, adopted to delay the dynamic stall, is an application of turbulators [
16] or vortex generators [
17,
18] on the surface of the blades. These technologies energize the boundary layer making the flow turbulent and reducing the size of the laminar bubble with a consequent rapid reattachment of the flow. Even though turbulators and vortex generators are cheap technologies, they present main cons: (i) the right position on the surface of the blades is difficult to find because it is influenced by the variation of the angle of attack; (ii) the turbulent boundary layer creates extra friction drag [
19,
20].
Another technology often used to reduce the variation in time of the angle of attack is an active pitch system. Paraschivoiu et al. [
21] and Rezaeiha et al. [
22] studied the effect of a fixed-pitch angle on the aerodynamic performance of a turbine. However, according to that approach, blade sections do not operate at an optimum angle of attack at each azimuthal position. Indeed, turbine performance can be improved further by utilizing a variable-pitch mechanism. Generally, a linkage mechanism is the most used for this purpose. It is characterized by main links that rotate around the main rotational point, and secondary links that rotate around an eccentric rotational point. This way, the blades are controlled in order to avoid the stall and also maintain a favourable angle of attack. Even if a linkage mechanism is able to improve power generation with respect to a fixed-pitch configuration, the blade sections operate at angles of attack which are not optimum in most azimuthal positions [
23,
24]. Moreover, the determination of link lengths and eccentric rotational point position requires an optimization process, namely, the best combination of variables whose performance improvement is not known a priori. Therefore, this system is not versatile because it is necessary to execute an optimization at any variation of the operating conditions.
Different methodologies to define a variable-pitch law are presented in literature. Hwang et al. [
25] proposed an optimization procedure to determine the optimal pitch angle at specific azimuthal positions. Computational fluid dynamics (CFD) was adopted to estimate the aerodynamic performance coupled with a genetic algorithm optimizer. The optimal variable-pitch law improved the power generation by 25% with respect to the one obtained with the cycloidal motion. Paraschivoiu et al. [
21] analysed a two-blade H-Darrieus turbine modelling a variable-pitch law with an analytical function whose coefficients were used as optimization process variables. The aerodynamic model used to determine the performance was a double-multiple streamtube (DMS), while the optimization was carried out by using a genetic algorithm. The results showed an increase in the power generation at different tip-speed ratio (TSR). Both methodologies reported above required a high computational effort and should execute at the variation of the operating conditions. Conversely, Staelens et al. [
26] proposed three different methodologies, computationally more efficient, to define a variable-pitch law. The first imposed the local angle of attack of airfoils equal to the stall angle value throughout the rotation of the turbine. Although they obtained an increase of efficiency for a higher wind speed, the variation in time of the angle of attack was physically impossible because it presented discontinuity at azimuthal positions
and
. The second methodology proposed to replace the value of the angle of attack with the value of the angle of stall when the former exceeded the latter. Although the aerodynamic performance decreased, the variation of the attack angle was smoother than the previous one but it still presented discontinuities. The last methodology proposed a sinusoidal correction function whose amplitude was equal to the maximum difference between the local angle and the static stall angle. This way, the local angle of attack had a continuous variation in time with respect to the cyclical condition. Although these methodologies allowed an improvement of the aerodynamic performance, the blade worked in an unstable condition because the angles of attack were close to the angle of stall, and aerodynamic performance could drop dramatically due to flow separation.
In the present work, a variable-pitch law that makes each blade section, namely, airfoil, able to operate at the conditions of the maximum steady aerodynamic efficiency, is presented. The proposed methodology is computationally efficient compared to equivalent approaches found in the literature. In fact, for eight azimuthal positions, the static polars of the airfoil are calculated using a potential solver, and the effective angles of attack that maximize steady efficiency are determined. The angle by which each blade section must rotate to operate at an effective angle of attack is the pitch angle. A cubic interpolation of pitch angles produces a variable-pitch law. The automatic procedure was implemented in Matlab software.
2. Aerodynamics of VAWT
A schematic representation of a VAWT is reported in
Figure 1. A polar reference system (
) having its origin in the centre of the shaft is used to represent the current blade position.
The flow field on a blade is characterized by different velocity types: (i) a tangential velocity component due to rotation
where
is the rotational velocity and
is the radius vector; (ii) an asymptotic wind velocity
; (iii) an induced velocity
. According to
Figure 1, the sum of those velocities returns the relative velocity of a single blade:
As the turbine rotates, both magnitude and direction of the relative velocity change cyclically. Therefore, the angle between the airfoil chord and the relative velocity, namely, the geometric angle of attack
, follows the same cyclical variation. The geometric angle of attack can be expressed, neglecting for simplicity the small induced velocity
, as a function of azimuthal position (
) as:
where
is the tip-speed ratio (TSR) defined by:
The parameter that gives knowledge about the three-dimensionality of the flow is the aspect ratio
, defined as:
where
b is the blade span while
c is the chord of the airfoil. By assuming an
greater than 7.5, three-dimensional effects can be neglecting [
27,
28]. The aeronautical convention is adopted for
, i.e., positive values of
are associated to anticlockwise rotations of the relative velocity with respect to the airfoil chord, as shown in
Figure 2. To ensure an effective operating condition of the turbine, the suction side of the i-th airfoil is oriented toward the rotor centre, as shown in
Figure 2.
An inverse proportionality relation links the TSR values with the values of
[
10]. Low values of the TSR relate to steep variations of the
and vice versa. Steep changes in time of
, and a high frequency oscillation of airfoils (pitching motion) create an unsteady flow field on turbine blades, known as the dynamic stall condition [
29]. As reported in the literature [
30], the dynamic stall is characterized by the development and interaction of vortical structures of different sizes generated on the airfoil’s suction surface. Blades experience an increase of the lift force for values of the angle of attack far beyond the static stall angle. This phenomenon occurs when the leading edge vortex (LEV) spans over the whole suction surface side. Consequently, when the airfoil approaches its maximum angle of attack, the LEV detaches from the airfoil causing a sudden lift stall and a consequent rapid increase of drag [
31,
32].
In
Figure 3, a schematization of the aerodynamic forces acting on an airfoil is reported. Tangential (
) and normal (
) forces are related to lift (
) and drag (
) forces, respectively, by:
The tangential force plays a major role in generating power; indeed, is the only force which is able to generate a torque at the shaft. As the lift contribution is greater than the drag one, the torque is oriented according to the direction of rotation generating power. Conversely, when the lift stall occurs, the lift contribution is negligible and the torque will be in disagreement with the direction of rotation.
6. Results
In this section, the 2-D numerical simulations are reported. In particular, the effects of the variable-pitch law on the VAWT performance are assessed. In
Figure 14, the instantaneous torque coefficient obtained with the pitch law (Case 1) is compared to a fixed pitch (Case 0).
As expected, the behaviour obtained in Case 0 was typical of a highly unsteady aerodynamic field characterized by the existence of vortical structures generated by dynamic stall. Instead, in Case 1, the dynamic stall was delayed as it appears by looking at the behaviour of
in
Figure 14, where the sudden loss of
is not present. Those different behaviours are highlighted by the contours of the pressure and velocity fields shown in
Figure 15 and
Figure 16, respectively. In both cases, in the range
, the static pressure fields around the airfoil showed the flow fully attached to the airfoil (
Figure 15a,b). Two main reasons caused this behaviour: (i) the airfoil worked at angles of attack far from dynamic stall conditions; (ii) the aerodynamic field in the upstream phase
was not influenced by the wake generated by other airfoils, as confirmed by the velocity field reported in
Figure 16a,b. Furthermore, pressure fields obtained at
, represented in
Figure 15a,b, showed an increment of the static pressure difference between the pressure side and suction side of the airfoil in Case 1. This caused the increment of
compared to Case 0, as shown in
Figure 14. In the remaining azimuthal range, the phenomena occurring in Case 0 and Case 1 were different from each other. In Case 0, the values of the angle of attack increased in the range
.
Moreover, a reversed flow occurred at the trailing edge due to the rapid airfoil motion, as shown in
Figure 15 and
Figure 16. In the same range, the LEV was already formed, delaying the lift stall to an angle greater than that of the static stall. The LEV increased the depression on the suction side raising the difference in static pressure between the surfaces of airfoil. In that way, the lift force increased compared to the static case. The maximum difference of static pressure was obtained at
since, as shown in
Figure 15d and
Figure 16d, the LEV spanned over the whole suction side of the airfoil. At this value of
, namely
, a peak of instantaneous torque coefficient was obtained. For
, the angle of attack achieved the dynamic stall angle value and the LEV detached from the airfoil causing a sudden loss of lift, as shown in
Figure 14. Since the variation of
was steep, there was a considerable delay in the flow reattachment. Only when the airfoil returned to the upstream phase did the flow reattach to the airfoil, as shown in
Figure 15e–i and
Figure 16e–i. Therefore, in that range, the
values oscillated around the zero value. On the other hand, in Case 1, the flow was attached to the body until the section blade was in the range
, as shown in
Figure 15b–f. Currently, only in the range
did separation occur at the trailing edge, and a simultaneous growth of LEV was observed. This behaviour is visible in
Figure 15f–h and
Figure 16f–h. In Case 1, the flow was quasi-steady, and the airfoils worked at a low angle of attack; at the remaining azimuthal positions, the flow reattached, producing an increase of
values, as shown in
Figure 15i and
Figure 16i.
As we expected, the maximum value of
was obtained in Case 0. This behaviour was due to the methodology used to determine the effective angle. In fact, the static polar used underestimated the maximum efficiency condition compared to the dynamic polar. Nevertheless, the improvement in performance achieved adopting the variable pitch was evident when observing the behaviour of the global torque coefficient of the three blades versus the azimuthal positions reported in
Figure 17. For Case 0, the average torque coefficient value
was equal to
compared to a
value obtained in Case 1 equal to
with an improvement of 0.27. The average value of the power coefficient
was defined as:
where
is the torque coefficient of the three blades. In Case 0,
was equal to
. As is known in the literature [
43,
44], the solidity, the Reynolds number, and the cord length identify a class of turbine with the same performance. In particular, Brusca et al. [
43] studied the performances of turbines characterized by airfoil section NACA 0018 at low Reynolds numbers, a high solidity and different TSRs. These geometrical and operational properties identified the same class of turbine model presented in this work. In the literature, there are no experimental studies on this class of turbines at low TSRs because they typically do not produce power but absorb it.
Consequentially, in order to validate the performance obtained with the proposed numerical modelling, we compared the
obtained in this work with the one obtained by Brusca et al. at the same TSR. The percent error committed was estimated by means of the following equation:
and it was equal to 12%.
Globally, the turbine without the pitch law did not produce energy but absorbed it, while, in Case 1, the turbine generated an average power equal to 23 W with equal to .
7. Conclusions
Vertical-axis wind turbines represent a suitable technology to convert kinetic energy, extracted from the wind, into electric energy. However, the presence of unstable aerodynamic phenomena such as dynamic stall, deep stall, and blade–wave interaction results in a lower efficiency of such turbines. Several methodologies have been proposed in literature to improve the aerodynamic efficiency of wind turbines, but they turn out to be computationally inefficient. In this study, a new efficient methodology to define a variable-pitch law which improves the performance of an H-Darrieus turbine was implemented using an automatic procedure. It was based on the determination of the angles of attack in eight azimuthal positions where the maximum steady aerodynamics efficiency condition was satisfied. In order to validate this methodology, the effects of a variable pitch on the torques of a 2D model of the H-Darrieus turbine were analysed. In particular, a U-RANS analysis with a turbulent model was carried out. The sliding-mesh technique was used to simulate the pitch motion during the cycle rotation.
A comparison of the results obtained by applying the pitch law as opposed to a no-pitch-law condition highlighted how the pitch law actually delayed the dynamic stall improving the aerodynamic performance considerably. The mean value of the power coefficient obtained by implementing the pitch law results significantly increased. Obviously, the global angle of attack obtained with the proposed methodology represents a suboptimal solution in the dynamic case because the dynamic maximum aerodynamic efficiency is obtained at values of the angle of attack greater than the static case. However, this methodology is more versatile than those already present in the literature because it does not require optimization and could be implemented in hardware to modify the pitch law in real time, following different asymptotic wind conditions.