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The Angle of Attack (AOA) of the Vertical Axis Wind Turbines (VAWTs) blades has a dominant role in the generation of the aerodynamic forces and the power generation of the turbine. However, there is a significant uncertainty in determining the blade AOAs during operation due to the very complex flow structures and this limits the turbine design optimization. The paper proposes a fast and accurate method for the calculation of the constantly changing AOA based on the velocity flow field data at two reference points upstream the turbine blades. The new method could be used to calculate and store the AOA data during the CFD simulations without the need for extensive post-processing for efficient turbine aerodynamic analysis and optimisation. Several single reference-points and pair of reference-points criteria are used to select the most appropriate locations of the two reference points to calculate the AOA and It is found that using the flow data from the two reference points at the locations 0.5 aerofoil chord length upstream and 1 chord away from each side of the aerofoil can give most accurate estimation across a range of tested AOAs. Based on the proposed AOA estimation method, the performance of a fixed pitch and the sinusoidal variable pitch VAWT configurations are analysed and compared with each other. The analysis illustrates how the sinusoidal variable pitch configuration could enhance the overall performance of the turbine by maintaining more favourable AOAs, and lift and drag distributions.


INTRODUCTION
In recent years, there has been a notable increase in the number of investigations on the Vertical Axis Wind Turbines (VAWTs) and this has given the VAWT technology a new rebirth. While the Horizontal Axis Wind Turbines (HAWTs) have acquired the major portion in the wind power market, the VAWT concept is estimated to play a dominant role in the next 2 3 decades [1]. In particular, the VAWTs feature many potential advantages, especially for operating in the urban environment and the offshore floating platforms [2]. However, in general VAWTs currently suffer from lower efficiencies in comparison with the HAWTs [3]. Therefore, an intensive research on improving the aerodynamics of the VAWTs have been observed in recent years.
The VAWTs can be classified as two types of configurations, i.e. the Savonius and Darrieus designs [4], [5]. The Darrieus designs rely on the lift generated from the aerofoil-profiled blades while the Savonius designs are driven by the drag from bucket shaped vanes [6]. Generally, Savonius turbines have lower efficiencies although they have better startup characteristics than the Darrieus turbines [4]. However, the Darrieus type VAWTs offer significant advantages over Savonius turbines, and have a much higher power coefficient and suitable for large scale operations [4]. Since the driving elements of Darrieus type VAWTs are the aerofoils-profiled blades, the turbine performance is strongly dependent on the incident angle of the flow relative to the blade chord, also is referred to as the baled Angle of Attack (AOA). Therefore, an accurate estimation of the incident flow direction and the AOA during turbine operation is critical for turbine design optimisation [7].
There is intensive research interest on improving the straight bladed VAWT efficiency through controlling the blade AOA during its rotation around the vertical axis, especially for high efficient operations at low Tip Speed Ratios (TSRs) which rely on the appropriate design of the turbine blade pitching angle [8] or applying the variable pitch to the blade control [9]. For example, the variable pitch based on the cycloidal kinematics has been widely investigated [10] [12]; Erickson et al. [13] obtained 35 % enhancement in the turbine efficiency using a first-order sinusoidal pitch; Liu et al.
[14] achieved a some improvement in the turbine performance using sinusoidal pitch with low amplitude; and Paraschivoiu et al. [15] found that the turbine annual energy production could be increased by about 30% using an optimized variable pitch based on a suggested polynomial of sinusoidal functions.
The interactions between the wind and the VAWT rotations lead to very complex time-variant aerodynamic phenomena around the spinning blades. The use of the overall turbine power as a function of turbine TSR is the most common method for the analysis of the aerodynamic performance of VAWTs, and it can illustrate the variation of the power coefficient at different TSRs.
Although several studies have analysed the instantaneous power and/or torque coefficient over one cycle [16] [22], a more detailed analysis is required to understand the reasons for the differences in the power generation efficiency between different turbine designs, in order to gain a better understanding of the complex aerodynamic characteristics of the VAWTs.
The interactions between the wind and the VAWT rotations lead to very complex time-variant aerodynamic phenomena around the spinning blades. Although several studies have analysed the instantaneous power and/or torque generation over one rotating cycle [16] [22], a more detailed aerodynamics analysis and in particular the effects of instantaneous AOA are required in order to obtain an in-depth understanding of the aerodynamic reasons for the differences in the power generation efficiency between different turbine designs of the VAWTs. A range of different-fidelity analyses has been used to investigate both fixed and variable pitch VAWTs and the estimations of the AOAs. These include the streamtube based models  [32]. However, the 2D CFD analysis, based on the Reynoldsaveraged Navier Stokes (RANS), is widely used because of its reasonable accuracy and moderate computational cost [28]. In the blade aerodynamics analysis, the AOA could be simply estimated assuming that the approaching wind velocity to the blade is constant and parallel to the undisturbed wind flow velocity. This simple calculation ignores the effects of the rotor on the flow and in particular the blade-wake interactions existing in the VAWT operation, which can lead to a significant error in the prediction of the performance of the turbine blades. While this simplified calculation of the AOA is widely used [18] [20], [33] [37], a more realistic estimation of the AOA is needed that takes into account the variation of the magnitude and direction of the approaching wind velocity vector to the blade at different azimuthal positions. Kozak [38] calculated the AOA based on the CFD data using two different methods and these are based on the calculated lift coefficient or the pressure ratio between the suction and pressure sides of the blades. However, his validation of these methods was limited to the study of a pitching motion with a geometric AOA between 0° and 8°. Bianchini et al. [39] used the CFD data for the estimation of the AOA based on the location of the pressure peak by comparing it to the location of the pressure coefficient peak obtained by the panel method. In order to account for the virtual camber effect, the original aerofoil coordinates are transformed to a virtual aerofoil and then the panel method is used for the pressure coefficient calculations [39]. Although this method has a good agreement with the Blade Element Momentum (BEM) results, it involves many intermediate tasks. Edwards et al. [7] presented an estimation method of the corrected AOA based on the cycle-averaged CFD velocity flow-field. This method involves discarding the distorted velocity near the blade trajectory then interpolating the flow-field. While this method provides a good estimation of the AOA, it ignores the instantaneous variation of the velocity flow field and involves many intermediate tasks. Gosselin et al. [17] claimed a good estimation of the AOA using CFD data based on the averaged velocity vector at a single point located on the tangential trajectory at a distance of two-chord lengths in front of the blade.
where the chord to radius ratio is quite high.
It is noted that most of the estimation methods of the AOA that are available in the literature have two common drawbacks, namely (i) the lack of a reference for comparison and validation of the methods and thus can lead to relatively large errors, and (ii) the need for extensive post-processing. This paper presents a new method for the estimation of the AOA which uses the CFD simulated flow field data at two well-selected reference points around the blade. The new method has a minimal error and more accurate estimation of the AOA compared to all the existing method tested. In addition, the new method could be integrated into the CFD solver to provide a computational inexpensive calculation in order to extract the instantaneous AOA variations along the blade flying path for efficient blade aerodynamic analyses and optimization. Finally, the new method has been applied successfully to the evaluation of the lift and drag coefficients for a fixed and a variable pitch two-bladed VAWT configurations in order to analyse the differences in the performance between the two configurations.

PROPOSED METHODOLOGY FOR AOA ESTIMATION
In this paper a new method of estimating the AOA for VAWTs based on the flow field at two reference-points upstream the turbine blade is proposed. Since the relative approaching flow velocity and direction to the blade are very complex and constantly changes, it is critical to select correct reference points where a representative incident flow direction can be obtained for the correct estimation of the blade AOA. In order to find these reference points around the blade that can result in an accurate and easy calculation of the AOA for VAWTs, the fluid flow around a static aerofoil with a range of set AOAs has been used as a test and validation case. CFD simulations have been performed to obtain the flow field data around the aerofoil at several selected referencepoints, and these data are used to calculate the AOAs around this static aerofoil and compared with the set AOAs. The normalized Root Mean Square Error (RMSE), based on the differences between the calculated and the prescribed AOAs, is calculated and used to select the most suitable referencepoint locations with a minimal error, as will be discussed in Sections 2.3. It is found that using the flow data from the two reference points at the locations 0.5 aerofoil chord length upstream and 1 chord away from each side of the aerofoil can give most accurate estimation across a range of tested AOAs. Then, these selected reference-point locations are used to estimate the angle of attack around a VAWT blade which have successfully predicted the AOAs with good accuracy, as discussed in Section 3.2. This proposed method for the estimation of the AOA is illustrated in Figure 1.

Estimation of the AOA for a static aerofoil
The flow around a static aerofoil is considered with a range of prescribed geometric AOAs as shown in Figure 2 (a) where the NACA0015 aerofoil with a chord length of 0.225 m is used. This range of AOAs includes the angles between 0° to 25° with 5° increment. The flow conditions and aerofoil geometry are chosen according to the reference VAWT case that is discussed in Section 3.1.1. The static aerofoil simulations are performed at different incident flow velocities of 7, 14, and 21 [m/s] and these correspond to the average relative velocities around a VAWT blade that operates at TSRs of 1, 2, and 3, respectively, when rotating across a mainstream flow with a velocity of 7 m/s. This range of TSRs is chosen to cover the optimum operation range of moderate solidity VAWTs. Due to the lack of experimental data at the chosen flow conditions, the data from the widely validated XFOIL software [40] is used to validate the CFD model. XFOIL has been developed for the prediction of the aerofoil characteristics at low Reynolds numbers. This is established by incorporating both the integral boundary layer and transition equations along with the potential flow panel method.
Morgado et al. [41] compared the XFOIL predictions to both of the experimental and CFD data for the aerofoil characteristics at low Reynolds number and they reported that XFOIL is an excellent analysis tool for aerofoils. The accurate predictions on using XFOIL makes it reasonable to be used for the verification of other numerical methods, including CFD, especially when there is a lack of experimental data for the desired flow conditions. A commercial CFD solver, namely ANSYS FLUENT, has been used with double precision in the steady mode in order to model the flow around the aerofoil. The pressure based coupled algorithm is used to solve the momentum and continuity equations while the SST kaccount for the turbulence effects. The second-order upwind interpolation scheme is employed for the discretization of the momentum and turbulence equations. The solution is iterated upon until the normalized residuals of the flow variables reduce by five orders of magnitude.
In order to impose the prescribed AOA, the whole computational domain is inclined with the prescribed AOA as shown in Figure 2 (b). The computational domain is extended to 40 chord length in the downstream direction and 20 chord lengths elsewhere in order to eliminate any effects of the domain boundaries on the flow around the aerofoil. The computational domain is divided into a circular subdomain around the aerofoil and a rectangular extended domain and a circular nonconformal interface is used to connect these subdomains. This two subdomain configuration assists in maintaining the same mesh structure and quality regardless of the changes in the imposed AOA.
A full structured mesh is constructed across the domain as shown in Figure 3 (a). Figure 3 (b) shows the mesh in the vicinity of the leading edge of the aerofoil where a fine resolution is maintained around the aerofoil using an inflation zone with 110 layers, a maximum dimensionless wall distance, y + <1 and a growth rate of 1.05. y+ represents the normalized distance perpendicular to the wall and has very important role in describing the near wall flow. By maintaining y+<1, the viscosity dominated region, including the viscus sublayer, is resolved and hence a better estimation of the aerodynamic forces could be achieved.  In order to test the solution sensitivity to the generated mesh, three levels of mesh refinement are constructed with a refinement factor of 2. These meshes include the coarse mesh, baseline mesh, and fine mesh with a total number of elements 116400, 354600, and 1198400, respectively. Figure   4 shows the distribution of the pressure coefficient around the aerofoil at AOA=10° for the largest flow velocity of 21 [m/s] for the three meshes and it is found that there is no significant difference observed in the results obtained. The reason why there are no obvious differences between different grids is that the dimensionless wall distance y+, first layer thickness has been kept less than 1, and the mesh growth rate perpendicular to the aerofoil profile is kept small in all the cases. The CFD predictions of the pressure distribution around the aerofoil depends mainly on the near wall treatment and y+. Therefore, when the computational mesh is reasonably fine, the mesh refinements in the spanwise direction or outside the inflation layer do not have a significant effect on the computational results. The baseline mesh with 354600 elements is considered for this static aerofoil study in order to reduce the computational cost while maintaining a fine mesh distribution and a reasonable accuracy, in spite of the fact that the coarser mesh could be used with adequate accuracy. However, optimizing the computational cost was not prioritized in this study. Figure 5 shows the comparisons between the CFD and XFOIL predictions of the pressure coefficient around the aerofoil for flow velocities of 7, 14, and 21 [m/s] at AOA=10°. It is observed that the differences between the CFD and XFOIL data are associated with the prediction of the laminar separation bubbles. The reason is that the SST k-model in the CFD simulation imposes a fully turbulent flow while the viscous boundary layer module in XFOIL accounts for the laminar to turbulent transition.
However, these small separation bubbles have a negligible effect on the velocity field around the aerofoil and hence do not affect the calculation of the AOA.

Selection of the Reference-points
The proposed estimation of the AOA using CFD is based on the calculation of the inclination of the absolute velocity vector in one or multiple reference-points while the most appropriate selection of the reference-points is essential for the accuracy of the estimated AOA. The inclined flow around a static NACA0015 aerofoil is considered with six geometric AOAs, namely 0°, 5°, 10°, 15°, 20°, and 25°. In addition, three different incident flow velocities are considered, namely 7, 14, and 21 [m/s].
Both the single reference-point and pair of reference-points criteria are considered, while two groups of reference-points are selected. The first group is distributed around the chordwise direction, regardless of the incident flow direction, as shown in Figure 6 (a), and in the second group, the points are distributed around the incident flow direction as shown in Figure 6 (b). Figure 7 shows the typical streamlines released from the first group of reference-points that are clustered around the aerofoil chordwise direction at AOA=10° and these streamlines show how the flow around the aerofoil is distorted, especially in the vicinity of the leading edge. The degree of distortion depends on the location of the chosen test point and hence the appropriate selection of the test point locations is essential for the accurate estimation of AOA. Several aspects are considered in the selection of the locations of the reference-points. The points should not be located in the wake of the aerofoil and the distance between each point and the aerofoil profile should not be too small to be affected by the flow distortion around the leading edge of the blade. In addition, this distance should not be too large to miss-represent the incident velocity vector.

Validation of the proposed method
In order to examine the accuracy of the AOA estimation using the flow field data at specific test point locations, the normalized RMSE is calculated based on the differences between the calculated values from the flow field data and the exact values of the six prescribed geometric AOAs. The test point locations are considered appropriate with acceptable accuracy if the corresponding normalized RMSE is less than 5%.

Model description
The wind tunnel experiments, carried out by Li et al. [42], are selected to validate the current CFD model. Their experimental data offers a good opportunity for the validation of 2D CFD simulations due to the inclusion of the torque contribution at the mid-span section of the blade, based on the integration of the instantaneous pressure data from a high-frequency multiport pressure scanner.
In addition, their data includes the pressure distribution around the wind turbine blade at different The flow across the mid-span plane the turbine is modelled using 2D CFD simulations, based on the RANS equations, and the SST k-turbulence model is utilized for the turbulence modelling as In contrast with the segregated algorithm, the coupled algorithm features a significant reduction in the computational cost by maintaining a stable solution at high Courant numbers [47]. This means, for a given mesh, a relatively larger time step size may be used and hence the computational cost could be reduced. The second-order implicit unsteady formulation is enabled for the temporal discretization due to its improved accuracy [47]. The second-order upwind scheme is implemented for the spatial discretization of the momentum and the turbulence model equations. The sliding mesh method is used to model both of the turbine rotation and blade pitch motion, when required.
The angular velocity of both the pitch motion and turbine rotation is imposed to the corresponding subdomain using an interpreted User Defined Function (UDF) that is integrated into the solver.
The solver is allowed to perform 30 iterations per each time step and this has been found to be sufficient for reducing all the normalized residuals to, at least, five orders of magnitudes, except the turbulence kinetic energy residuals that usually reduce by, at least, four orders of magnitude. Each simulation includes five complete revolutions, while the data is recorded for post-processing at the fifth cycle to eliminate any effect of the starting unsteadiness and to ensure that a time-periodic solution is obtained. This has been found to be sufficient, under the current setup, to reduce the differences in the cycle-averaged torque coefficient between the successive cycles to less than 1.0%.

Computational domain and meshing topology
The flow around the two-bladed VAWT mid-span plane is modelled by a C-shaped computational domain. Figure Figure 11 The mesh topology of (a) the stationary subdomain and (b) region around the blade.

Mesh and time step size sensitivity study
The proper selection of the temporal and spatial resolution is paramount for the modelling of the unsteady flow around VAWTs. Using the baseline mesh with a total of 701600 elements and 1260 nodes around each blade, three temporal resolutions are investigated namely, 360, 540, and 1080 time steps per cycle that correspond to resolving each degree of the azimuthal angle by 1, 1.5, and 2-time steps, respectively. Table 1 shows the effects of the different temporal resolutions in both of the single-blade cycle-averaged and the peak torque coefficient over the fifth cycle. The differences in the torque coefficient are considered to be fairly insignificant as shown in Table 1  For all the tested meshes, a fine wall-normal mesh resolution is maintained with a maximum dimensionless wall distance, y + of 2.5 and an average y + <1. Although there are small differences in the prediction of the peak torque coefficient, as shown in Table 1, the baseline mesh with 701600 elements is considered to be sufficient for the analysis of the turbine performance with a reasonable computational cost.

Estimation of the AOA for a VAWT Blade
T AOA direction of the approaching wind velocity, V, are constants and equal to that of the undisturbed flow velocity, V∞. Figure 14 (a) illustrates the theoretical velocity triangle at the blade mount point for a zero fixed pitch turbine in an arbitrary azimuthal position, , and the blade rigid body velocity is represented by TSR* V∞. Therefore, the local incident relative velocity, Vr, and the AOA, are simply defined as follows: This calculation of the AOA is referred to as the theoretical AOA. However, in the real flow conditions, there are several phenomena that result in some distortion in both the magnitude and direction of the approaching wind velocity vector. These include the streamtube expansion, the flow deceleration in front of the turbine, and the blades wake interactions. A more realistic relative velocity triangle could be obtained by considering the variation of the magnitude and direction of the approaching wind velocity as shown in Figure 14 (b). A simple aerodynamic analysis could not achieve an accurate prediction of the AOA and the relative velocity magnitude. However, detailed CFD data could provide a good estimation of these quantities that could facilitate a better understanding of the turbine performance. and the relative velocity magnitude. Firstly, the absolute velocity component is calculated at the suggested reference-points while the rigid body velocity of the blade is calculated at the blade mount point. Then, the AOA and the relative velocity magnitude are calculated and averaged between the pair of reference-points. Figure 15 shows a comparison between the theoretical AOA obtained using equation (2) and the estimated AOA based on the proposed method, over one cycle, for both zero fixed pitch (ZFP) and 6° fixed pitch (6° FP) configurations operating under the same condition as the validated test case. It is clear that the differences between the theoretical AOA and the estimated AOA are relatively smaller in the upstream part of the cycle, i.e. from 0° to 180° of azimuthal angle, and these differences are expected to be due to the streamtube expansion phenomenon. However, the differences in the downstream part of the cycle are dramatically higher due to the complexity of the turbine wake.

APPLICATION TO VARIABLE AND FIXED PITCH CONFIGURATIONS
The proposed AOA estimation method is used to evaluate the lift and drag coefficients for two test cases, namely the zero fixed pitch configuration and the sinusoidal variable pitch configuration. For simplicity, and from this point onward, zero fixed pitch and sinusoidal variable pitch will be referred to as ZFP and sinusoidal-VP, respectively. These pitch-configurations are simulated under the same geometrical and dynamical specifications as the validated test case in Section 3.1.4. The magnitude of the sinusoidal-VP is 11.9° and it is selected to equate to the difference between the maximum theoretical AOA of a ZFP turbine at a TSR of 2.29 and the favourable AOA of NACA0015 aerofoil that is assumed to be 14°. Five key characteristics have been considered and these include the instantaneous power coefficient, the AOA, the relative velocity magnitude, lift coefficient, and drag coefficient. Figure 16 illustrates the variations of these characteristics over a complete cycle, from 0° to 360° of azimuthal angle, for both of the ZFP and sinusoidal-VP configurations. The quantities plotted with the dashed lines and marked as theoretical are based on equations (1) and (2) and these are only plotted in the subfigures 16 (b) and (c). The VAWT under consideration has a twobladed design but only one blade is considered for the analysis of the instantaneous power coefficient. The single-blade instantaneous power coefficient represents the generated power from a certain blade regardless of any other turbine component and hence could give more information on the performance of the blade in contrast with the instantaneous power coefficient of the whole turbine. The variation of the single-blade power coefficient for both of the ZFP and sinusoidal-VP are shown in Figure 16 (a). It is clear that the single-blade power coefficient of the sinusoidal-VP configuration is significantly higher in the period between 90° and 180°. However, the ZFP configuration has a higher single-blade power in the downstream part of the cycle, i.e. between 180° and 360°. The cycle-averaged single blade power coefficient is found to be 0.151 and 0.182 for the ZFP and sinusoidal-VP, respectively. Hence, the sinusoidal-VP is able to improve the overall performance of the turbine by about 20% compared with the ZFP configuration under the current setup. and this is characterized by a high AOA followed by a sudden reduction in the lift and drag coefficients. Although the ZFP blade has a higher lift coefficient in the period between 90 and 140°, its power coefficient is dramatically lower. The reason is that the large drag forces in this period act to significantly reduce the tangential force and hence reduce both the driving torque and the power 33 coefficient. On the other hand, the sinusoidal-VP blade operates at a favourable AOA and produces a lower lift coefficient while maintaining a significantly lower drag coefficient and this is in contrast with that of the ZFP blade in the upstream part of the cycle. This enables the sinusoidal-VP blade to achieve a better power coefficient distribution over the upstream part of the cycle. However, in the downstream part of the cycle, the ZFP blade obtains a higher lift coefficient with a negligible drag due to its better AOA potential. These favourable lift and drag coefficients illustrate how the ZFP configuration obtains a higher power coefficient over the downstream part of the cycle. The distribution of the AOA over the entire cycle has significant effects on the overall power coefficient of the VAWT and this makes the proper estimation of the AOA essential. Due to the very complex flow across the VAWT, several ways of selecting the reference-points for the estimation of the AOA have been investigated and it is found that using the flow data from the two reference points at the locations 0.5 aerofoil chord length upstream and 1 chord away from each side of the aerofoil can give most accurate estimation across a range of operating conditions. The new method could be used to calculate and store the AOA data during the CFD simulations without the need for extensive post-processing. In comparison with existing method in the literate, the proposed method can reduces the RMSE by as much as an average of 33.8% for the three tested flow velocities relent to small wind turbine conditions. Based on the proposed AOA estimation method, the performance of a fixed pitch and the sinusoidal variable pitch VAWT configurations have been analysed and it illustrated how the sinusoidal variable pitch configuration could enhance the overall performance of the turbine by maintaining more favourable AOAs to maximise the lift and reduce the drag generation. The method could be used for efficient turbine aerodynamic analysis and optimisation of VAWT.

Mohamed M. Elsakka would like to express his gratitude to the Egyptian Cultural Affairs and
Missions Sector along with Port Said University for their financial support.