Descriptions of Attack Angle and Ideal Lift Coecient for Various Airfoil Proles in Wind Turbine Blade

The angle of attack is highly sensitive to pitch point in the airfoil shape and the decline of pitch point value induces smaller angle of attack, which implies that airfoil profile possessing closer pitch point to the airfoil tip reacts more sensitively to upcoming wind. The method of conformal transformation functions is employed for airfoil profiles and airfoil surfaces are expressed with a trigonometric series form. Attack angle and ideal lift coefficient distributions are investigated for various airfoil profiles in wind turbine blade regarding conformal transformation and pitch point. The conformed angle function representing the surface angle of airfoil shape generate various attack angle distributions depending on the choice of surface angle function. Moreover, ideal attack angle and ideal lift coefficient are susceptible to the choice of airfoil profiles and uniform loading area. High ideal attack angle signifies high pliability to upcoming wind, and high ideal lift coefficient involves high possibility to generate larger electric energy. According to results obtained pitch point, airfoil shape, uniform loading area, and the conformed airfoil surface angle function are crucial factors in the determination of angle of attack.


Introduction
According to growing greenhouse emissions and global warming the exploitation of renewable and pollution free energies is significantly required to alternative fossil fuel energy.
Representative sources of renewable energy are solar and wind. Wind energy has stood in the spotlight progressively due to the increased demand of scientific and industrial advancements, and wind supports possibly more that 20% of global electricity in the year of 2030 [1]. By the advantage of superior efficiency, wider operational range, and reliability, horizontal axis wind turbine (HAWT) is an example structure as a mass power production during the last several decades.
Flow field behavior around wind turbine are very complicate owing to turbulence generation, vortices, and stall flow. Numerous researches have investigated the aerodynamics occurring around wind turbines to understand the detailed process of kinetic energy extraction from wind. Blade element momentum (BEM) theory, Vortex method (VM), and Computational fluid dynamic (CFD) are widely known approaches. BEM is the combination of momentum and blade element theories by dividing the wind turbine blade into independent elements. The method is not appropriate for heavy loaded states, that is, the axial induction factor is greater than 0.5 [2,3]. The assumption of the flow field is inviscid is added to explicate VM. The wind turbine blade and wake are regarded as vortex particles. The influences of tip vortices can be predicted using VM, while it is difficult to investigate the effects of flow separation with VM [4,5]. CFD is a mathematical process for solving the numerical solutions of complex nonlinear partial differential equations involving fluid flow. CFD enables to describe the physical phenomena of laminar, transitional, and turbulent flows. Moreover, the behavior of streamlines, pressure, and velocity contours as actual flow around a wind turbine can be described with the help of CFD [6], and CFD has been used widely to predict HAWT performances.
The determination of airfoil shape for wind turbine blades is fundamental task for designing wind turbine rotors [7]. Airfoil design in the wind turbine is to accomplish a high aerodynamic performance that produces high electric energy in the designed wind turbine. By virtue of computer technologies many numerical tools are developed to investigate the optimization states of airfoil shapes. Hicks et al. [8], Eppler [9], Tangler and Somers [10], are early authors to compose numerical optimization in airfoil design. Conformal mapping method and velocity distribution are taken into account to design airfoil shape by researchers [11,12]. Echjijem and Djebli [13] considered the correction of the axial and tangential induction factors in the design of wind turbine blade, and Lim and Kim [14] studied multi-objective airfoil shape optimization using adaptive hybrid evolutionary algorithm. Lim and Kim found that the cases of lift maximum and high angle of attack show multi-modality in the design spaces.
However, the separated design process of airfoil and blade fails to grant an optimum state, and integrated design of airfoil and blade is required. 2D airfoil and 3D integrated design approach is addressed by Sartori et al. [15]. The airfoil shape is designed to increase computational efficiency, whereas limited accessible airfoil geometry is appeared. Wang et al. [16] suggested an integrated method that describes airfoil profiles with a trigonometric series form and series of polynomial equations using conformal transformation. The method yields direct and accurate shapes of airfoil and airfoil shape is controlled by the coefficients in series.
Wei et al. [17] presented an integrated method for designing airfoil families of large wind turbine blades. They included rotor diameter, tip speed ratio, and local speed ratio to express optimal airfoils, and a shape perturbation function is applied to the geometry of the prescribed airfoils. Kumar et al. [18] simulated airfoil shape for optimum wind characteristics using a computer program JAVAFOIL to drive the relationship of angle of attack with coefficients of lift and drag. But, most of previous works have analyzed numerically the influences of critical factors in the aerodynamic performance of specified airfoil shapes.
In the present study, by the method of conformal transformation functions for airfoil profiles and airfoil surfaces are expressed with a trigonometric series form, which depicts various airfoil profiles with the appropriate control of the coefficients of series. The attack angle distributions are described analytically around the circumference of various airfoil shapes and compared. Even, pitch point is controlled to clarify the differences of attack angle distributions. Ideal angle of attack and ideal lift coefficients are expressed by dint of the area of uniform for various airfoil profiles.

Airfoil Profiles
Any airfoil profiles can be obtained by a conformal transformation of a circle and expressed with Fourier expansions [19, p.46]. Let us define the coordinates of and ′ by, respectively, Due to the general transformation of ′ relating to plane given by the and can be represented with With the values at the specified 2 equally spaced intervals in the range 0 ≤ ≤ 2 , that is, 0, , 2 , ⋯ , (2 −1) , the λ, , and of the function are determined as and the function of is described with Note that the function ( ) describes the surface angle of airfoils. The conformal mapping from ′-plane to -plane caused by the transformation yields a circle in the -plane into a curve resembling a wing section in the -plane, where = 4 (see Figure 1). The c represents the airfoil chord length. The coordinates of ξ are defined as ξ = x + iy, which provides the expression of the coordinates of the airfoil with Then the angle of attack can be written by (a) (b) Figure 1. Airfoil profile transformed of a circle by conformal mapping: (a) angle of surface, (b) relative flow angle onto blade.

Ideal design of mean lines
The mean line is regarded as the locus of midway points between the upper and lower surfaces of the section. Mean lines under the action of a uniformly distributed loading from ⁄ = 0 to ⁄ = * and decrease linearly to zero at ⁄ = 1 have been calculated in Ref [19, p.74], and the ordinates are expressed as The implies the ideal lift coefficient. The ideal angle of attack for these mean lines is

Results and discussions
Three types of ( ) and five types of ( ) are chosen to describe of angle attack and ideal lift coefficient for various airfoil shapes. The functions chosen are exhibited below, and the airfoil shapes depicted by the chosen functions are displayed in figure 2.         Figure 4 portrays the attack angle distributions of Type_3 airfoil profiles described by 1 ( ). As shown in Figure 4(a) and 4(b), the attack angle distributions surge with further fluctuation as RCP value decreases, which signifies that airfoil shape with smaller RCP value is more susceptible to the upcoming wind. But, the magnitude of attack angle is getting smaller with the decay of RCP, and the variation of angle attack meager in Type_3 airfoil of RCP ≥ 4. Figures 5 and 6 represent the angles of attack for NACA64418 and S809 airfoil profiles, respectively. The magnitude of attack angle distribution is getting larger as the value of RCP grows on both NACA64418 and S809 airfoil shapes. But, unlike Type_3 airfoil, there is no variation in the number of distribution fluctuation. On the whole the decline of RCP value induces smaller angle of attack in magnitude, which implies that airfoil profile possessing pitch point close to the airfoil tip reacts weakly to upcoming wind.

Conclusions
In this study analytical analysis has been developed to investigate attack angle and ideal lift coefficient distributions for various airfoil profiles in wind turbine blade. Pitch point and uniform loading area are taken into account, and airfoil shape functions of wind turbines and surface angle functions of airfoil are expressed with series using conformal transformations.
Angle distributions of attack appear with various forms depending on the conditions of airfoil shape, pitch point, and surface angle function. Ideal attack angle and ideal life coefficient are very sensitive to the variation of uniform loading area. Through results obtained the determination of proper pitch point, loading area, and airfoil shape are essential factors to control the angle of attack, and the results may be useful in designing wind turbine airfoils.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The author declares that there are no conflicts of interest.