Aerodynamic characteristics of two-bladed H-Darrieus at various solidities and rotating speeds

Energy xxx (2015) 1e13

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Aerodynamic characteristics of two-bladed H-Darrieus at various solidities and rotating speeds Sungjun Joo a, 1, Heungsoap Choi b, 2, Juhee Lee c, * a

Department of CO-OP, Hoseo University, Chungnam 336-795, Republic of Korea Department of Mechanical & Design Engineering, Hongik University, Sejong 339-701, Republic of Korea c Department of Mechatronics Engineering, Hoseo University, Chungnam 336-795, Republic of Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 October 2014 Received in revised form 15 May 2015 Accepted 7 July 2015 Available online xxx

Three-dimensional unsteady numerical analysis has been performed in order to analyze the aerodynamic characteristics of an H-Darrieus vertical axis wind turbine with two straight blades. The reliability of the numerical models has been demonstrated through good agreement between the calculated and measured efficiency of an H-Darrieus. Flow characteristics are closely investigated according to tip speed ratios and solidities. A comparison of aerodynamic characteristics at various operational conditions, including the maximum power point is performed. The direction of the free stream approaching the blade is considerably bent following the interaction between blade-to-blade and blade-to-free stream. Even though, the peak value of a torque increases as solidity increases, the blockage and interaction also increase, and thus, increasing the solidity alone does not improve the performance of the H-Darrieus. On the other hand, decreasing the solidity can reduce the effect of blockage and interaction, but the selfstarting features via the negative torque at the low tip speed ratio becomes lost. Therefore, a theoretical model such as the DMST (double multiple stream tube) is not suitable for predicting the performance of H-Darrieus with a high solidity. The blockage by blades in the upwind revolution and the interaction between blades significantly change the magnitude of an incidence velocity, and the angle of attack. Thus, the tip speed ratio of the operation point (i.e. the highest power coefficient point) is found to be lower than it is expected. © 2015 Elsevier Ltd. All rights reserved.

Keywords: H-Darrieus wind turbine CFD (computational fluid dynamics) TSR (tip speed ratio) Solidity Blockage

1. Introduction With the ever-growing concern over the effects of fossil fuels on global warming, there is presently a resurgence of interest in the use of wind turbines as a clean, sustainable and renewable source of energy. This interest is reflected in a corresponding increase in research activity among academic institutions and a proliferation of designs based on several aerodynamic models. Wind turbines are categorized into two types based on the orientation of the rotational axis: the HAWT(horizontal axis wind turbine) and the VAWT(vertical axis wind turbine). Although the HAWT is more common in the wind turbine industry worldwide and its technology better developed, the VAWT, (e.g. the Savonius, Troposkein

* Corresponding author. Tel.: þ82 41 540 9669; fax: þ82 41 540 5808. E-mail addresses: [email protected] (S. Joo), [email protected] (H. Choi), [email protected] (J. Lee). 1 Tel.: þ82 41 540 9966. 2 Tel.: þ82 44 860 2864.

Darrieus and H-Darrieus types) has been gaining increasing attention for a number of its advantages, the primary being a simplicity of design which allows for energy conversion at any wind angle. Moreover, low starting torque makes it an ideal candidate for use in urban areas where wind speeds are relatively low and changes rapidly [1,2]. The modern Darrieus VAWT was invented by French engineer George Jeans Mary Darrieus, with his U.S. patent submitted in 1931 [3] for both the “eggbeater (curve-bladed)” and “straight-bladed” types. In addition to the general advantages of VAWTs over HAWTs, the straight-bladed H-Darrieus type, consisting of two or three straight blades (airfoils), has the following further advantages: 1. low noise owing to the low rotating speed; 2. self-starting and low maintenance costs; 3 ease of design and construction owing to the use of simple straight airfoils.

http://dx.doi.org/10.1016/j.energy.2015.07.051 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

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Nomenclature C CD CL CP D H k L l n p P Re r R

Chord length [m], 2 SÞ], Drag coefficient [D=ð0:5rU∞ 2 Lift coefficient [L=ð0:5rU∞ SÞ], 3 l2RÞ], Power coefficient [P=ð0:5rU∞ Drag [N], Height of blade [m], Turbulent kinetic energy [m2/s2], Lift [N], Span or height of blade [m], Number of blades, Pressure [N/m2], Power [Watt] Reynolds number [rU∞C/m], Rotational radius [m], Rotational radius of blades [m]

The energy conversion achieved by wind turbines occur through aerodynamic forces on the blade such as lift and drag, an operational principle analogous to the aerodynamics of the wings of an airplane in part. The key factors affecting the aerodynamic performance of the H-Darrieus are the radius of the rotor, number of blades, chord length, blade height, tip speed ratio, and installation angle. As air flows over the airfoil, forces are exerted which may generally be divided into its lift and drag components: the drag force (D) is parallel to the wind, while the lift force (L) is perpendicular to the drag force. For most airfoils the lift-to-drag ratio (L/D) increases with an increasing angle of attack (a) until a critical angle of attack is reached where the air ceases to flow smoothly over and separates from the leading edge. This critical condition is referred to as a stall. Once this stall point is reached, L/D decreases with continued increases in a, thereby obstructing the aerodynamic performance of the wind turbine significantly. For optimal aerodynamic performance, therefore, stalls must be avoided or minimized. There have been a number of studies to improve the performance of VAWTs. In particular, in the 1970s and 1980s, Sandia National Laboratories in the USA performed a wide range of studies on the eggbeater-shaped VAWT including investigation of basic physics [2], wind tunnel experiments [4e7], field tests [8,9], design and theoretical performance predictions [10e14], and structural analyses [15e17]. Oler et al. [5] analyzed the unsteady aerodynamic characteristics of the Darrieus VAWT using a doublet panel method to model an airfoil's surface, an integral boundary layer scheme to model the viscous attached flow, and discrete vortices to model the detached boundary layers that formed the airfoil wake. However, while they were able successfully to predict the lift and drag coefficients in steady state for several airfoils, they were unable to predict the unsteady behavior of the Darrieus turbine, owing to a coupling problem related to the viscid and inviscid calculations on the blade's surface. Since the VAWT operates at local Reynolds numbers and angles of attack seldom encountered in aeronautical applications, there is currently limited airfoil data available to describe the aerodynamics of the turbine blades. Sheldahl and Klimas [7], in a study involving seven symmetric airfoils with different thicknesses such as NACA0012 and NACA0025, obtained some sectional data for wide angles of attack (up to 180 ), and aerodynamic forces for both increasing and decreasing angles of attack that showed aerodynamic hysteresis or dynamic stall. Klimas [6] developed a new

S T TSR u i, u j U∞ w xi

a b

ε

r f m mt s u

Blade area [Cl, m2] Torque [Nm], Tip speed ratio [Ru/U∞], Velocity component [m/s2], Free stream velocity [m/s2], Incidence velocity or relative velocity in Fig. 4 [m/s], Coordinate system [m], Angle of attack in Fig. 4 [ ], Installation angle in Fig. 4 [ ], Turbulent dissipation rate [m2/s3], Density [kg/m3], Azimuth angle in Fig. 4 [ ], Viscosity [kg/ms] Turbulent viscosity [kg/ms], Solidity [nC/R], Angular velocity [rad/s],

tailored airfoil to reduce the COE (cost-of-energy) and improve fatigue life, which was analyzed using conformal mapping technique when the flow was attached and wind tunnel data when the flow was highly separated. He performed a wind tunnel test on the new airfoil with a small-scale wind turbine, and found that the tailored airfoil that was modified NACA0015 reduced the drag by the drag pocket. Klimas obtained nearly constant efficiencies over a wide range of TSR (tip speed ratio). However, the maximum power and peak efficiency were slightly lower than that of the original airfoil section. He insisted that these constant efficiencies led less change in power obtaining and easy to control. The free stream coming into the turbine fluctuates randomly because of intrinsic turbulence, and the performance of the VAWT can be affected by this random fluctuation. Homicz [14] used a numerical code called VAWT-SAL (VAWT Stochastic Aerodynamic Load), which used a DMST (double-multiple-stream tube)) method and numerically estimates the random loads created by atmospheric turbulence. Homicz found the output power to decrease slightly with increased turbulence intensity, while for TSRs near the maximum power coefficient, power increased as the turbulence level was raised. Since an eggbeater-type VAWT with flexible blades is subject to periodic and dynamic loads, it needs to be structurally stable. Thus, knowledge of the modal frequencies and mode shapes is essential in predicting the structural response and fatigue life. Lauffer et al. [16], in a study involving a prototype wind turbine, found two modal frequencies that were close to integer multiple times of the operating TSR, and which were a major cause of resonance. There are some numerical methods to predict the performance of the VAWT and to help in the designing. Islam et al. [18] summarized the theoretical models for the predicting the performance of a straight-bladed wind turbine: the doublemultiple stream tube model, the vortex model and the cascade model. Each of the models has its own strengths and weaknesses. Vortex models were considered to be the most accurate models but they are computationally expensive and some cases suffer from a convergence problem. Islam et al. suggested that the doublemultiple stream tube model was not suitable for high tip speed ratios and high-solidity VAWT. The VAWT has an inherently unsteady aerodynamic behavior owing to the variation of the angles of attack with respect to the azimuth angles. Dynamic stall is one of its intrinsic phenomena of the VAWT at low tip speed ratios, thereby affecting both loads and power. Numerical investigations have been performed using the

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vortex panel method [19], and the finite volume method [20], and the experimental investigation with visualization by a PIV (particle image velocimetry) [21]. Ferreira et al. [20] investigated the effect of accurately modeling the separated shed wake from a dynamic stall. The structure and magnitude of the wake were validated with PIV results. To precisely model the 2-D (two-dimensional) configuration of the PIV experiments, a 0.4 m diameter single-bladed VAWT was used in a wind tunnel experiment. Four different turbulence models were compared: the one-equation SeA (SpalarteAllmaras), RNG (renormalized group) k  ε, DES (detached eddy simulation), and LES (large eddy simulation). Among the turbulence models, the DES model was able to predict the generation and shedding of vorticity and its convection for a dynamic stall. The LES model did not perform as well as the DES model, because of a less accurate modeling of the wall region. Ferreira et al. suggested that the SeA model did not provide a sufficient instantaneous flow field and eddies from the blade, but they did not compare the ensemble-averaged results after reaching the periodic state. This is because RANS (Reynolds-averaged NaviereStokes) equations such as kε only provided the time-averaged flow field. Castelli et al. [22] proposed a new performance prediction method for an H-Darrieus using a CFD (computational fluid dynamics) code instead of BEeM (blade elementemomentum) theory. They provided detailed aerodynamic characteristics according to the azimuth angles of the blade, showing that the maximum torque values were generated during the upwind revolution of the turbine and in azimuth positions at which rotor blades experience very high relative angles of attack. The azimuthal positions of maximum power extraction were located between the fourth and fifth octants, owing to the combination of energy extraction exerted by the rotor blade and a relative high rotating arm with respect to the rotor axis. Castelli et al. suggested that the distribution of aerodynamic forces as a function of rotor azimuth angles should be investigated further. That is, the key to the H-Darrieus is the aerodynamic performance of the blade; thus, the airfoil section is critical. Mohamed [23] performed a parametric study using 2-D CFD (computational fluid dynamics) in order to maximize the output torque coefficient and output power coefficient (i.e. efficiency) with 20 different symmetric and non-symmetric airfoils. The optimal airfoil among 20 airfoils was found to be the S-1046, and the maximum power coefficient was 0.4051 for a tip speed ratio of 10 and the solidity of 0.1. Mohamed thus recommended using a low solidity wind turbine to obtain a wide range of operation and high performance. However, low solidity shows low torque at low rotating speeds, which can lead to a self-starting problem. Yang and Shu [24] optimized an airfoil section of a helical vertical axis turbine parameterized with Bezier curves with the use of a genetic algorithm, hydrodynamic constraints, and mechanical constraints. They were able to improve the power coefficient was improved to 41.2% while the pulsation of its power coefficient and torque could be maintained at a low level. The energy conversion in H-Darrieus, mainly occurs through aerodynamic forces on the blades. The purpose of this study is to elucidate the aerodynamic forces and characteristics of the HDarrieus, through a close investigation of the interactions between blade-to-blade and blade-to-free stream and the blockage using CFD methods. The design of an H-Darrieus requires both an effective angle of attack and relative velocity to the blade, but these vary widely according to the interaction of and blockage by the blades. The blade in the upwind revolution can only extract the aerodynamic energy from the free stream, while the blade in the downwind revolution cannot extract aerodynamic energy because of the blockage from the blade in the upwind revolution. The effective angle of attack is changed because of the flow interaction between blade-to-blade and blade-to-free stream, and the

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operational TSR (the TSR of the highest efficiency) is reduced according to changes in the angle of attack. The blade in the downwind revolution accelerates the air and this consumes the energy obtained in the upwind revolution. As a result, the torques in these regions are reduced rapidly and become negative. The large solidity not only leads to the large torque between blade and air, but also the blockage, the flow disturbance by the blades, and the interaction between blades also increase, which in turn leads to a low operational TSR and low efficiency by energy loss. 2. Physical model 2.1. Governing equations around H-Darrieus Fig. 1 shows the computational domain and boundary conditions of the H-Darrieus considered in this study. The flow around the blade is assumed to be three-dimensional, turbulent, and in the unsteady state with incompressible fluid (i.e., air). The turbulent flow of the air is described by RANS (Reynolds-averaged NaviereStokes) equations for mass and momentum, which can be expressed in tensor notation as follows:


vr v þ ruj ¼ 0 vt vxj

(1)


vðrui Þ v vp þ þ si ruj ui  tij ¼  vt vxj vxi

(2)

Fig. 1. Computational domain including boundary conditions (non-scaled) and grid: a) computational domain and b) locally refined grid.

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2 vu tij ¼ mt Sij  mt k dij  ru0i u0j 3 vxk

(3)

where xj are the Cartesian coordinate vectors, uj are the mean velocity components, si are the momentum source term, ru0i u0j is the Reynolds stress tensor; and mt and Sij are the turbulent viscosity and the modulus of the mean strain rate tensor, respectively. In this study, the realizable k  ε model proposed by Shih et al. [25] is applied to model the turbulent flow around the blade. The realizable k  ε model contains a new transport equation for the turbulent dissipation rate ε. Also, a critical coefficient of the model, Cm, is expressed as a function of mean flow, and turbulence properties, rather than assumed to be constant as in the standard k  ε model. The realizable k  ε model can resolve complex flows located close to the blade surface, and is substantially better than the standard k  ε model. 2.2. Numerical methods and H-Darrieus geometry Three-dimensional incompressible unsteady state NaviereStokes equations are solved using computational methods. To avoid the blockage effects and disturbances of outer boundaries, the computational domains are extended sufficiently as follows: the inlet region is far from the leading edge by 20 m, and the exit in the downwind direction is located at a distance of 60 m from the rotating center of the H-Darrieus as shown in Fig. 1. The top and bottom outer boundaries in the z-direction are extended by 15 m in each direction from the center of the blade. For the calculation of aerodynamic characteristics such as lift, lift-to-drag ratio, and flow field, a commercial CFD package, STAR-CCMþ [26], is used. Details about computational methods including moving grid can be found in Ref. [26]. The accuracy and efficiency of the computational results also depend on how the grid system is constructed. For accurate and effective computations, polyhedral grids with a nonlinear distribution (i.e., refined next to the blade) are used. The computational analysis is carried out for a nominal Reynolds number of 1.61  105 based on the chord length (C) of blade, and a free stream velocity of 7 m/s is applied. No-slip boundary conditions on the airfoil surface, and a pressure boundary condition at the exit (x-direction), are specified. A slip-wall boundary condition is prescribed at the top, bottom, and side outer boundaries. The solutions are considered to be converged when all of the residuals (inner iteration) for the continuity and momentum equations are less than or equal to 104. To obtain the periodic solutions, the computation is performed for more than three revolutions. The moving grid methods are used to resolve the rotating blade effect. An AMD Opteron TM processor (2.2 GHz) with Linux is used for the computations, and requires more than several days to obtain at least four revolutions of the rotor blades. The number of revolutions is varied according to the rotational speed of the wind turbine.

Table 1 Mesh dependency test (relative error of Torque).

Coarse Base Refine01 Refine02 Refine03 Refine04

No. of cells

Torque

Relative error

Computational time (hour)

250,000 490,000 690,000 880,000 1,200,000 1,700,000

2.935 3.023 3.001 2.983 2.964 2.965

0.34% 2.62% 1.91% 1.29% 0.65% e

25 45 65 88 143 285

increases, the relative errors against “refine04” (1.7 M cells) decrease, and the value of torque converges as shown in Table 1. When the number of cells is less than 490 K (“base”), the relative error is large. The coarse mesh (“coarse”) with 280 K cells shows the smallest error, but this cannot reproduce the torque peaks properly, as shown in Fig. 2. In Fig. 2, the torque during one revolution after three revolutions is plotted. Under 490 K cells, the peak values are slightly lower than the others. However, when the number of cells is larger than 1.2 M (“refine03”), the relative errors versus those of the 1.7 M cells are less than 1%, and the peak values of the torque versus the azimuth angles are in good agreement. Considering computational accuracy, the mesh generation parameter of “refine03” (the case of 1.2 M cells) is subsequently employed. It takes 143 h of computational time to finish four revolutions in calculation.

2.4. Numerical model validation To validate the numerical models used in this study, a comparison of computational results with those of wind tunnel experiment [27] is performed with various TSR s. The wind turbine consists of three 0.4m-length blades and the Reynolds number based on the chord length (0.1m) and free stream velocity (5.07m/s) is about 35,000. The section of the blade (NACA0022) is thick, which is favorable to structural stability and performance at a low rotating speed. To directly compare the results between computational methods and experiment directly, the cross sections of the computational domain and the experiment are identical in dimension (1.2 m  1.2 m). The non-slip wall boundary condition is used on the blade of the wind turbine, but the slip wall boundary condition on the wall of the wind tunnel is used since the shape of the wind tunnel such as contract ratio, length of the test section, and surface roughness are unknown. Howell et al. [27] listed the

2.3. Mesh dependency test A mesh dependency test is performed with TSR ¼ 1.79, s ¼ 0.7, and R ¼ 1 m. The nominal Reynolds number is 1.61  105, and the Reynolds number based on the tip speed is 2.89  105. to determine the proper number of meshes, the calculations are performed with different numbers of cells, as shown in Table 1. Basically, a nonuniform distribution (small cells next to surface to resolve the boundary layer), and four layers of prism mesh with a high aspect ratio along the rotor blade are employed. The torque is the most important feature in a wind turbine analysis; thus, we compare the torque according to the number of cells. As the number of cells

Fig. 2. Mesh dependency test (torque vs. azimuth angle).

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Fig. 3. Comparison power coefficients between computational result and experimental result [27].

Table 2 Specifications of H-Darrieus. Chord (m) Sectional profile Installation angle (o) Radius (m)

0.1~0.35 NACA0012 3 1

Solidity Number of blades Free stream (m/s) Blade height (m)

0.2~0.7 2 7 1

readings from the brake torque, which included the mechanical losses. They performed an experiment as well as two-dimensional and three-dimensional computational analyses. In Fig. 3, the power coefficient of this study is shown to be slightly higher than that of the experiment because the numerical results only consider the aerodynamic forces and thus, the torque excludes mechanical losses. The gradual increment, maximum point, and sharp decline of the efficiency agree well with those of the experiment.

3. Results and discussion In spite of the simple geometry of the H-Darrieus, the aerodynamic characteristics are highly complicated due to a large variation of the angle of attack and the relative flow velocities into the blades. In this study, the aerodynamic forces are investigated according to various solidities and TSR s. The specifications of the HDarrieus used are listed in Table 2. Detailed schematic configurations (geometry, angles, and forces) of the H-Darrieus are show in

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Fig. 4. Fig. 5 shows the performance variation of the wind turbine according to the solidities. Interestingly, the variation of the torque per unit solidity (T/s) is relatively small in Fig. 5(c), whereas the variations of CP and T according to TSR s in Fig. 5(a) and (b) are large. The torque is proportional to the square of the incidence velocities (w) to the blade. However, the actual torque is not proportional for various reasons. The design parameters of the H-Darrieus include solidity, the sectional shape of blades, the installation angle and so on. When the angular velocity (operational TSR) and sectional shape are given, there is an optimal solidity and an installation angle according to the given design variables. In general, when the installation angle or solidity decreases, an optimal TSR increases and vice versa. The solidity (s) defined as the ratio of a total sectional area (occupancy) of blades to a half frontal area. A large solidity expects the large interaction between blade and air, and the large energy transfer from air to blade. However, the blockage, the flow disturbance by the blades, and the interaction between blades also increase. It leads a low operational TSR and a low efficiency by energy loss. The H-Darrieus used in this study shows the typical tendency of increased TSR with decreased solidity. As shown in Fig. 5(a), the maximum efficiency (CP ¼ 0.23) is found to appear at s ¼ 0.5 and TSR ¼ 2.69. At s ¼ 0.3 or 0.2, the efficiency is slightly lower but the TSR is somewhat higher (TSR ¼ 3.1 and 4.0), whereas the maximum efficiency of s ¼ 0.7 appears at TSR ¼ 2.4. In Fig. 5(b), the largest value of torque occurs at s ¼ 0.7 instead of s ¼ 0.5, at which is found the maximum efficiency. These two cases with relatively high solidities of s ¼ 0.5 and 0.7 show maximum torque at a relatively low TSR whereas the two cases with low solidities of s ¼ 0.2 and 0.3 show maximum torque at a relatively high TSR. At s ¼ 0.2, the entire torque is low and negative values can be seen at the low TSRs of 1.8 and 2.2. This negative torque of the low solidity H-Darrieus makes self-starting of the wind turbine difficult [14], thereby requiring considerable time to reach the operational TSR. Therefore, the low solidity H-Darrieus is not suitable as a self-starting wind turbine, but suitable as an induction-type wind turbine. In Fig. 5(c) representing the torque per unit chord length, the overall shape of the torque curve is similar to the curves shown in Fig. 5(a) and (b), but the differences in the maximum values of torque per unit solidity are small. That is, the torque per unit chord length is nearly constant irrespective of the solidity. In this study, with the exception of the chord length (C), the height of the blade (H), free stream velocity (U∞), radius of rotation (R), and the number of blades (n) are same, and thus, torque is linearly proportional to the tangential force of the wind turbine. The tangential force is determined by the angle of attack (a) and incidence velocity (w) of the blade. To improve the performance of

Fig. 4. Geometry and definition of angles and forces.

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Fig. 5. Torque and efficiency with respect to solidity: a) efficiency, b) torque and c) torque/solidity.

wind turbine, an increase in torque per unit chord length is required. The difficulties of designing an H-Darrieus owe significantly to the fact that a large torque leads to a high blockage effect by the blades and flow interaction between the blade and free stream. That is, the incidence velocity and angle of attack are severely affected by the blockage and interaction from blade-toblade and blade-to-free stream, causing the angular velocity at maximum performance (operational TSR) to decrease and the flow at downwind revolution also to decrease, resulting in poor performance of the wind turbine. Therefore, both torque and angular velocity are critical to the performance of the H-Darrieus. Fig. 6 shows the torque during the last one revolution at the maximum performance for each solidity. As the wind turbine has two rotating blades in this study, two peaks can be observed during each revolution of the blade. The azimuth angle (f) is measured in the counter-clockwise direction as shown in Fig. 4 and thus, one blade is placed at f ¼ 0 , so the other is placed at f ¼ 180 . When the blade is placed at f ¼ 0 or f ¼ 180 , the lift does not produce the torque as it acts in radial direction, while the drag has a significant effect on negative torque. Consequently, the torque in this area is negative, and the blade passes through this region with inertia. Interestingly, at each blade a large torque in the upwind revolution can be observed, but a small or nearly-zero torque is observed in the downwind revolution. The largest instantaneous torque is recorded at f ¼ 90 . The azimuth angle of the maximum torque angle varies slightly according to differing solidities. At an angular velocity above the maximum efficiency, the torque in downwind revolution decreases rapidly and becomes negative. The difficulty in designing the H-Darrieus mainly comes from this decrease in torque. An increase in the torque in the upwind revolution is the cause of blockage in the downwind revolution, resulting in a decrease in torque in the downwind revolution. Since the blade in the downwind revolution cannot extract the power from the wind, the total torque in the upwind revolution is identical to that of a single blade, as shown in Fig. 6(a). A similar tendency can also be found in the other results (Fig. 6(b)e(d)). That is, maximum performance can be obtained when the obstruction of flow from the blade in the downwind revolution is at a minimum. From this, we deduce that if the TSR is larger than the maximum operation point, the blade in the downwind revolution consumes the energy obtained by the blade in the upwind revolution, much like a fan, accelerating the surrounding air in the direction of the rotation. The installation angle (b) is another important design factor. In this study, the installation angle of 3 is used. In the case of s ¼ 0.2(Fig. 6(a)), incidence velocity (w) is large owing to its high TSR, and the flow disturbance is small owing to low solidity. It is expected, therefore, that s ¼ 0.2 would show the highest performance among the various solidities considered in this study; however, it is not because of the installation angle, which changes the incidence angle of attack. The installation angle of 3 reduces the effective incidence angle of attack in the upwind revolution, but increases the angle in downwind revolution. Hence, stall is retarded in the upwind revolution, but advanced in the downwind revolution. To improve performance when s ¼ 0.2, therefore, the smaller installation angle should be used. But the smaller installation angle would enforce a negative torque at rotation startup (low TSR). On the other hand, a torque at s ¼ 0.7 is the largest as shown in Fig. 6(d), but the performance (efficiency) is not the most optimal because of low rotating speed. The negative torque in the downwind revolution increases because of the blockage effect by the blades in the upwind revolution. Thus, at s ¼ 0.7, the interaction between the upwind stream and the blades decrease the angle of attack and flow speed rapidly. It is a s ¼ 0.5, as shown in Fig. 6(c),

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Fig. 6. Torque variations according to solidities: a) s ¼ 0.2, TSR ¼ 4.5, b) s ¼ 0.3, TSR ¼ 3.6, c) s ¼ 0.5, TSR ¼ 2.7 and d) s ¼ 0.7, TSR ¼ 2.2.

that the performance becomes the highest. The sum of the torque becomes slightly larger than the torque of the single blade since the blade placed in the down stream revolution has positive torque. This implies that the air in the area has momentum in the x-direction with actual energy. The optimum design with chord length, and installation angle depends on the given free stream velocity, angular velocity, and sectional wing shape. Since the blade of a VAWT rotates 360 with respect to the free stream, the variation of the angle of attack is very large and the performance also changes considerably. Therefore, the theoretical performance prediction of a VAWT with large solidity is difficult because of the flow interaction from blade-to-blade and free stream-to-blade [18]. Due to the interaction among the blades and the free stream, the positive y-direction flow is induced at 0 < f < 90 whereas the negative flow occurs at 90
than the stall angle). The torque of TSR ¼ 0.9 (60 rpm) suddenly decreases at f ¼ 50 instead of f ¼ 90 as shown in Fig. 7 (s ¼ 0.2). This implies that the high angle of attack leads to a stall in the area and, the lift sharply decreases at f ¼ 50. Therefore, the torque becomes asymmetric, and drops sharply after the maximum point, as shown in Fig. 7. Similar asymmetric torque distributions and a significant decrease in torque due to stall can be observed for low TSR in Figs. 8 and 9. On the other hand, the angle of attack decreases as the rpm (TSR) increases. The stall disappears, and the torque in the upwind revolution increases gradually but the torque in downwind revolution decreases due to the blockages of the upwind stream at the same time. When the rotational speed is larger than that of the operation point (see Fig. 9 (f), (g), and (h)), the shape of the torque is symmetric. The peak values of the torque in these TSRs decrease due to the low angle of attack and low magnitude of the relative velocity. The airfoil section used in this study is symmetric, and there are some similar studies in 1970s [4,8,10]. However, in the present study, operational TSR s do not reach as high as the TSRs in previous studies. High solidity and installation angles improve the self-starting characteristics, but degrade the performance. It is interesting that the azimuth angle of the peak torque shifts to high azimuth angles according to the increase in both TSR and solidity. The maximum torques for s ¼ 0.2, 0.5, and 0.7 occur at f ¼ 90.1, 95.0, and 97.1 respectively. The torque, angle of attack, and relative velocity for various solidities are plotted in Fig. 11. The angle of attack and the relative velocity are obtained for every 4 of azimuth angle along the blades trajectory. The values of w and a are recorded during one revolution of the blade, and the averaged values at the same location are calculated. However, the values of some points cannot be obtained

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Fig. 7. Torque with respect to azimuth angle at s ¼ 0.2.

properly because of the passage of the rotor blade (there is no flow field where the blade is placed). These values, including two additional points next to the blade are ignored when the average values are calculated. The detailed processes used in this study can be found in Ref. Raciti et al. [23]. The “no blade” case shown in Fig. 11 presents the geometrically calculated values from the free stream, and tangential velocity. We ignore the blade effect at the same azimuthal position. The values between the with-blade and without-blade cases shown in Fig. 11(d) (s ¼ 0.2 and TSR ¼ 2.2) are very similar. This implies that the solidity is low, and thus the free stream is not disturbed while the blade is rotating. However, the torque is small because of the stall at the early stage of rotation (f > 60 ). The solidity and rotational velocity are small, and so the effect of the blockage by the blade placed in the upwind evolution is negligible. Consequently, the angle of attack and relative velocity for s ¼ 0.2 and TSR ¼ 2.2 are similar to the values for the case without the blade. In the case of s ¼ 0.2 and TSR ¼ 4, the difference between the case with blade and without blade, however, is small in the upwind revolution, but the difference in the downwind revolution is considerable. Despite its low solidity, the blockage of the blade in the upwind revolution is significant, and the free stream velocity decreases in the downwind revolution. The velocities between the

case with blade and without blade appear similar, but a decrease in the free stream velocity (x-direction velocity) is significant in the downwind revolution shown in Fig. 11(e). However, the velocity changes in downwind revolution cannot be seen clearly since the angular velocity that is part of the relative velocity is large as shown in Fig. 4. Considering a velocity triangle, the relative velocity is the vector sum of the free stream and rotational velocities. Even though, the free stream is small, the tangential velocity is large and the relative velocity is also large. In addition, the blade is in stall, and thus, the near-zero torque can be observed in the downwind revolution. In the case of s ¼ 0.7, the gap of the angle of attack, and the incidence velocity to the blade between the cases with blade and without blade becomes large in downwind revolution as well as in the upwind revolution, as shown in Fig. 11(f). Since the solidity is large, the interaction between blades and upwind flow is significant. Thus, the induced velocity by the blade in the free stream is also large, and the angle of attack is considerably reduced. In the downwind revolution, the x-directional flow velocity having energy significantly decreases. The footprint of the optimal solidity (s ¼ 0.5) is relatively large, and thus, the reduced angle of attack between the cases with blade and without blade is large as shown in Fig. 11(a)e(c). The difference increases as TSR increases. Both TSR ¼ 2.2 (150 rpm) and 2.7

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Fig. 8. Torque with respect to azimuth angle at s ¼ 0.3.

(180 rpm) show a similar tendency in terms of angles and velocities. In the case of TSR ¼ 2.2, the angle of attack is too large at f > 90 and the torque suddenly decreases due to stall. However, for TSR ¼ 2.7, the angles of attack do not cross the stall angle and the torque is smooth and decreases gradually at f > 90 , and it shows the largest torque in the wind revolution. Despite the x-direction velocity decreases by the blockage, the torque is still non-negative in the downwind revolution as shown in Fig. 11(b) and it maximizes the torque in the upwind revolution. Interestingly, for TSR ¼ 4 (270 rpm), the incidence velocity in the downwind revolution is constant (slightly weaving), and the angle of attack is nearly zero as shown in Fig. 11(c). Note that there are no relative velocities between rotating blade and air flow in the downwind revolution. The air lost the energy containing x-direction velocity is just accelerated by the blade. Therefore, the energy cannot extract during the downwind revolution; rather, the blade accelerates the air around like a fan. The torque in downwind revolution becomes negative with increasedTSR, and energy cannot be extracted in this range of TSR values. The air flow changes its direction and magnitude according to the solidity and TSR. The cube of the magnitude of the velocity represents the energy stored in the air stream, and the direction of the air flow represents the angle of attack. Therefore, the vector of the air stream is very important. In Figs. 12 and 13, the velocity

magnitudes are plotted at three consecutive locations: r/R ¼ 0.5, 1.0 (blade), and 1.5. The two blades are located at f ¼ 0, and 180 . Thus, the large disturbance can be seen in both Figs. The velocity variations according to the TSR and solidity can be observed in Figs. 12 and 13, respectively. At a glance, the difference in velocity between the upwind and downwind revolution can be observed easily, and intensifies as the TSR increases. In Fig. 12(c), and (d), the velocity magnitude in the downwind revolution is very small and the flow direction is changed to the y-direction. In upwind revolution, the direction of the air flow also changed to the y-direction (see in Fig. 12(c) and (d)), which changes the flow characteristics of the wind turbine severely. The blockage and interaction of the blades and free stream are very important as mentioned previously. The solidity shows similar blockage effects, as shown in Fig. 13. Although, the TSRs in Fig. 13 are identical, there are few disturbances in both the upwind and downwind revolutions in Fig. 13(a) where the solidity is small whereas there are considerable disturbances in Fig. 13(d) where the solidity is large. The flow in the downwind revolution is accelerated in the rotational direction by the blade, and it consumes the energy obtained in upwind revolution as shown in Fig. 13(d). Fig. 14 shows the vorticity distribution at the center of the blades (z ¼ 0), and also shows the flow characteristics of the VAWT according to the value of the TSR. The two blades are placed at f ¼ 0

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Fig. 9. Torque with respect to azimuth angle at s ¼ 0.5.

and 180 in Fig. 14. The counter-clock-wise rotation of the vorticity represents positive and vice versa. As the TSR increases, the movement of the vortex generated in the upwind revolution is getting slow, and the number of traces inside the rotational radius also increases. In the case of a large TSR (TSR ¼ 4.0), the free stream inside of the rotational radius becomes extremely slow and flow in this area is stagnated, as shown in Fig. 14(d). The wake area behind the rotor increases, and we expect that the energy could not be extracted properly at this high TSR. We also observed low air velocity due to the blockage, and flow interaction between the blade and free stream in upwind revolution. The large turbulent dissipation, which is not plotted in this study, can be observed on the left-hand side (i.e., inner rotating area) in Fig. 14(d).

4. Conclusion and future work In this study, a computational analysis with a moving grid to mimic rotating blades has been performed with various solidities (s) and TSRs (tip speed ratios). Depending on the s s and TSRs, the aerodynamic characteristics of an H-Darrieus wind turbine are significantly varied. Blades with s ¼ 0.7, and s ¼ 0.2 show the largest torque and the highest angular velocity at their operational

TSRs, respectively. However, the largest efficiency (CP ¼ 0.23) occurs at s ¼ 0.5. The flow characteristics can be clearly explained by two primary effects: the blockage, and the flow interaction between the blades and free stream. The relative velocity and angle of attack with respect to the blade rotating change significantly due to the above two effects. In upwind revolution, an energy transfer from the free stream is active, whereas in downwind revolution, the transfer is inactive. Furthermore, the blade in the downwind revolution accelerates the air in the rotational direction, and consumes the energy instead of extracting power in the free stream. For large TSR s such as, TSR  3.7 and s ¼ 0.5, a deep decrease in torque can be observed because of a weak free stream in the downwind revolution due to blockage. On the other hand, the direction of the free stream approaching the blade is considerably bent in the y-direction due to the interaction. Thus, the angle of attack of the blade also decreases. The difficulty in the design of an H-Darrieus comes from this decrease in angle of attack and the change of flowdirection. In addition, an increase in torque in the upwind revolution is the cause of blockage in the downwind revolution, and results in a decrease in torque in the downwind revolution. When the TSR is low such as TSR  2.2 and s ¼ 0.5, the angle of attack reaches its critical angle (stall) during the upwind revolution, and the

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Fig. 10. Torque with respect to azimuth angle at s ¼ 0.7.

Fig. 11. Comparison torque, relative velocity, and angle of attack between with and without blade.

torque suddenly decreases, whereas when the TSR is high such as TSR  3.7 and s ¼ 0.5, the angle of attack dose not reach its critical angle, but the torque becomes small because of its small angle of attack. In the downwind revolution, since the air is accelerated by the blade, the incidence velocity tends to be constant, and the angle

of attack decreases to zero. These physical phenomena appear in advanced TSR as the solidity increases. In terms of the aerodynamics, the difference between the twoand three-dimensional analyses can be observed in a few studies [22,23,27]. The efficiency in two-dimensional analysis is more than

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Fig. 12. Velocity variation according to TSR at r/R ¼ 0.5, 1.0 and 1.5.

Fig. 13. Velocity variation according to solidity at r/R ¼ 0.5, 1.0 and 1.5.

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Fig. 14. Vorticity distributions at z ¼ 0 and s ¼ 0.5: a) TSR ¼ 2.2, b) TSR ¼ 2.7, c) TSR ¼ 3.1 and d) TSR ¼ 4.0.

40% but the efficiency in three-dimensional is only 10e25% [27]. To improve the design of the H-Darrieus, extensive three-dimensional effect such as wing tip vortex or else will be closely investigated in the near future.

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