eic climate change technology conference 2013

EIC Climate Change Technology Conference 2013

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CFD Investigation of 2D and 3D Turbulence Structures Effects on a Small Straight-Blade Vertical Axis Wind Turbine

CCTC 2013 Paper Number 1569697193

S. Naghib Zadeh1, M. Komeili1 and M. Paraschivoiu1 1 Concordia University, Quebec, Canada

Abstract

This paper investigates the accuracy of a CFD model to capture the complex flow around a small vertical axis wind turbine (VAWT) on 2D and 3D grid. Therefore, an Unsteady Reynolds-Averaged Navier-Stokes analysis is performed with parallel OpenFOAM solver based on the Spalart-Allmaras (SA) turbulence model. A grid convergence study is conducted on 2D grids to examine our CFD model sensitivity to grid resolution. Moreover, a 3-D grid of the VAWT is modeled in order to explore the influence of the 3D effects on the aerodynamic performance of the turbine.

Keywords: grid convergence, Spalart-Allmaras, tip vortex, Vertical Axis Wind Turbine

Résumé Cette étude a pour objet l’évaluation de l’exactitude d’un modèle CFD (dynamique des fluides numérique) conçu pour représenter l’écoulement complexe autour d’une petite éolienne à axe vertical (SVAWT) munie de quatre pales, sur des grilles 2D et 3D. La complexité aérodynamique de l’écoulement est attribuable essentiellement à la variation rapide de l’angle d’attaque de chaque pale. L’écoulement obtenu comporte des zones de décollement important, un décrochage dynamique et une interaction sillage-pale. Une analyse instationnaire est effectuée à l’aide des équations de Navier-Stokes en moyenne de Reynolds (URANS), avec calculs en parallèle au moyen du solveur OpenFOAM® en se basant sur le modèle de turbulence Spalart-Allmaras (SA), et cela en vue de saisir ces phénomènes complexes. La méthode numérique fait appel au schéma implicite d’Euler et au schéma amont pour la discrétisation du temps et de l’espace, respectivement. En outre, une technique d’interface glissante sert à interpoler les valeurs à la limite entre les domaines mobile et stationnaire. Une étude de convergence de grille est réalisée sur la grille 2D afin d’évaluer la sensibilité de notre modèle CFD à la résolution de grille. On utilise en conséquence un résiduel de grille moyenne de y+ >30, ainsi qu’un traitement de paroi afin de représenter les structures de l’écoulement dans les régions voisines de la paroi. En outre, on modélise une grille 3D de l’éolienne à axe vertical (VAWT) dans le but d’examiner l’influence des effets 3D sur les performances aérodynamiques de l’éolienne. L’étude est spécialement axée sur la visualisation de la formation de vortex aux pointes et sur leurs répercussions sur les charges et la puissance de la VAWT.

Mots-clés : convergence de grille, Spallat-Allmaras, vortex aux pointes, éolienne à axe vertical


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1. Introduction

Global concern about environmental issues is a significant motivation to look for a “green” alternative to generate electricity. Wind energy is considered as a competitive choice among the renewable sources of energy. The growing demands and investments in wind farms, both onshore and offshore, demonstrate the growth of the wind energy industry. However, the lack of vast suited lands in populated areas that provide the desirable wind speed and continuous wind flow during a year, as well as the high cost of installation and maintenance of grid connections are two formidable obstacles to the development wind turbine energy in the urban areas and small farms. The small and medium sizes of wind turbines have the potential to fill this gap. However, the question arises what type of wind turbine is fitted best with already stated conditions?

Wind turbines are categorized based on the axis of their rotation, into two types: Horizontal Axis Wind Turbine and Vertical Axis Wind Turbine. Some features of VAWT such as: lower cost of the maintenance, smaller wake behind the VAWT, and insensitivity to yaw angle make them a better choice compared to horizontal one in urban areas. Choosing a suitable turbine at a specific site is an important task for wind consultation companies. Therefore, a precise simulation of air flow through turbines and investigation of aerodynamic parameters would be significant and helpful to optimize the design of turbines. Many influential aerodynamic phenomena that may cause a reduction or increase in the efficiency of vertical axis wind turbines are related to turbulence features of flow. Therefore, developing an accurate turbulence simulation of flow is a key factor for prediction of the aerodynamic forces on the blades.

For HAWT, the blade design has benefited from enormous research [1] [2] [3][4][5]. However, the VAWT simulation is relatively new and most simulations are carried out in the last two decades. Therefore, there are still many uncertainties regarding their aerodynamic behaviours. McLaren et al. [6] used commercial CFX solver to estimate the dynamic loading of the blades of a small-sized straight blades VAWT. They utilized SST model in combination to the transition model, introduced by Menter et al. [7], to calculate lift and drag forces on the blades and they declared that the transitional method captured more features of the flow. Ajedegba [8] investigated the power coefficient of special type of small vertical axis wind turbine (Zephyr) at different tip speed ratios. He used the commercial software FLUENT with multiple reference frame (MRF). The simulations were carried out on a 2-D domain using model along with standard wall function. Although there was a relatively good agreement between CFD results and stream tube model at low tip speed ratio range, the deviation from experiments is observed

for higher tip speed ratios. The weakness of model in simulating intense adverse pressure gradients flows and complex geometries are the main reasons behind the discrepancy with experimental data.

Camelli and Lohner [9] introduced a new computation model that combined Baldwin-Lomax with Smagorinsky (BLS) to capture the separation on a circle and cylinder. They also analysed the turbulence structures inside the wake behind the obstacles. Their model utilized Baldwin-Lomax in the near wall region and the Smogorinsky model for far-wall region. They concluded that both Balwin-Lomax and BLS produce similar results and that their predictions are closer to the experimental data compared to when Smagorinsky model is applied alone due to coarse grid in the near-wall region. Zhi et al. [10] used BLS to investigate the effects of the number of blades and tip speed ratio on the power coefficient of a small-sized straight Darrieus wind turbine. Although their results demonstrated, qualitatively, the effect of solidity and the number of blades on the power coefficient of turbine, they could not capture the stall on the blades. The drawback correlates to the fact that BLS has the same deficiencies as BL model in the solution of flow


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including the large separation region and significant curvature effects. Ferreira et al. [11] simulated the air flow through the VAWT turbine with different turbulent models and validated the results with experimental observations obtained with particle Image velocimetry (PIV) technique. They investigated, in specific, the part of rotations in which dynamic stall happens. Of

the CFD models that they used (RANS with Spalart-Allmaras, RANS with , LES with Lilly SGS model, DES with S-A near-wall region), DES resulted in the best agreement with the experiment. Although using LES or DES simulation with sufficient grid resolution may result in more accurate flow perdition compared to URANS models, a significant obstacle to use these methods in practical wind turbine problems is computer resources availability. The problem gets more severe for simulations aimed at capturing the 3D turbulence features such as tip vortices. Consequently, the only affordable turbulence model for 3D VAWT simulation is tied to URANS models.

In the present research the Spalart-Allmaras turbulence model is used to simulate 2D and 3D vertical axis wind Turbine. For 2D simulations the total power coefficient and the torque generated by each blade at different tip speed ratio (TSR) are calculated. A grid convergence is conducted to estimate the sensitivity of the model to the grid resolution. A 3D simulation is also performed to demonstrate the three-dimensional turbulence impacts on the torque and the power coefficient. Hence, in the following sections the URANS governing equations and the numerical methodology are described. Second, the numerical domain, boundary conditions and grid properties are demonstrated. Then, convergence methodology, based on 2D results, is conducted. Finally, the differences between two dimensional and three dimensional results are addressed.

2. Governing Equations

The simulation of flow over VAWT is studied by solving the Unsteady Reynolds-Averaged Navier-Stokes equations (URANS) with the Spalart-Allmaras (SA) as its turbulence model. Applying Reynolds decomposition and taking time-average of the continuity and momentum equations yields the following URANS equations for incompressible flows,

(1)

( )

(2)

where, and are the average pressure and velocity components, respectively. is the kinematic viscosity and is the fluid density. Furthermore, is the specific Reynolds Stress

tensor and can be described as,

(3)

This is an additional symmetric tensor which has six independent components and expresses the correlation between the fluctuating velocities. Basically, for three dimensional flows, there are four equations and ten unknowns including six components for Reynolds stress, one pressure and three velocities. Therefore, in order to close the system, more equations are needed.


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2.1 Turbulence Modelling

The next approach is to employ Spalart-Allmaras (SA) turbulence model as an additional transport equation. In this model the kinematic eddy viscosity term is expressed by the following equation:

(4)

where and are modified kinematic eddy viscosity and closure function, respectively and can be expressed by,

(5)

(6)

Here, is kinematic molecular viscosity and is a constant. The transport eddy viscosity equation in SA model can be defined by the following formula:

(

)

[( )

]

(7)

In this equation, the first three terms on the right hand side are the production, destruction and diffusion of the kinematic eddy viscosity, respectively. The diffusion term includes both

the molecular viscosity and the turbulent structures effects. represents the production term which is,

(8)

where corresponds to the magnitude of vorticity and is the point field distance to the nearest wall, and

( ) (9)

( )( )

(10)

The destruction function can be defined as,

(

)

(11)

where,

( ) (12)

[

] (13)

The constants correspond to the SA model are listed in Table 1.


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Table 1. SA model coefficient

2.2 Wall treatment

In order to capture the near-wall regions flow structures, standard wall function for SA model is employed as follows,

(

) (14)

where Kappa ( ) and is the additive constant utilized in the logarithmic law of the wall. Using wall function allows us to locate the first point adjacent to the wall in logarithmic zone instead of sub-layer. Therefore, the averaged y+ calculated should be around 30 in all the simulations.

3. Numerical Implementation

In the following sections the details of the finite volume technique that is used to discretize the governing equation is described and the linear solver is explained.

3.1 Discretization Schemes In order to discretize convection terms in the velocity and turbulence equations, Gauss scheme with upwind interpolation is used. Furthermore, Euler implicit scheme is employed for the time discretization. Pressure gradient is discretized by using Gaussian integration followed by the linear interpolation scheme wherein second-order central differencing is used. Viscous terms

are discretized by Gaussian integration with linear interpolation of the diffusion coefficient ( ) with a surface normal gradient scheme, which is an explicit non-orthogonal correction. It should be mentioned that the surface normal gradient is computed at the face of the cell and is a component which is normal to the face of the values of gradient at the centers of the two cells that is connected by the face [12]. The SIMPLE algorithm is mostly used for transient-state simulations to couple the velocity and pressure in OpenFOAM. This method has the capability of having larger time steps compared to the PISO algorithm [13]. 3.2 Numerical Solution In order to solve the linear system of equations, the Krylov Subspace Solvers (KSS) are employed. Preconditioned Bi-Conjugate Gradient (PBiCG) approach followed by the diagonal-based incomplete lower-upper (LU) preconditioner for asymmetric matrices is used for velocity and turbulence equations. For the pressure a preconditioned conjugate gradient (PCG) solver with a diagonal-based incomplete Cholesky preconditioner is employed [12].


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4. Computational domain, Grid generation and Boundary Condition

In the present work, unsteady flow simulations are performed for 2D and 3D computational domains of a VAWT with the geometry specifications given in Table 2. Gambit software is used as a tool for creating the geometry and mesh generation.

Table 2. Specification of the geometry

Number of rotors 1

Number of blades 4

Blade chord 0.4445 m

Rotor Diameter (D) 5.395 m

Angular velocity of the rotor 90 rpm

The diameter of the rotating zone 1.053 D

The computational domain of a VAWT includes two main zones, namely, stationary zone, which is considered for the far field flow and the rotating zone that rotates with four NACA 0018 blades with the given angular velocity. A rectangular domain, which corresponds to the stationary zone, is utilized, for 2D grid, with a distance of 10 rotor diameter from the axis of rotation to top, bottom and left boundaries (Figure 1.a.) Also there is 15 rotor diameter distance from the axis of rotation to upwind. Figure 1.a. and Figure 1.b. illustrate the mesh of the computational domain as well as the mesh around an airfoil, respectively. Bottom-Up approach is used to make the 3D grid. In this technique the vertices, edges, faces are first created in 2D and thereafter 3D meshes and volumes are constructed. We extend 2D domain by 1.71 chords in span-wise direction and after that the grid is extended another 1.71 chord from the tip of blade to the front in z-direction Figure 1.c.

In order to conduct CFD analysis, it is necessary to specify initial and boundary conditions for the domain. Fixed value is set for velocity at the far-field boundary in upstream of rotor and atmospheric pressure is specified for the pressure at the downstream boundary. Other air properties at far-field are listed in Table 3. No-slip boundary condition is assumed on the blades and at the top and bottom of the domain symmetry condition is employed. In the case of 3D domain the symmetry boundary is chosen for the back face, where the blades are attached, and slip boundary conditions are chosen at the opposite face. The modified kinematic eddy

viscosity ( ) is chosen as at the inlet and at the walls, based on the reference [14]. Similarly, turbulent viscosity ( ) is computed from Equation 4. and set to be at the inlet.

In the present study, General Grid Interface (GGI), developed by M. Beaudoin and H. Jasak [15], is used to couple stationary and rotating zones. The methodology is to utilize weighted interpolation to compute and transmit the flow variables in the interface region.


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a b

c

Figure 1. Mesh; a. Computational domain; b. Blade; c. Span

Table 3. Far-field air flow properties

Properties Value

Kinematic viscosity ( ) (m2/s)

Density ( ) 1.1774(kg/m3)

5. Vertical Axis wind turbine Simulation In this section first the grid convergence methodology is described and then it is calculated based on results from 2D simulations. Afterward, comparison and differences between 2D and 3D simulations are studied. 5.1 Grid convergence study A consistent manner in reporting the results of a grid convergence was developed by Roache [16]. The Grid Convergence Index (GCI), which is based on Richardson Extrapolation (RE) is considered to be the most acceptable and recommended method employed for the discretization error estimation. In this study, grid convergence analysis has been performed among three meshes with the specifications given in Table 3. The first step is to provide a grid or mesh size ( ) as follows,


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[

∑( )

]

(15)

Here, is the area of the th cell, and is the number of elements used in the computation. In this study, the coefficient of power ( ) is the key variable and has been the objective of the

numerical simulations. Grid refinement factor is given by,

(16)

Based on the experience it should be greater than 1.3 and the grid refinement procedure is carried out systematically. For a case where three meshes are employed, the calculation of the apparent order of the method is based on and,

(17.a)

(17.b)

where and are the grid refinement factors for the first-second and second-third meshes, respectively, and is expressed by the following equation,

( )| | | ( )|

(18)

( ) (

) (18.a)

( ) (18.b)

where , and corresponds to the key variable in the th mesh. It

should be mentioned that for , ( ) . On the other hand, extrapolated values of the objective variable can be calculated using the following equation,

( )

( )

(19)

In order to estimate GCI for the fine mesh, the approximate and extrapolated relative errors are described as follows,

(20)

|

| (20.a)

|

| (21)

where is the safety factor, and for comparison among three meshes [16]. The calculation procedure for three chosen meshes is shown in Table 4.


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Table 4. Discretization error for TSR=2 and TSR=3

Characteristics

Coefficient of Power ( )

TSR=2 TSR=3

Fine Mesh Elements ( ) 2 569 426 2 569 426

Medium Mesh Elements ( ) 1 429 690 1 429 690

Coarse Mesh Elements ( ) 798 919 798 919

1.34 1.34

1.34 1.34

0.3168 0.2809

0.3067 0.2796

0.2784 0.2527

3.52 5.36

0.3224 0.2866

3.1 % 1.9 %

1.7 % 0.48 %

2.2 % 0.64 %

Generally, the overall accuracy of the numerical solution can be improved by refining the grids. As a result, according to table 3, the numerical uncertainty in the fine grid GCI for Power Coefficient ( ) in TSR two and three are reported as 2.2 and 0.64 percent, respectively.

5.2 2D Simulation In this section, it is intent to define the key parameters in wind turbine analysis. One of the most important parameters in wind turbine investigations is Tip Speed Ratio (TSR), which is defined as the ratio of tip speed of the blade to the wind speed,

where is radius of rotor, is angular velocity of turbine, and is wind velocity. Power coefficient ( ) is considered to be another important parameter, which can be derived by

applying Pi-Buckingham theorem, and is given by,

(23)

Here, is power, is the wind density, is the swept area, and is the free stream velocity. In order to calculate the coefficient of power for various tip speed ratios, a fixed angular velocity of 90 rpm as well as different wind velocities are employed. The torque is calculated by means of the developed code in OpenFOAM, defined as the sum of the forces acting on each blade. In order to compute the coefficient of power a one meter span is assumed for the swept area calculation. The vorticities contours, at different blade locations, are depicted in Figure 2 for TSR = 2 and TSR = 3. It is observed that clockwise vortices in top and counter-clock-wise vortices in bottom are generated and travelled downstream by the mainstream flow and gradually dissipated.

Figure 2 also shows the blade-vorticity interaction. The interaction is more obvious for TSR =2

(22)


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compared to TSR = 3. This is because, the blades are encountered higher angle of attacks in lower TSR ratios.

Figure 2. Comparison of vorticity contours for TSR = 2 and TSR=3

TSR = 2 TSR = 3

(a)

(b)

(c)

(d)


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Moreover, increasing the angle of attack advances the separation of flow from the suction side of the blades, left blade in figure 2.a. The shed vorticities from the trailing edge of the blades may collide with the other blades, Figure 2.b to Figure 2.d. It is also demonstrated that the lower the TSR, the wider extension of the wake formed behind the VAWT. Figure 3. illustrates the results with respect to time for TSR = 3. It should be stated that the

value of demonstrates a transient behavior before reaching a periodic pattern.

Figure 3. Cp versus time for TSR = 3

Figure 4. illustrates the validation of the numerical results for three meshes with experimental

data, quantitatively. For all three grids the maximum power coefficients are obtained at . However, simulations on coarse mesh underestimate power coefficients in respect to medium and fine cases. In contrast, results from medium and fine grid are in a perfect fit. All three numerical results overestimate the values. The discrepancies lie on existence of the

rotor hub and blades connections that result in reducing the power coefficient of real turbine by effecting on flow turbulence features around the blades. In addition, tip vortex and other 3D turbulence features influence the aerodynamics forces of the blades.

Figure 4. Comparison of CFD and experimental results for Cp with respect to TSR

Figure 5. and Figure 6. depict the torque variations of one blade versus azimuth angle for two TSR’s on the fine mesh. In this study a blade is rotated counter-clock wise and the angle of attack is changed accordingly. At the time in which the chord and far-field flow are in the same direction ( zero and 180 degrees) rotor is experienced the minimum Torque from the blade. By increasing the azimuth angle from 0, torque keep increasing until it reaches its pick at 90 degrees. Herein, the maximum angle of attack of the blades is occurred. By passing this point the torque is decreased and then the second pick is observed at backwind in 270 degrees. The

0

0.1

0.2

0.3

0.4

0 1 2 3 4

Cp

TSR

Coarse Mesh

Medium Mesh

Fine Mesh

Experimental Data


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results of and almost show the same pattern, except the maximum pick, which is higher in .

Figure 5. Comparison of the torque variation versus azimuthal angle, TSR=2

Figure 6. Comparison of the torque variation versus azimuthal angle, TSR=3

5.3 3D simulation

Herein, 3 dimensional simulation results are represented. Figure 7 illustrates the tip vortex of the blades in x and y directions. It is seen that the tip vorticities are moved outwards the center of rotor. Figure 7.a and Figure 7.b demonstrate the tip vortex, captured in the simulation, vividly affects the flow pattern on and around the blades.

-50

0

50

100

150

200

250

300

0 45 90 135 180 225 270 315 360

Torq

ue

Azimuthal angle

-40

-20

0

20

40

60

80

100

120

0 45 90 135 180 225 270 315 360

Torq

ue

Azimuthal angle

Flow Direction 90

0

270

180

Flow Direction 90

0

270

180


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Figure 8 shows the variation with respect to time for the 3-D mesh at . It should be

mentioned that the value of follows a transient behavior before reaching a periodic

convergence.

(a) (b)

Figure 7. Tip vorticity of the blades. a) X-Vorticity [1/s] b) Y-Vorticity [1/s]

Average power coefficient for 3D dimensional ( ) simulation is less than the 2D

simulation ( ) for the same TSR. Therefore, by taking into account the 3 dimensional

structures in 3D simulation the power coefficient is closer to experimental value. The discrepancy comes back to capturing the 3 dimensional eddies and vorticities on the blade including tip voticities shown in Figure 7.

Figure 8. variation with respect to time for the 3-D mesh at

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5

Cp

Time (Sec)


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Exerted torque on each blade is demonstrated in Figure 9. As it is expected, the 3D simulation results demonstrate the reduction of the turbine efficiency, compared to 2D simulation, Figure 5 shows that the torque curve in the second half of revolution for 3D and 2D solutions are completely different. The main reason lies on the different prediction of separation of the flow from the blades as well as the various blade-vorticity interactions between 2D and 3D simulation.

Conclusion

The flow around the vertical axis wind turbine is simulated for both the 2D and 3D cases. In 2D

simulation, first, a grid convergence study was conducted to ensure the independency of solution

to the grid resolution. The study showed the results for all the three chosen mesh are highly

compatible. Afterwards, the power coefficients at different tip speed ratios were calculated in

order to predict the efficiency of wind turbine at different free-stream condition and angular

velocity speed. In this CFD research the maximum efficiency is obtained at TSR = 2.4. Finally,

the 3D simulation proved that capturing the 3-dimensional turbulence, including the tip vortex,

reduces the efficiency of wind turbine. References

[1] Monteiro J. M. M., Páscoa1 J. C., Brójo F. M R. P., "Simulation of the Aerodynamic Behaviour of a Micro Wind Turbine," in International Conference on Renewable Energies and Power, Valencia, 2009.

[2] Duque E. P.N., Johnson W., VanDam C.P., Cortes R., and Yee K., "Numerical Predictions of Wind Turbine Power and Aerodynamic Loads for the NREL Phase II Combined Experiment Rotor," AIAA, no. 2000-0038, 2000.

[3] Duque E. P.N., Burklund M. D., Johnson W., "Navier-Stokes and Comprehensive Analysis Performance Predictions of the NREL Phase VI Experiment," AIAA, no. 2003-035, 2003.

[4] Tachos N.S., Filios A.E., Margaris D.P. and Kaldellis J.K., "A Computational Aerodynamics Simulation of the NREL Phase II Rotor," The Open Mechanical Engineering Journal, vol. 3, no. 1, pp. 9-16, 2009.

[5] Tongchitpakdee, Ch., Benjanirat, S., and Sankar, L.N., "Numerical Simulation of the Aerodynamics of Horizontal Axis Wind Turbines under Yawed Flow Conditions," Journal of Solar Energy Engineering, vol. 127, no. 4, pp. 464-474, 2005.

[6] McLaren K., Tullis S., Ziada S., "CFD simulation of dynamic thrust and radial forces on a vertical axis wind turbine blade," in the 15th Annual Conference of the CFD Society of

0 45 90 135 180 225 270 315 360

-40

-20

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140

Azimuthal Angle

Torq

ue

Flow Direction 90

0

270

180


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Canada, Toronto, 2007.

[7] Menter F.R., Langtry R.B., Likki S.R., Suze, Y.B., Huang P.G., and Völker S.A., "Correlation - Based Transition Model Using Local Variables- Part I: Model Formulation," Journal of Turbomachinery, vol. 128, no. 3, pp. 413–422, July 2006.

[8] Ajedegba J. O., "Effects of Blade Configuration on Flow Distribution and Power Output of a Zephyr Vertical Axis Wind Turbine," University of Ontario Institute of Technology, Oshawa, Master Thesis 2008.

[9] Camelli F., and Lohner R. L, "Combining the Baldwin Lomax and Smagorinsky Turbulence Models to Calculate Flows with Separation Regions," AIAA, no. 2002-0426, January 2002.

[10] Jiang Zh., Doi Y., and Zhang Sh., "Numerical Investigation on the Flow and Power of Small-Sized Multi-Bladed Straight Darrieus Wind Turbine," Journal of Zhejiang University - Science A, vol. 8, no. 9, pp. 1414-1421, 2007.

[11] Ferreira C. J. S., Bijl H., Bussel G van, Kuik G. van, "Simulating Dynamic Stall in a 2D VAWT: Modeling strategy, verification and validation with Particle Image Velocimetry data," Journal of Physics Conference Series , vol. 75, no. 012023, 2007.

[12] (2011-2012) OpenFOAM Foundation. [Online]. openfoam.org

[13] Auvinen Mikko., Ala-Juusela Juhaveikko., Pedersen Nicholas., Siikonen Timo. (2012) Turbulence Modeling Resource. [Online]. turbmodels.larc.nasa.gov/spalart.html

[14] (2012) Turbulence Modeling Resource [Online]. turbmodels.larc.nasa.gov/spalart.html

[15] Beaudoin Martin., Jasak H., "Development of a Generalized Grid Interface for Turbomachinery simulations with OpenFOAM," , 2008.

[16] Roache, P.J., "Verification and Validation in Computational Science and Engineering," , New Mexico, 1998.

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