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Research Papers: Flows in Complex Systems

# Development of the Dual Vertical Axis Wind Turbine Using Computational Fluid DynamicsOPEN ACCESS

[+] Author and Article Information
Gabriel Naccache

Department of Mechanical
and Industrial Engineering,
Concordia University,
Sir George Williams Campus,
1515 Ste-Catherine Street West,
e-mail: gabriel_naccache@hotmail.com

Marius Paraschivoiu

Department of Mechanical
and Industrial Engineering,
Concordia University,
Sir George Williams Campus,
1515 Ste-Catherine Street West,
e-mail: marius.paraschivoiu@concordia.ca

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 7, 2016; final manuscript received July 16, 2017; published online September 7, 2017. Assoc. Editor: Francine Battaglia.

J. Fluids Eng 139(12), 121105 (Sep 07, 2017) (17 pages) Paper No: FE-16-1355; doi: 10.1115/1.4037490 History: Received June 07, 2016; Revised July 16, 2017

## Abstract

Small vertical axis wind turbines (VAWTs) are good candidates to extract energy from wind in urban areas because they are easy to install, service, and do not generate much noise; however, the efficiency of small turbines is low. Here-in a new turbine, with high efficiency, is proposed. The novel design is based on the classical H-Darrieus VAWT. VAWTs produce the highest power when the blade chord is perpendicular to the incoming wind direction. The basic idea behind the proposed turbine is to extend that said region of maximum power by having the blades continue straight instead of following a circular path. This motion can be performed if the blades turn along two axes; hence, it was named dual vertical axis wind turbine (D-VAWT). The analysis of this new turbine is done through the use of computational fluid dynamics (CFD) with two-dimensional (2D) and three-dimensional (3D) simulations. While 2D is used to validate the methodology, 3D is used to get an accurate estimate of the turbine performance. The analysis of a single blade is performed and the turbine shows that a power coefficient of 0.4 can be achieved, reaching performance levels high enough to compete with the most efficient VAWTs. The D-VAWT is still far from full optimization, but the analysis presented here shows the hidden potential and serves as proof of concept.

## Introduction

The most common way of extracting energy from the wind is through wind turbines. There have been incredible technological advancements and even until now, there is still plenty of room for improvements and development. Wind turbines have gotten more efficient and much bigger since they were first invented, thanks to advancements in aerodynamic, structural, and material design.

Wind turbines of large scale are used for onshore and offshore farms while the small-scale turbines are used for urban applications. So far, the large wind turbines have been favored as they were typically more efficient and produced significantly higher amounts of power. However, recent advancements in small-scale wind turbines have made them more attractive, especially since distributed energy production is quite an attractive concept, as it is a much cheaper solution because power can be produced locally or near where it would be consumed. Therefore, typical problems faced with large-scale turbines such as transportation, transmission cables, and maintenance costs can be avoided.

Wind energy can be harnessed by a variety of turbines, which are classified in terms of their axis of rotation. This includes the horizontal axis wind turbines (HAWT), which have their axes of rotation parallel to the incoming wing, and vertical axis wind turbines (VAWT), which have their axes of rotation perpendicular to the incoming wind. Typically, HAWTs are used more for large-scale energy production, while VAWTs are used for small- and large-scale applications. The two most common types of VAWTs are the Savonius turbine, which is a drag-based turbine, and the Darrieus wind turbine, which is a lift-based turbine. There are also a number of different types of Darrieus VAWTs, the most common being the rotor Darrieus, H-rotor Darrieus (H-Darrieus), and helical Darrieus. All the Darrieus turbines have an airfoil profile for their blade cross section, but differ in their blade shape.

Each of the HAWTs and VAWTs has a number of advantages and disadvantages. A comparison between HAWTs and VAWTs has been investigated by Eriksson et al. [1]. HAWTs are among the most efficient turbines and can be more easily scaled up in size for higher amounts of energy production; however, they are highly dependent on the wind direction, needing to face the wind for optimal performance. A yawing and pitching mechanism can be added to increase their flexibility at the cost of higher complexity and financial cost. The manufacturing of their blades is more expensive than VAWTs since the blade cross-sectional profile varies along the span, while a number of VAWTs have the same blade profile along the span. Because HAWTs' gearboxes, generators, and other mechanical components are at the top of the tower, their maintenance is more difficult, expensive, and dangerous. They are also known to be quite noisy since they operate at high tip speed ratios (TSRs).

Vertical axis wind turbines offer a number of advantages despite some drawbacks addressed in the previous paragraph. They produce much less noise since their operating speeds are lower than HAWTs. Their maintenance is simpler since all components (the gearbox, generator, etc.) are placed on the ground. They typically do not require a yaw control mechanism as their performance is independent of the incoming wind direction. Also, because they have smaller wakes than HAWTs, they can be packed quite closely together, resulting in higher turbine density per unit area. However, VAWTs are generally less efficient than HAWTs. They are structurally more challenging to design since the loads on the blade continuously change throughout the turbine rotation. The constant change in the blade incident angle puts them more at risk of failure due of fatigue loads. The majority of VAWTs also lack the ability to self-start, except for the Savonius type turbines which typically have low power efficiency.

In this paper, a new turbine based on the classical H-rotor Darrieus VAWT is proposed, which utilizes the advantage of Darrieus turbines where the most power is produced when the blade is perpendicular to the incoming wind direction. The analysis of the dual vertical axis wind turbine (D-VAWT) is performed through computational fluid dynamics (CFD) means using the finite volume commercial code ANSYS fluent 14.5 [2].

The following two dimensionless parameters are commonly used to describe the performance and operating condition of a VAWT. The first is the tip speed ratio, which is the ratio of the blade speed at the tip to the incoming wind speed Display Formula

(1)$λ=TSR=ωRU∞$

where $ω$ is the angular velocity of the turbine, R is the radius of the turbine, and $U∞$ is the freestream velocity. The second is the coefficient of power (CP), which is the ratio of extracted power to the available power in the incoming wind Display Formula

(2)$CP=P12ρU∞3A$

where P is the total extracted power from the turbine, $ρ$ is the fluid density, and A is the turbine swept area.

###### Literature Review.

Using a similar idea of extending the maximum power region of a VAWT, Ponta et al. [3,4] analyzed a variable geometry oval-trajectory Darrieus wind turbine using the double-multiple streamtube model, where they showed a very small improvement in efficiency over a classical H-Darrieus VAWT. They also showed that the turbine performance was independent of the number of blades, but highly sensitive to the incoming wind direction.

A new concept turbine was investigated by Kinsey et al. [57], where it consists of a pair of oscillating hydrofoils moving in a sinusoidal path. Kinsey and Dumas presented a computational methodology in Ref. [5] that agreed very well with their experimental data shown in Ref. [7]. They used ANSYS fluent [2] to solve both two-dimensional (2D) and three-dimensional (3D) simulations of the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. After studying different turbulence models to compute the turbine performance, they showed that the one-equation Spalart–Allmaras (SA) model performed very similarly to the two-equation shear-stress transport (SST) k–ω model. To simulate the oscillating motion, nonconformal sliding meshes were used inside of a dynamically moving mesh. Sliding meshes are used for the simulation of the pitching motion, while the dynamic mesh is used for the heaving motion. In Ref. [6], they showed it is possible to limit losses appearing in 3D simulations, such as tip vortices, from their 2D prediction to about 10% with the use of endplates and a blade aspect ratio larger than ten. Gauthier et al. [8] investigated the blockage effect on the same oscillating-foil hydrokinetic turbine using the finite volume code cd-adapco star ccm+ with the overset mesh technique. They showed that the increase in blockage effect and extracted power is linearly related for up to 40% blockage as well as providing a correlation factor to account for that said blockage effect.

Delafin et al. [9] compared the performance of a rotor Darrieus turbine using 3D CFD simulations of the SST k–ω model with other aerodynamic models, such as the double-multiple streamtube and vortex models. They showed that the 3D simulations accurately predicted the turbine behavior, while the aerodynamic models overpredicted the power for all TSR values.

Gosselin et al. [10] studied the effects of various parameters for a three bladed H-Darrieus turbine. They showed that for a turbine operating at high TSR values, the choice of turbulence model had little effect on the turbine behavior predictions, while for low TSR values, significant differences in behavior were found for different turbulence models. The result for the CP can be seen in Fig. 1 for a single blade analysis. They also showed that the SA strain/vorticity-based model produced ten times less turbulent viscosity than the SST k–ω and transition SST models. Using the SST k–ω with $y+$ ∼ 1, they compared the turbine performance in 2D and 3D with blade aspect ratios of 7 and 15. The power obtained in 3D for an aspect ratio of 7 and 15 are 41.8% and 69% of the 2D power, respectively. This shows how much 2D simulations overestimate the turbine performance and that increasing the aspect ratio increases the turbine performance as the aerodynamic losses such as wingtip vortices affect a smaller portion of the blade. Beves and Barber [11] investigated the effects of dimples on a wing with an endplate. They performed experiments on a wing, of an inverted Tyrrell026 airfoil profile with an endplate, with and without the presence of dimples. They showed that the presence of the dimples increased the strength of the wingtip vortex by 10% and more importantly reduced the region of high turbulence in the wake by 50%. They suggest that the net improvement of the flow in the wake is a result of reduced drag and improved aerodynamic efficiency of the dimpled wing.

Mohamed et al. [12] investigated 25 different airfoil profiles, using the SST k–ω model in 2D, for an H-Darrieus configuration. The best airfoil boosted the turbine performance by 10% when compared to the NACA 0018, which is a commonly used airfoil profile and is often used as a baseline for comparison. Yamazaki and Arakawa [13] showed a performance improvement in VAWTs through the shape optimization of airfoil profiles by maximizing certain characteristics of the airfoil. The shape optimization was performed using a Kriging response surface approach; then 2D simulations were performed on the optimized shapes to quantify the improvement from the profile optimization. Xiao et al. [14], using the realizable k–ε model in 2D, studied the impact of fixed and oscillating flaps and showed a performance improvement of 28%.

Lim et al. [15] and Chong et al. [16] performed experimental tests and 2D simulations, using the SST k–ω model, to optimize an H-Darrieus VAWT using an omni-direction-guide-vane. They showed it improved the self-starting capability of the turbine by 182% and its performance by 58% from the original configuration.

Balduzzi et al. [17], after compiling a list of others' work and performing some of their own investigation, they recommend the SST k–ω model, $y+$ ∼ 1, and most importantly to have a convergence criterion for the torque variation from to cycle to cycle of less than 0.1%, instead of the commonly used value of 1%. They found that a variation of 1% can continue for up to ten cycles, leading to a large overestimation from the actual torque value.

McNaughton et al. [18] compared the standard form of the SST k–ω with the SST k–ω with a correction for low Reynolds number effects. They tested the models, in 2D with a $y+$ < 1, for a turbine operating at Reynolds number of 150,000. They showed an improvement in performance prediction with the low Reynolds correction model. Lanzafame et al. [19] compared, in 2D with a $y+$ < 1, the two-equation SST k–ω with the four-equation transition SST model. The transition SST showed much better agreement with experimental results than the SST k–ω; however, the transition SST model is more computationally expensive and required a series of tests to calibrate the local correlation parameters with the experimental values in order to get accurate results.

Though 3D simulations are well known to provide more realistic performance as it is possible to capture secondary flows, wingtip vortices, and aerodynamic losses from structural components such as the supporting arms and central shaft, the computational power, and time needed are significantly higher than that of 2Ds. For that reason, few simulations in literature are done in 3D. Siddiqui et al. [20] compared a 2D Darrieus turbine performance's predictions, using the realizable k–ε model, with 3D by simulating the support arm and central shaft. They found that 2D can overestimate the actual turbine performance by up to 32%. Castelli et al. [21,22] first performed full 3D flow simulations, using the Realizable k–ε model, to find the loads on the blades, followed by a structural analysis using a finite element method code to find the stresses and deformation on the blades. Howell et al. [23] performed 2D and 3D simulations at low Reynolds number and found that 2D largely overestimated the extracted power, while 3D showed reasonable agreement with their experimental results. Rossetti and Pavesi [24] investigated the self-starting capabilities of H-Darrieus VAWTs using blade element momentum, 2D and 3D methods at TSR = 1. They found that effects only captured in the 3D simulations, such as secondary flow and tip vortices, had a positive effect on start-up.

Ferreira et al. [25] compared the simulation results in 2D, for turbine cases where dynamic stall occurred, with experimental results from particle image velocimetry. They found the model that agreed the most with their experimental results was the detached eddy simulation model, followed by the large eddy simulation model, and finally the two URANS models, the SA and the k–ε models. These results were expected as the Eddy models are known to be more accurate but their drawbacks are higher computational costs.

Salim and Cheah [26] investigated the $y+$ strategy for turbulent flow for a few simple cases. They suggested that resolving the log-law layer was sufficiently accurate (30 < $y+$ < 60) without the need to fully resolve the viscous sublayer ($y+$ < 5) and to avoid resolving the buffer region (5 < $y+$ < 30) as neither wall functions nor near-wall modeling accounted for it accurately.

Almohammadi et al. [27] investigated three mesh independency techniques: the general Richardson extrapolation, grid convergence index, and the fitting method. The study was performed in 2D for an H-Darrieus VAWT using the renormalization group k–ε and transition SST models.

Lee and Su [28] performed experiments on airfoils undergoing pure heaving, pitching, and combined motions at Reynolds number of 36,000 to better understand the behavior of unsteady boundary layers (BLs) on airfoils. With accurate surface pressure measurements, smoke-wire flow visualization and typical data of lift, drag and moment, CFD validation can be performed with a high level of accuracy because of the broad spectrum of data available for comparison. The airfoil performance was captured during stall and hysteresis as well, providing a complete range for comparison purposes.

###### Dual Vertical Axis Wind Turbine Concept.

The new turbine concept, with a similar blade shape as the H-Darrieus, would utilize the location of maximum power and have it extended as much as possible in an attempt to increase the overall aerodynamic efficiency of VAWTs. The position of maximum power can be seen in Fig. 2 for the upwind blade at θ = 90 deg, and the region of maximum power generation would be when the blade is traveling perpendicular to the incoming wind direction. Since a larger portion of the turbine cycle is spent where the blade is producing the most power possible, this new design should improve the overall efficiency of VAWTs. This motion can be achieved if the blades turn around two axes, hence the name dual axis in D-VAWT. Figure 3 shows a top view of the turbine and a 3D CAD model to give a better idea of the geometry and mechanism of the D-VAWT. It should be noted that the mechanism will not be present in any simulations. It is shown here for illustration purposes only.

###### Geometry.

The idea of the D-VAWT lies in extending the regions where the most power is extracted from a conventional H-Darrieus type VAWT. The D-VAWT blade path is shown in Fig. 3(a), where the axes spacing, L, is the distance between the two axes of rotation and R is the radius of rotation. The original dimensions of the investigated turbine are presented in Table 1.

For the current analysis, the ratio of L/R is set as 4. Further investigation will be needed to find the optimal ratio of L/R. For the current analysis, a D-VAWT with only a single blade is investigated as the mesh and motion methodology needed for more than one blade is more complex and will greatly increase the simulation time, while the current purpose is to first investigate the methodology and feasibility of this new design. The selected airfoil profile is a NACA 0018 for its high lift characteristics. The mounting point of the blade is at 1/3C away from the leading edge; however, this will also need investigation to find whether it is the optimal mounting point.

###### Parameters for Dual Vertical Axis Wind Turbine.

Based on their original definitions, some of the previously mentioned parameters are modified to be used appropriately for a D-VAWT and others are newly defined. Since the equation for the TSR is the same for VAWTs and D-VAWTs, it will not be presented again here.

###### Swept Area.

The swept area is defined as the projected area that is normal to the incoming wind. For the D-VAWT case, the swept area becomes a function of the incoming wind direction Display Formula

(3)

where h is the height of the turbine, which is equal to unity for a 2D analysis, and $φ$ is the incoming incident wind angle, where $φ$ = 0 deg is for the case shown in Fig. 3(a) with the flow normal to the longitudinal side of the turbine or the line connecting the two axes (L).

###### Coefficient of Power.

The coefficient of power is defined as the ratio of extracted power to the available power in the incoming wind. The typical method to calculate the CP for a VAWT is based on the torque produced by the blade. However, since the D-VAWT blade follows a noncircular path, it is not appropriate to use torque in regions where the blade is not rotating. The CP is defined below as: Display Formula

(4)

where TRotating is the sum of the torque produced when the blade is rotating, FTranslating is the sum of the tangential force produced when the blade is translating for a single cycle, CP,Torque is the coefficient of power for the rotating sections, and CP,Force is the coefficient of power for the translational sections.

In the regions where the blade is rotating, the CP,Torque is used since it is the torque that produces power in those regions. In the regions where the blade moves in a straight path, the CP,Force is used as the torque would not be appropriate to use in those sections since part of the torque would be seen as stresses on the mechanism and would not help produce power. An example of the resultant CP curve obtained from the combination of the two methods can be seen in Fig. 4.

## Methodology Validation in Two-Dimensional

An approximate calculation of the CP in the upstream translational region predicted a theoretical value of 1.41. The steps for the theoretical calculation are shown in Ref. [31]. This calculation was based on the airfoil data for the coefficient of lift and drag found in Ref. [32] at the angle of attack of 12.5 deg and blade Reynolds number of 500,000, which are the conditions the D-VAWT blade sees when it moves straight. This range of CP is extremely high for a wind turbine considering that the highest average CP per cycle that conventional VAWTs can draw is between 0.2 and 0.4. For this reason, extending the position of maximum should help increase the overall aerodynamic efficiency of Darrieus type VAWTs. It should be noted that the predicted theoretical value should only be compared with that of 2D simulations, since the values for the coefficient of lift and drag are that of an “infinitely” long airfoil.

###### Numerical Setup.

The analysis of the D-VAWT is performed through CFD means using the finite volume commercial code ANSYS fluent 14.5 [2] to solve the URANS equations. The value for the density of air, $ρ$, is set to and the dynamic viscosity, $μ$, is . The semi-implicit method for the pressure linked equations (SIMPLE) algorithm is selected for the pressure–velocity coupling. For the spatial discretization, second-order schemes are used for the pressure, momentum, and turbulent viscosity calculations. As for the transient formulation, a first-order implicit scheme is required due to the limitations imposed by fluent while the dynamic mesh capabilities are active. The dynamic mesh is needed to simulate the unconventional blade path of the D-VAWT. All simulations are performed with an absolute convergence criterion of 10 − 3 for the continuity, velocity components, and all turbulent properties. Though a convergence criterion of 10−5 was tested, the results were near identical with the advantage of having lower computational time. The size of the time step is Δt = 0.5585 ms or about one thousandth of a period (T/1136). The steps taken to select the time step are outlined next.

First, it was decided to have 500 time steps if the blade was to do one full rotation in a circular path, or equivalently, the blade moves about 360 deg/(500 time steps) = 0.72 deg per time step. Based on the angular velocity of 22.5 rad/s (obtained from TSR = 4.5 and U = 4 m/s), the time step is $Δt=(2π/(22.5×500))=0.5585 ms$. With this value, the number of time steps it takes to complete one side of the translational region equals to $(L/(Vblade×Δt)=318$ time steps and the total number of time steps to complete a full cycle of a D-VAWT is $500+318×2=1136$.

For the boundary conditions, shown in Fig. 5, uniform and constant velocity inlets are defined for the left, top, and bottom boundaries with incoming wind speeds of 4 m/s, which is a typical wind speed value for urban applications. The right boundary is defined with a pressure outlet condition. The incoming turbulent boundary conditions parameters are defined with a turbulent intensity of 5% and a turbulent viscosity ratio $(νt/ν)=5$, where the kinematic viscosity, ν, of air is $1.4607×10−5 m2/s$. As for the blade boundary condition, a no-slip condition is applied.

###### Domain Size Study.

The domain size and motion prescription study were performed using the one-equation SA with strain/vorticity-based production turbulence model with a $y+$ ∼ 30 as it was seen from literature that for high TSR values, the choice of model should have little effect on the results. The purpose of this test was to determine the smallest possible domain size that would not affect the results. The mesh used in this study was finer than needed, which was only realized after the mesh convergence study. The domain contained 135,000 elements, mainly focused around the airfoil. The summary of the results and domain sizes tested are shown in Table 2, where it can be seen that changing the domain from 150C to 250C resulted in a change of less than 2%. It was then decided to use a domain of 150C for the rest of the simulations as it provided enough accuracy for the current objectives. For all 2D simulations presented in this paper, all elements across any interface are matched in size to reduce interpolation errors.

###### Motion Prescription.

In this section, three possible methods to prescribe the motion of the blades are presented, followed by their results comparison. For all motion types of the D-VAWT blade, the motion is prescribed in fluent through the use of user-defined functions. A domain size of 150C is used for all three cases based on the domain size investigation.

###### Motion Type 1.

For this motion type, the domain is composed of three subdomains: a rotating domain, a translating domain, and a static deforming domain. In Fig. 6, the three domains, their interfaces, and the motion of each are presented. A sliding mesh approach is employed for the motion of the rotational domain to allow the blade to rotate at both ends of the path. The translating domain is needed to simulate the translational motion by moving up and down perpendicularly to the flow. The outer static deforming domain acts as a buffer region where its elements would deform to account for the motion of the translating domain. The mesh used for this simulation can be seen in Fig. 7, where higher refinement is seen around and behind the tail of the blade. It is the same mesh used for the domain size study.

###### Motion Type 2.

The second type of motion is very similar to type 1, except the previously named static deforming domain moves along with the translating domain as one domain with no relative motion in between them. In this type of motion, there is no element deformation. The representation of the motion is shown in Fig. 8. The same domain and mesh as for type 1 motion are used as to only have the type of motion different between the two cases and directly compare them. Special care is required when applying the boundary condition for the inlets and outlet, where they had to be specified to be independent of the mesh motion.

###### Motion Type 3.

The last type of motion is presented here, and is significantly different from the previous two. The domain is composed of three subdomains: a dynamic domain, a deforming domain, and a static domain. The representation of the domain motion is shown in Fig. 9. In this case, the dynamic domain is the smallest of the domains and contains inside it the blade that moves in the D-VAWT trajectory, both rotating and translating accordingly. As the dynamic mesh moves, the deforming domain will have its elements both deformed and remeshed during the motion of the blade. The static domain remains unchanged during the simulation in this case. The starting mesh used for this simulation can be seen in Figs. 10 and 11.

###### Results and Discussion.

The results for the average CP are shown in Table 3. The small difference in type 3 could be due to the fact that the starting mesh was not identical to the type 1 and 2 meshes. Another reason for the possible difference is that during the simulation, there is constant remeshing occurring, so the final mesh would be different than the one shown at the beginning of the simulation.

Based on the CP, the results for the three motion types still came to be very close to each other. The advantage of type 1 and 2 motions is shorter simulation time when compared to type 3 since no remeshing occurs during the simulation; however, both type 1 and type 2 are limited to only having a single blade in the simulation. With type 3 motion, it is possible to have as many blades as needed in the simulation. However, the simulation time is longer and due to the fact that the mesh deforms and remeshes throughout the simulation, additional errors are introduced. For the remainder of the 2D investigation, type 1 is used, where possible, for a single blade analysis as it is among the simplest and avoids any potential problems or errors from having the domain boundaries move. Type 2 will be used for 3D simulations to maintain the speed of the calculation but more importantly is to avoid certain complication with fluent in 3D when using type 1 motion. Type 3 will only be used in future work for a multiblade turbine analysis.

Future studies can also consider using a similar approach proposed by Remaki et al. [33]. Remaki proposes a new algorithm, the virtual multiple rotating frame, for flow simulations with rotating machines. The proposed algorithm, based on the multiple rotating frame method, allows the use of a single mesh for the entire domain and defines a virtual interface between the rotating and stationary domain at the solver level. This method simplifies the tasks performed at the CAD level as there is no need to create geometrical interfaces and decompose the domain into multiple mesh zones. This method can be especially beneficial for cases with complex parts connection. They performed 2D fan and pump and 3D wind turbine test cases where they showed that the new algorithm gave near identical answers to the one provided by the standard multiple rotating frame algorithm with the benefit of simplicity of application and reduced overall computational cost.

###### Mesh Convergence Study Using Shear-Stress Transport k–ω Model With $y+$ ∼ 1.

Previously, a mesh of 135,000 elements was used for the initial testing of the turbine concept. This mesh was based on the finest mesh of other's simulations with the same Reynolds number of 500,000 [5]. However, at this point, it was necessary to perform a grid convergence study to determine the necessity of using either a coarser more efficient mesh or perhaps a more refined mesh to more accurately estimate the D-VAWT performance. In this section, a mesh independence study is performed to determine the most efficient mesh. Since it is necessary to perform 3D simulations, having the coarsest mesh possible in 2D will significantly reduce the element count in 3D and improve computational time. For this study, the SST k–ω turbulence model with low Reynolds number correction is used with a $y+$ ∼ 1 as it is known to have higher accuracy, especially since the entire boundary layer is resolved instead of being approximated with a wall function. In Sec. 2.5, a turbulence model comparison is done with the selected mesh from this study to determine the effect of turbulence model choice on the results.

###### Meshes.

Based on the previous investigation, type 1 motion is used with the domain sizes of 150C, 75C, and 2D for the static, translating, and rotating domains, respectively. Three meshes are employed for this study. An initial mesh of 45,000 elements, named mesh 1, is used for the coarsest case. Instead of doing two levels of mesh refinement with a factor of 1.35, as recommend by Roache [34] based on the grid convergence index method developed from the theory of the generalized Richardson extrapolation, the intermediate mesh was skipped. The finest mesh is refined with a factor of 2 or ∼ 1.42. However, two meshes are tested (mesh 2 and mesh 3) at this refinement factor of 2 relative to the starting mesh (mesh 1), with the difference being mesh 2 excludes the refinement of the BL elements. This approach makes it possible to determine the refinement effect from the mesh outside the BL and then the effect of the BL refinement separately. The mesh details are summarized in Table 4 and the meshes used can be seen in Figs. 1214. One interesting technique used for the generation of the inflation layer can be seen in the trailing edge view of Fig. 14, where one can see that the trailing edge is rounded and not included in the inflation layer. Typically, the inflation layer encapsulates the entire airfoil profile; however, the current technique results in higher overall element quality for the BL elements as it avoids stretching the BL elements to have them meet at the tailing edge. High element quality, specifically high element orthogonality, is of the utmost importance for correct simulation of the BL flow.

###### Mesh Convergence Study Results.

All simulations performed in 2D are now run for 15 cycles. Even though at the tenth cycle, the cycle-to-cycle convergence was only less than 1%, the CP was still dropping. However, after the 15th cycle, the CP only changed by about 0.01%, thus reaching satisfactory convergence. The convergence of the average CP per cycle for mesh 1 can be seen in Fig. 15. The same trend is noticed for all the other cases. Therefore for 2D simulations, 15 cycles are sufficient and the last three cycles are averaged to be used as point of comparison.

Table 5 shows the results from the mesh convergence study, where the percent difference with respect to the original mesh (mesh 1) is presented. Even comparing the coarsest mesh (mesh 1) with the finest (mesh 3) with the finer time step, the difference between them is less than 1%. Based on this result, the coarsest mesh and time step size are determined to be adequate for further investigation in 2D as well as 3D since the cheapest mesh possible is needed for 3D purposes to reduce its high computational cost.

###### Turbulence Model Study.

In this section, the D-VAWT is simulated using different turbulence models to investigate the accuracy of the results and justify the selection of the turbulence model. Three turbulence models are considered in four cases: the one-equation SA strain/vorticity-based production model with a $y+$ ∼ 1 and ∼30, the two-equation SST k–ω model with a $y+$ ∼ 1, and the four-equation transition SST model with a $y+$ ∼ 1. It should be noted that the SA strain/vorticity-based production model is referred to as SA strain in all graphs and tables in this paper and as SA strain/vorticity in the text. The domain/mesh used in this study is the coarsest mesh (mesh 1) with Δt = T/1136. Table 6 shows the CPU times for all cases performed in the turbulence model study. Also, note that these simulations were performed on a local computer with 3.50 GHz processing frequency and seven out of eight cores fully dedicated to the simulation. The CPU times in Table 6 are calculated by multiplying the simulation times by the number of cores. As expected, the fastest simulation is the SA strain/vorticity model with $y+$ ∼ 30 since it is the lowest equation model with a coarser mesh to achieve a $y+$ ∼ 30, while the most expensive is the transition SST model since it is the highest equation model with a more refined mesh to achieve a $y+$ ∼ 1. It should be noted that to help reduce the transition SST simulation time, the number of Newton subiterations was limited to 20 for the first five cycles and then that limit was removed to allow the residuals to converge; however, even with the limited subiterations in first five cycles, the transition SST model took nearly 60% more time than the SST k–ω model. The most interesting comparison is between the SST k–ω model and the SA strain/vorticity model with $y+$ ∼ 1, where the SST k–ω model was only slightly slower for the last 1000 time steps, but slightly faster for the complete simulation time.

The results for the average CP for the different cases are shown in Table 7 and Fig. 16. The transition SST model predicts the highest CP value, while the SA strain/vorticity for $y+$ ∼ 30 predicts the lowest. The biggest difference is noticed between the cases using the mesh of $y+$ ∼ 1 and $y+$ ∼ 30. Though the trends shown in Fig. 16 for all cases are similar, the SA strain/vorticity with $y+$ ∼ 30 underpredicts the upstream side of the translational region and the following rotating region or at the normalized times between 0.22 and 0.72. It can be seen that the SA strain/vorticity model at $y+$ ∼ 30 consistently underpredicts the CP value in that region, while for the other models at $y+$ ∼ 1 show better consistency in value throughout the cycle. The results obtained here are also supported by Gosselin et al. [10], where they showed at high TSR, the choice of turbulence model with the same $y+$ has little effect on the results.

Comparing the average value of CP in the upstream portion of the translational region, the SST k–ω model predicted a value of 1.38, while the theoretical results predicted a value of 1.41 [31]. This is only a difference of about 2%, which further supports the analysis and the selected methodology.

## Airfoil Validation With Experimental Results

Using the presented methodology, an experimental case is simulated and the results are compared with their respective experimental values. The case is that of a 2D simulation of a static airfoil. Using the same methodology to generate the mesh, a NACA 0018 airfoil is simulated with an identical numerical setup as the D-VAWT. The only difference from the D-VAWT case is the inlet boundary condition values for the turbulent intensity, which were matched with the experimental setup to ensure a proper comparison. The Reynolds number is 500,000, which is in the same range as what the D-VAWT's blade experiences in the upstream translational region. The results are validated with the experimental case performed by Timmer [35].

The purpose of these simulations is to further validate the choice of turbulence model and accuracy of the CFD setup with existing experimental results. The same four cases from the turbulence model study are repeated here, which are the SA strain/vorticity model with a $y+$ ∼ 30 and $y+$ ∼ 1, the SST k–ω and transition SST models with a $y+$ ∼ 1.

###### Results.

Figures 1720 show the results for the coefficient of lift (CL) and coefficient of drag (CD) for all simulations cases as well as the experimental values. Figure 17 shows that the transition SST model predicts the most accurate results as it captures the experimental curve the closest. The SST k–ω model is the second most accurate model tested, with the advantage of being computational cheaper than the transition SST. The SA strain/vorticity model with $y+$ ∼ 1 overpredicts drag, as seen in Fig. 18, while the SA strain/vorticity model with $y+$ ∼ 30 greatly overpredicts drag and shows premature stall behavior that is not seen with any of the other models at $y+$ ∼ 1. This further explains the underprediction of the SA strain/vorticity model with $y+$ ∼ 30 in the results of Fig. 16, where the blade experienced higher drag and lower lift than it should have. This finding further supports the results of the turbulence model study.

The average error for the ratio of CL to CD for the range of angle of attacks that the D-VAWT experiences (from 0 deg to 12.5 deg) at TSR = 4.5 for the cases of SA strain/vorticity model using a $y+$ ∼ 30 and $y+$ ∼ 1, the SST k–ω and transition SST models with a $y+$ ∼ 1 are 44.9%, 18.6%, 2% and 0.6%, respectively. These values confirm that the transition SST model provides the most accurate solution, closely followed by the SST k–ω model, while the SA strain/vorticity model provides the least accurate results for both $y+$ cases. The ratio of CL to CD is directly related to the predicted output power of the turbine. Seeing that the SA strain/vorticity with $y+$ ∼ 30 consistently underpredicts that said ratio, this explains the underprediction in power for the case of the D-VAWT.

Based on the turbulence model study and the validation with experimental results, the SST k–ω model at $y+$ ∼ 1 and the SA strain/vorticity model at $y+$ ∼ 30 provide an upper and lower bound estimates for the CP, where the lower bound will represent a more conservative estimate for the D-VAWT's performance. The upper and lower bounds can be clearly seen in Fig. 19. The reason the SST k–ω model was chosen over the other two models at $y+$ ∼ 1 is that it is faster than the transition SST model and it is more accurate at resolving the BL flow in the near-wall region than the SA strain/vorticity model. The value predicted from the SST k–ω model is expected to represent actuality more closely than from the lower bound estimate because it is a higher equation model capable of accurately resolving the flow in the near-wall region. For airfoil and turbine simulations where there is no shedding of vortices, accurately capturing the boundary layer flow is of the utmost importance, which is exactly what the SST k–ω model is good at resolving.

## Three-Dimensional Investigation of Dual Vertical Axis Wind Turbine

This section presents 3D simulations of the D-VAWT at TSR = 4.5 with the same geometrical parameters presented in Table 1 and the methodology presented in Sec. 2. The blade aspect ratio (AR = span/chord) of 5 and 15 are investigated using the SA strain/vorticity ($y+$ ∼ 30) and SST k–ω ($y+$ ∼ 1) turbulence models to provide an upper and lower bound estimates of the CP. Note that the blade aspect ratio will be referred to as aspect ratio for the remainder of the paper.

###### Domain.

The domain used for the simulation with AR = 5 is shown in Fig. 21. Further details are included in Table 8 for both domains of AR = 5 and 15. It should be noted that only half of the blade is simulated by making use of a symmetry plane boundary condition. This essentially allows the domain to be cut by half, which significantly reduces the computational cost.

Type 2 motion is used for all 3D simulations since fluent encountered many problems with the deforming mesh of type 1 motion. Based on the comparison of motion types, the results should be identical for both motions. The coordinate system is shown in Fig. 21, where the incoming wind is coming in the positive X-direction.

###### Mesh.

Two turbulence models are tested, the SA strain/vorticity and SST k–ω, with different $y+$ strategies for each. There are a total of four cases performed, comprising of two aspect ratios with two turbulence models. The details of the mesh used are shown in Table 9, which are based on the 2D mesh study. The mesh for the AR = 5 with the SA strain/vorticity model is shown in Figs. 2225, while the one used for AR = 5 with the SST k–ω model is shown in Figs. 2628. The main difference between the meshes of the two turbulence models is the first layer height to obtain the desired values, while the main difference between the meshes of the two different aspect ratio is the number of elements in the span wise direction. Elements across the rotating interface are matched in size to reduce interpolation errors across the interface. The elements in the refinement region around the blade are hexahedron elements formed from quadrilateral elements that have been swept in the spanwise direction. It should be noted that the elements swept have a bias toward the blade tip, meaning the elements near the blade tip are smaller than the ones near the blade center (at the symmetry plane). The purpose of the bias is to better capture the flow and pressure drop near the blade end from the tip vortex. All the elements outside of the refinement regions are tetrahedrons.

###### Numerical Setup for Three-Dimensional Simulations.

The SIMPLE algorithm is selected for the pressure–velocity coupling. For the spatial discretization, second-order schemes are used for the pressure, momentum, and turbulent viscosity calculations. First-order implicit is still used for the transient formulation due to the use of the dynamic mesh capabilities. All simulations are performed with an absolute convergence criterion of 10−3 for the continuity, velocity components, and all turbulent properties. The time step size is Δt = 0.5585 ms or about one thousandth of a period (T/1136). The SA strain/vorticity model and the SST k–ω model with low Reynolds number correction are used for turbulence modeling.

The boundary conditions are shown in Fig. 29, where constant and uniform velocity inlets are defined everywhere, except for the bottom plane, which is defined with a symmetry condition and the right most boundary, which is defined as a pressure outlet. The velocity of incoming wind is 4 m/s with a turbulent intensity of 1% and a turbulent viscosity ratio of . The turbulent intensity is reduced from 5% to 1% from the 2D to 3D simulations to speed up the convergence of the simulation. This change would have marginal effect on the results as both turbulence intensities were tested and compared in 2D. As for the blade boundary condition, a no-slip condition is applied.

###### Results.

All 3D simulations of the D-VAWT at TSR = 4.5 were run for 10 and 12 cycles for the aspect ratio of 5 and for 15, respectively. The CP cycle convergence can be seen in Fig. 30, where it can be noticed that the simulations with AR = 5 converge faster than the ones with AR = 15. The cycle-to-cycle convergence criterion used for all 3D simulations is 0.2%, since matching that of 2D's of 0.01% was difficult to reach in 3D. Table 10 presents the summary of the averaged CP in the last three cycles and the ratio of 3D to 2D CP values for each case, respectively. It can be seen that for the AR of 5, only about 40–44% of the power in 2D is captured, while for an AR of 15, this increases to about 70%. The 3D to 2D ratio value for the AR = 15 case is also supported by Gosselin et al. [10], where a ratio of 69% is obtained for the same AR. It is well known that as the AR of airfoil blades is increased, the 3D performance approaches that of 2D. For a short AR, the 3D losses, especially because of the formation of a wingtip vortex, dominate and a larger portion of the blade sees a large decrease in performance. The formation of the wingtip vortex leads to a decrease in performance as a result of the pressure drop on the blade surface when approaching the blade tip. The wingtip vortex allows the flow to “leak” over the blade tip and reduces the built up pressure from the airfoil profile, which in turn reduces the lift of the blade portion affected by it.

The instantaneous CP curves are shown in Fig. 31 for all 3D cases. For the same AR, there is a gap in performance between the SA strain/vorticity model and SST k–ω model, where the SA strain/vorticity model with $y+$ ∼ 30 still underpredicts the CP values, most noticeably in the upstream translational region and the following rotational region. The CP trends between the blade of AR of 5 and 15 are similar throughout most of the cycle, except for the fact that the blade of AR of 15 produces more power as expected. An interesting behavior between the two is noticed in the downstream portion of blade path, especially in the translational region. For the AR of 15, CP behavior is very similar to that of the 2D one, but for the AR of 5, looking at the normalized time from 0.72, the CP is initially slightly higher than for the AR of 15, but after t/T = 0.88, a dip in power occurs that is not seen in either the 2D or 3D with AR = 15 cases.

In Fig. 32, the normalized velocity deficit () is shown at a cross-sectional plane that is half a chord away from the symmetry plane as it is preferable not to visualize the flow on the symmetry plane. The normalized velocity deficit shows how much velocity is either reduced from losses or extracted by the turbine. Figures 32(a) and 32(b) are for the SA strain/vorticity model of AR = 5 and 15, respectively. It should be noted that the blade tip effect is stronger for the case of AR = 5 at this plane because this plane is only two chords away from the tip while for the AR = 15, it is seven chords away. A plane that is two chords away from the blade tip was visualized for the case of AR = 15; it was then noticed that the behavior of the flow highly resembled that of the AR = 5 shown here. This means that at a distance of two chords from the blade tip for either AR = 5 or 15 cases, the performance of the blade is highly affected and reduced by the presence of the wingtip vortex, while at seven chords away for AR = 15, this effect dissipates and the performance approaches that of 2D. To better grasp the effect of AR on the blade performance, Fig. 33 shows the static pressure contour on the blade surface during the upstream translational region for the SST k–ω model for both AR = 5 and 15. One can see the pressure drop starting from around one chord's length from the blade tip and because about the same length of the blade is affected for both ARs, it means that the blade with AR = 5 has a larger portion affected.

As shown in Fig. 32, after the upstream pass of the blade, about 35–40% of the freestream velocity is extracted or lost, while after the downstream pass, the freestream velocity drops to 75–80% of its initial value. This means that with a single turbine, it is possible to extract about 75% of the incoming wind's energy from the two passes of the blade in the translational region. This can also be supported from the CP values seen in those regions. However, with such a strong velocity deficit, the wake of this turbine will be very strong and will take some time to recover and return to its original freestream velocity. This means that if another turbine were to be placed downstream of it, it would have to be significantly far away from the first.

Another detail to note from Fig. 32 is that for the same AR in Figs. 32(a) and 32(c), the flow for the SA strain/vorticity and the SST k–ω models have similar wake structure, but for the SST k–ω model, the velocity deficit in the wake is higher, which can be explained by the higher power extraction seen in the CP Curve. This can also be seen again in Figs. 32(b) and 32(d). The difference between the models can be seen in the size of the BL in Fig. 34, where the turbulent viscosity ratio ($νt/ν$) is shown at the same cross-sectional plane as before, but at the normalized time t/T = 0.33, which is approximately midway in the translational region. In Fig. 34, the AR does not seem to affect the size of the BL, but the choice of $y+$ between the two models does indeed affect it, resulting in a BL for the $y+$ ∼ 30 to be almost twice as thick as the one for $y+$ ∼ 1. The recirculation zone on the suction side to the airfoil (right side of blades in Fig. 34) is also much larger for the cases with $y+$ ∼ 30, leading to lower lift and higher drag as seen before in the results of the experimental validation case of a static airfoil, which explains the lower extracted power in this region.

Finally, Fig. 35 compares the 2D and 3D results for all cases simulated with the SST k–ω models. It can be seen that as the AR increases, the power curve approaches the 2D results in value and behavior, which is the expected behavior since 2D is considered to be a blade that is infinitely long. This further supports the analysis and the methodology transition from 2D to 3D.

###### Discussion of Dual Vertical Axis Wind Turbine Performance.

Considering the D-VAWT performance for an AR of 15, the lower and upper bounds of the CP are 0.34 and 0.404, respectively. The actual value would be closer to the upper bound as the simulations with $y+$ ∼ 30 (the lower bound) do not capture the BL flow as accurately as the ones with $y+$ ∼ 1. This finding is supported from the results of the turbulence model study and the experimental airfoil case study. Although 3D simulations are much more accurate than 2D simulations, they will still have some level of overprediction of the turbine performance. One should remember that this study is based on a numerical approach, which represents an idealized system where a number of losses are not taken into account. The unconsidered losses include the generator and mechanical losses as well as aerodynamic losses from the lack of simulating the structural components of the turbine. Nonetheless, the predicted CP values for a straight blade Darrieus type turbine is high, considering this range of CP is usually seen for the more efficient rotor Darrieus turbines and HAWTs. However, the D-VAWT is still far from being fully optimized. A simple improvement would be to use endplates for the blade, which would reduce the 3D aerodynamic losses and the CP would further approach the 2D approximation. Based on the results of this study, the D-VAWT concept of extending the region of maximum power production did improve the overall turbine performance. Most performance improvement studies on VAWTs are conducted on optimizing blade profile or other geometric parameters, while the D-VAWT concept is among the few studies that investigated a new and unconventional blade path for a turbine.

Future work will continue the investigation of the D-VAWT to further understand the flow and turbine behavior for different geometrical parameters and operating conditions. Note that the D-VAWT aerodynamic efficiency will strongly depend on the choice of blade geometry (chord and twist). The optimal blade shape for D-VAWT applications can be found via an inverse design, similar to those performed by Rosenberg and Sharma [36] and Lee [37], to further improve efficiency. Future studies of the D-VAWT can be performed in 2D only as the trends and behaviors predicted will still be valid for future designs, while the results can be obtained in a matter of days instead of months. To put in perspective, the longest 3D simulation was performed on a cluster of 24 cores with 2.67 GHz processing frequency, which took over 8 weeks to complete.

## Conclusion

Using CFD as the design tool, 2D simulations were used for the methodology validation of the D-VAWT, while 3D simulations were used to obtain a more realistic performance prediction. The validation of the methodology was performed for a single blade in 2D, where the mesh convergence, domain size, turbulence model, $y+$ strategy, and blade motion prescription were investigated. The methodology was also validated with experimental results of a static airfoil in 2D. The SA strain/vorticity turbulence model was shown to be a lower bound estimate of the coefficient of power while the SST k–ω turbulence model gave a more accurate prediction of the performance.

The novelty of the D-VAWT design is that it is among the few studies that modified a turbine's blade path in an attempt to improve the aerodynamic efficiency. In this paper, it was shown that the D-VAWT design did indeed succeed at improving the aerodynamic efficiency of small-scale turbines, which has been a challenge to overcome for many decades. The performed tests have shown the great potential of the D-VAWT by reporting a high CP performance of 0.4 for AR = 15 based on the 3D simulation of a single blade. For a common straight blade Darrieus type turbine, a CP of 0.4 is very high and difficult to achieve. However, the proposed D-VAWT is still far from being fully optimized. It is also important to note that because the D-VAWT requires the wind to be perpendicular to its longitudinal side to have optimal performance, this turbine now becomes dependent on wind direction, which is a drawback not encountered in traditional VAWTs. Also, the mechanism of the D-VAWT is more complex to design compared to more conventional VAWTs. Therefore, there are tradeoffs in obtaining higher efficiency, but further improvements in the performance could greatly outweigh those drawbacks.

## Funding Data

• Le Fonds Quebecois de la Recherche sur la Nature et les Technologies (FQRNT).

• The Concordia Institute for Water, Energy and Sustainable Systems (CIWESS).

• The Natural Sciences and Engineering Research Council of Canada (NSERC) for partial funding of this project through the Collaborative Research and Training Experience (CREATE) program.

## Nomenclature

• $A$ =

turbine swept area

• $AR$ =

• $C$ =

chord

• $CD$ =

coefficient of drag

• $CL$ =

coefficient of lift

• $CP$ =

coefficient of power

• $CP,Force$ =

coefficient of power for the translational sections

• $CP,Torque$ =

coefficient of power for the rotating sections

• $D$ =

turbine diameter

• $FTranslating$ =

sum of the tangential force produced when the blade is translating for a single cycle

• $h$ =

• $L$ =

axes spacing

• $P$ =

total extracted power

• $R$ =

• Re =

Reynolds number based on blade velocity

• $T$ =

period

• $TRotating$ =

sum of the torque produced when the blade is rotating for a single cycle

• $U∞$ =

freestream velocity

• =

• $Δt$ =

time step size

• $λ$ =

tip speed ratio

• $μ$ =

dynamic viscosity

• $ν$ =

kinematic viscosity

• $νt$ =

turbulent viscosity

• $ρ$ =

fluid density

• $φ$ =

incoming incident wind angle direction

• $ω$ =

angular velocity

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## References

Eriksson, S. , Bernhoff, H. , and Leijon, M. , 2008, “ Evaluation of Different Turbine Concepts for Wind Power,” Renewable Sustainable Energy Rev., 12(5), pp. 1419–1434.
ANSYS, 2011, “ ANSYS FLUENT 14.0 User's Guide,” ANSYS Inc., Canonsburg, PA, accessed Aug. 14, 2017,
Ponta, F. L. , Seminara, J. J. , and Otero, A. D. , 2007, “ On the Aerodynamics of Variable-Geometry Oval-Trajectory Darrieus Wind Turbines,” Renewable Energy, 32(1), pp. 35–56.
Ponta, F. L. , and Lago, L. I. , 2008, “ Analysing the Suspension System of Variable-Geometry Oval-Trajectory (VGOT) Darrieus Wind Turbines,” Energy Sustainable Dev., 12(2), pp. 5–16.
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## Figures

Fig. 1

Turbulence modeling behavior at high TSR (λ = 4.25) (top) and low TSR (λ = 2.55) (bottom) [10]

Fig. 2

Top view of a typical H-Darrieus VAWT with velocity vectors and forces, where θ is the azimuthal angle, U is the freestream velocity, Vblade is the blade velocity, Urelative is the relative velocity seen by the blade, α is the effective angle of attack, D is the drag force, and L is the lift force

Fig. 3

Example of D-VAWT: (a) top view and (b) 3D CAD model

Fig. 4

Instantaneous CP curve based on the combination of both the force- and torque-based methods

Fig. 5

Boundary conditions on domain

Fig. 6

Type 1 motion illustration

Fig. 7

Rotating domain mesh for type 1 and 2 motions

Fig. 8

Type 2 motion illustration

Fig. 10

Deforming domain mesh view for type 3 motion

Fig. 11

Dynamic domain mesh view for type 3 motion

Fig. 9

Type 3 motion illustration

Fig. 12

Rotating domain mesh view for mesh 1

Fig. 13

Refinement region mesh around blade for mesh 1

Fig. 14

BL views for mesh 1 at leading edge (top), zoomed at midchord (middle), and trailing edge (bottom)

Fig. 15

Average cycle CP convergence plot for mesh 1

Fig. 16

Instantaneous CP plots of the 15th cycle for the turbulence model study using mesh 1 at TSR = 4.5

Fig. 17

Coefficient of lift versus angle of attack for static airfoil at Re = 500,000

Fig. 18

Coefficient of drag versus angle of attack for static airfoil at Re = 500,000

Fig. 19

Ratio of coefficient of lift to coefficient of drag versus angle of attack for static airfoil at Re = 500,000

Fig. 20

Coefficient of lift versus coefficient of drag for static airfoil at Re = 500,000

Fig. 21

3D domain for D-VAWT with AR = 5

Fig. 22

Rotating domain mesh at symmetry plane for AR = 5 and y+ ∼ 30

Fig. 23

Refinement region mesh at symmetry plane for AR = 5 and y+ ∼ 30

Fig. 24

BL mesh view at symmetry plane for AR = 5 and y+ ∼ 30

Fig. 25

Cross section of mesh around the blade AR = 5 and y+ ∼ 30

Fig. 26

Refinement region mesh at symmetry plane for AR = 5 and y+ ∼ 1

Fig. 27

BL mesh view at symmetry plane for AR = 5 and y+ ∼ 1

Fig. 28

Cross section of mesh around the blade for AR = 5 and y+ ∼ 1

Fig. 29

Boundary conditions for 3D domains

Fig. 30

Average CP per cycle convergence for 3D simulations

Fig. 31

Instantaneous CP plots of the last cycle for 3D simulations

Fig. 32

Normalized velocity deficit ((U∞−U)/U∞) plots on a plane of half a chord away in the spanwise direction from the symmetry plane at t/T = 0.68 for cases: (a) SA strain/vorticity (y+ ∼ 30) with AR = 5, (b) SA strain/vorticity (y+ ∼ 30) with AR = 15, (c) SST k–ω (y+ ∼ 1) with AR = 5, and (d) SST k–ω (y+ ∼ 1) with AR = 15

Fig. 33

Static pressure contour on half of the blade surface for SST k–ω (y+ ∼ 1) model at t/T = 0.33 for (a) AR = 5 and (b) AR = 15

Fig. 34

Turbulent viscosity ratio (νt/ν) plots on a plane of half a chord away in the spanwise direction from the symmetry plane at t/T = 0.33 for cases: (a) SA strain/vorticity (y+ ∼ 30) with AR = 5, (b) SA strain/vorticity (y+ ∼ 30) with AR = 15, and (c) SST k–ω (y+ ∼ 1) with AR = 5, and (d) SST k–ω (y+ ∼ 1) with AR = 15

Fig. 35

Comparing instantaneous CP for 2D and 3D with AR = 5 and 15 using the SST k–ω model (y+ ∼ 1)

## Tables

Table 1 D-VAWT geometrical characteristics
Table 2 Results summary for domain size study
Table 3 Comparison of results for different motion types
Table 4 Details of each mesh used for mesh study
Table 5 Mesh and time convergence study results
Table 6 Comparison of the simulation times for the turbulence model study using mesh 1 at TSR = 4.5
Table 7 Results for the turbulence model study using mesh 1 at TSR = 4.5
Table 8 3D domain characteristics
Table 9 3D meshes details
Table 10 CP results summary for 3D simulations

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