This document describes a computational fluid dynamics (CFD) simulation of a 2D vertical axis wind turbine. It includes: 1) A description of the turbine geometry, governing equations, computational domain, and boundary conditions for the CFD model. 2) Details of the meshing approach and solver settings used in the CFD software Fluent. 3) Results and discussion of the simulation including validation, time-averaged forces and moments on the blades, flow visualization of velocity and vorticity fields, and effects of varying wind speed.

1. 1
Modeling 2D Vertical Axis Wind
Turbine
ME 505 Term Project
Ali Salman Alhamaly
201406980
Project Advisor: Dr. Rached Ben Mansour
King Fahad University of Petroleum and Minerals
May 2016

2. 2
Table of Contents
Abstract........................................................................................................................................... 4
1. Introduction................................................................................................................................. 4
2. Description of the Problem......................................................................................................... 6
2.1 Geometry Description ...................................................................................................... 6
2.2 Governing Equations........................................................................................................ 7
2.3 Computational Domain and Boundary Conditions .......................................................... 7
3. Details of the Method of Solution............................................................................................... 8
3.1 The Mesh.......................................................................................................................... 8
3.2 Solver Settings................................................................................................................ 10
4. Results and Discussion ............................................................................................................. 10
4.1 Grid and time sensitivity results and validation of the CFD simulation........................ 10
4.1.1 Grid sensitivity........................................................................................................ 10
4.1.2 Time step sensitivity ............................................................................................... 13
4.1.3 Validation................................................................................................................ 14
4.2 Forces and moment on the blades .................................................................................. 15
4.3 Flow Visualization ......................................................................................................... 20
4.4 Results of varying incoming wind speed ....................................................................... 25
5. Conclusion ................................................................................................................................ 28
References..................................................................................................................................... 29
Table of Figures
Fig. 1 Generic H-Darrieus (straight bladed) turbine showing the main geometry and the axis of
rotation............................................................................................................................................ 6
Fig. 2 Slice cut normal to the turbine axis of rotation. The figure shows the resulting 2D
geometry when the turbine is sliced................................................................................................ 7
Fig. 3 Computational domain (not to scale) showing the stationary fluid (white) and rotating
fluid (grey). The letter D refers to the diameter of the turbine. (1) inlet, (2) outlet, (3) interface
between stationary and rotating regions, and (4) no slip walls....................................................... 8
Fig. 4 Rotating fluid zone mesh...................................................................................................... 9
Fig. 5 The mesh around the airfoil.................................................................................................. 9

3. 3
Fig. 6 Stationary fluid zone mesh that is near to the rotating mesh.............................................. 10
Fig. 7 Torque coefficient variation with time and different mesh refinement for one revolution.12
Fig. 8 Averaged torque coefficient for one revolution for different mesh refinement................. 12
Fig. 9 Torque coefficient variation with time and different time steps for one revolution. ......... 13
Fig. 10 Averaged torque coefficient for one revolution for time step.......................................... 14
Fig. 11 Comparison between current CFD simulation and CACTUS showing the instantaneous
and averaged torque coefficient over one revolution.................................................................... 15
Fig. 12 Definition of the angle θ and the normal and tangential directions. ................................ 17
Fig. 13 The instantaneous normal force coefficient for blade 1 during one revolution. .............. 18
Fig. 14 Schematic of the ideal variation of angle of attack and forces on the blade as a function
of azimuthal angle. Figure taken from [9]. ................................................................................... 18
Fig. 15 The instantaneous tangential force coefficient for blade 1 during one revolution........... 19
Fig. 16 The instantaneous and mean torque coefficient for the whole turbine during one
revolution...................................................................................................................................... 19
Fig. 17 Contour of velocity magnitude at different time showing the wake of the turbine. (Values
are in m/s) ..................................................................................................................................... 22
Fig. 18 Contour of vorticity field at different time showing the vortex shedding and interaction.
Positive values are counterclockwise. (White regions are values outside the range of the figure
scale) (Values are in 1/s)............................................................................................................... 23
Fig. 19 Contour of vorticity field following the top blade at differ time instances. The figure
shows the evolution of the leading edge vortex on the surface of the blade. (Values are in 1/s). 24
Fig. 20 Incoming wind speed variation with time for three turbine revolutions. ......................... 25
Fig. 21 The instantaneous normal force coefficient for blade 1 during wind speed transient
plotted for total of three revolution............................................................................................... 26
Fig. 22 The instantaneous tangential force coefficient for blade 1 during wind speed transient
plotted for total of three revolution............................................................................................... 27
Fig. 23 The instantaneous and mean torque coefficient during wind speed transient plotted for
total of three revolution for the whole turbine.............................................................................. 27
Table of Tables
Table 1. Details of mesh refinement. The number of boundary refers to Fig. 3.......................... 11

4. 4
Abstract
Computational fluid dynamics (CFD) simulation of two dimensional vertical axis wind
turbine is presented. The governing equations are the unsteady incompressible continuity and
momentum. These equation were solved numerically using Fluent 15.0 software. The
computational domain is composed of a rotating fluid region and stationary fluid region in which
the rotation of the turbine is simulated using the sliding mesh model in Fluent. Validation of the
CFD simulation has been performed with the Sandia Laboratory CACTUS code. Validation shows
the current CFD simulation capture the revolution-averaged torque coefficient with error less than
9%. Emphasis is placed on characterizing the main performance parameters of the turbine and
show their variation as a function of time. Namely, normal and tangential forces on the blades and
the overall aerodynamic torque of the turbine is investigated in great details. In addition, the
velocity and vorticity fields around the turbine is visualized and their behavior is related to the
energy exchange process from the wind to the turbine. Differences of the flow field behavior
between the upwind and downwind blade is examined and shown to have great effect on the time
variation of the main performance parameters of the turbine. Time variation of the incoming wind
speed is studied and results show that for linear decreasing wind speed with time, the main effect
is to reduce the peak turbine torque. This reduction in the peak torque led to decrease in the
revolution-averaged torque by 7 % when the wind speed goes from 12 m/s to 10.5 m/s during three
turbine revolution.
1. Introduction
Wind turbines are devices used to convert wind energy into useful power by means of
converting the kinetic energy in the wind into rotational energy in the turbine rotor. There are two
main types of wind turbines: the familiar horizontal axis wind turbine in which the axis of rotation
is parallel with the direction of wind, and vertical axis wind turbine in which the axis of rotation
is perpendicular to the direction of wind.
Wind energy is one of most important renewable energy source available today. In fact, at
the end of 2014 wind energy global electric power capacity has reached 370 GW contributing to
about 56% of the total global renewable power generation (excluding large hydro power plants)
[1]. Also, in 2014 the total new installation of wind power reached about 51 GW making it the
largest renewable technology in terms of that total added power generation in 2014 [1]. These
figures show how important wind energy is and how it is growing very fast. Since wind energy is
exploited using wind turbines, the ability to assess the performance of wind turbines is very crucial
to the power industry. For this reason, wind turbine simulation are great tool to quantify different
performance parameters (such as: total power, torque, rotational speed, number of blades, shape
of the blades, and the geometry of the overall turbine) for wind turbine and how the wind speed
affect those performance parameters. In this report, we study different simulations of the flow field
around 2D vertical axis wind turbine to quantify some of the important performance parameters of
such a turbine.
Great amount work has been done in the literature regarding numerical simulation of
Vertical Axis Wind Turbine (VAWT) using Computational Fluid Dynamics (CFD). The analyses
include both 3D and 2D simulations of the turbine flow field. Great emphasis of these numerical
studies has been on investigating the turbine performance response due to changing main design

5. 5
parameters of the turbine. In addition, some studies are dedicated to study the dynamic stall
phenomena and its corresponding effect on performance of VAWT. Few studies used CFD
simulation in optimization in which the performance of the turbine is obtained from such
simulations.
Mohamed et.al. [2] conducted CFD simulation of 2D straight bladed VAWT (H-Darrieus)
with three blades. Two turbulent models were considered in the simulation: realizable 𝑘 − 𝜖 and
𝑘 − 𝜔 𝑆𝑆𝑇. Mohamed et.al. [2] showed that 𝑘 − 𝜔 𝑆𝑆𝑇 turbulence model is more stable than the
realizable 𝑘 − 𝜖, and hence 𝑘 − 𝜔 𝑆𝑆𝑇 was used for all the analysis presented. The emphasis of
the work is to investigate the effect of changing airfoil type on the performance of the turbine.
Hence, the performance of the turbine were simulated for 25 different airfoil at different tip speed
ratios. Benini et al. [3] performed 2D and 3D CFD simulations of straight bladed VAWT that has
3 blades with NACA 0021 airfoil. Results of 2D simulation were used to propose a simple
aerodynamic performance model that correlated the blade angle of attack to the blade forces.
Ferreira et al. [4] presented 2D CFD simulation of a straight single bladed VAWT. the blade used
NACA 0015 airfoil. The aim of the work is to study dynamic stall and vortex shedding from the
blade and compare the CFD simulation with PIV experimental data. Four different turbulence
models were used : Spallart-Almaras , 𝑘 − 𝜖, Large Eddy Simulation (LES), and Detached Eddy
Simulation (DES). Results show the DES model gives results that is closest to experiments since
the model was able to predict the generation, shedding, and convection of vortices over the blade
surface. Spallart-Almaras and 𝑘 − 𝜖 models predicted a vorticity field that is continuous over the
blade surface, whereas experimental data shows that the vorticity is composed of distribution of
several small vortices. Untaroiu et al. [5] performed 2D and 3D CFD simulation of straight bladed
VAWT with 3 NACA 0018 blades. The aim of the study is to predict the ramp-up transient of the
turbine and to see whether the simulation can predict the operation speed of the turbine compared
with experimental data. Because it is desired to study transient behavior of turbine speed, the
rotational speed of the turbine is changing for every time step and is dependent on the value of
aerodynamic torque generated by the turbine. 2D simulations over predict the final operating
speed with 12% of the experimental value, while the 3D simulations under predict the final
operating speed with 15% of the experimental data. Both 2D and 3D simulation fail to predict the
transient behavior of speed ramp-up since they show very rapid acceleration to the final operating
speed. Carrigan et al. [6] studied torque optimization of VAWT by changing the airfoil shape and
chord length at a constant tip speed ratio. The torque of the turbine is obtained from 2D CFD
simulation of the turbine. The optimization is based on maximizing the average torque for one
complete revolution. In recent paper, Ferrari et al. [7] presented critical issues in CFD modeling
of 2D VAWT. The paper summarized the VAWT CFD modeling approach of many recent works
and concluded that poor agreement between studies was found in terms of the simulation
parameters. The simulations parameters of interest are: turbulence models, methods of
discretization, time step size, number of complete revolutions, distance of the computational
domain boundaries, and number of mesh elements. Guidelines on the former mentioned simulation
parameters to insure high accurate results were recommended by the authors. For more CFD
studies on VAWT, the reader is referred to the references section of [7] .
The objective of this work is to perform 2D CFD simulation of VAWT to quantify the main
performance parameters of the turbine. Specifically, it is of interest to investigate how the
aerodynamic forces and torque on the blades vary with time. In addition, the global unsteady flow
field around the turbine will be investigated to understand the energy exchange process. Unsteady

6. 6
vortex shedding and the wake of the turbine is also to be visualized. Such investigation can help
in the design of wind turbine and also the design of wind farms configuration.
2. Description of the Problem
2.1 Geometry Description
The problem at hand is to simulate the flow over a straight bladed vertical axis wind turbine
(VAWT). The straight bladed VAWT is typically called H-Darrieus turbine. The H-Darrieus
blades are characterized as being straight and not curved. Figure 1 shows a typical geometry of the
H-Darrieus turbine. Due to the fact that H-Darrieus turbine has blades that are straight, any slice
cut of the turbine normal to its axis of rotation will give same 2D geometry on that slice. The 2D
geometry resulting from this slice is 2D airfoils locating on circle with diameter equals the
diameter of the turbine. Figure 2 shows the resulting 2D geometry from such a slice.
2D VAWT like the one shown in Fig. 2 contains much of the physics that caverns the full
3D VAWT and shows the details of the unsteady flow field and the wake development. For this
reason and for the fact that 2D simulation is simpler and computationally less expensive than a full
3D simulation, 2D VAWT simulation is perused in this project. So the aim of this project is to
simulate the unsteady flow field through a 2D VAWT like the one depicted in Fig. 2
Fig. 1 Generic H-Darrieus (straight bladed) turbine showing the main geometry and the
axis of rotation.

7. 7
2.2 Governing Equations
The equations that govern the flow field of a 2D VAWT is the unsteady incompressible
Navier-Stokes equations without body forces. Since the flow is turbulent, Reynolds averaged form
of the equations is used:
∇⃗⃗ . (𝜌 𝑉⃗ )) = 0 (1)
𝜕𝜌𝑉⃗
𝜕𝑡
+ ∇⃗⃗ . (𝜌 𝑉⃗ 𝑉⃗ ) = −∇⃗⃗ 𝑝 + ∇⃗⃗ . 𝜏 − ∇⃗⃗ . (𝜌 𝑉′⃗⃗⃗⃗ 𝑉′⃗⃗⃗⃗̅̅̅̅̅̅̅) (2)
𝜏 = 𝜇 (∇⃗⃗ 𝑉⃗ + (∇⃗⃗ 𝑉⃗ )
𝑇
) (3)
The Reynolds stresses (last term in Eq. (2)) is modeled using the Boussinesq approach in which
the turbulent viscosity is modeled using Spalart-Allmaras one equation turbulence model.
2.3 Computational Domain and Boundary Conditions
The computational domain is composed of: stationary fluid region and rotating fluid region.
The rotating fluid region is used to model rotating 2D VAWT. The computational domain with the
boundary conditions is shown in Fig. 3. In the rotating region, 2 NACA-0015 airfoils are used to
represent the blade of the turbine each with chord length of 0.6m. The diameter of the turbine is
4m. The turbine is rotating counter-clockwise with angular velocity of 12 rad/s (114.6 rpm). The
inlet wind speed is set to be 12m/s and is directed in the horizontal direction. The outlet condition
is at zero pressure gauge. The inlet turbulent viscosity ratio is set to be 10.
Fig. 2 Slice cut normal to the turbine axis of rotation. The figure shows the resulting 2D
geometry when the turbine is sliced

8. 8
3. Details of the Method of Solution
The simulation of the 2D VAWT was done using the CFD commercial software Fluent 15.
Fluent uses finite volume method to discretize the governing equations and converting them from
partial differential equations to algebraic equations that can be solved. The flow variables are
solved at discrete points in the computational domain in which location of these points are given
by the mesh. In this section, the mesh used in simulation is shown and the different Fluent solver
settings are summarized.
3.1 The Mesh
The mesh was done using Gambit 2.4. The mesh used is unstructured triangular mesh through the
whole domain. In the rotating fluid zone, the number of volume elements are 193K, in stationary
4
4
3
1
7D3D
2
20D
1.25D
Fig. 3 Computational domain (not to scale) showing the stationary fluid (white) and
rotating fluid (grey). The letter D refers to the diameter of the turbine. (1) inlet, (2) outlet,
(3) interface between stationary and rotating regions, and (4) no slip walls.

9. 9
fluid zone the elements are 68.8K. This particular mesh was chosen after doing mesh sensitivity
study in which the results of the study indicate that this mesh is suitable. The details of the mesh
sensitivity study is given in the next section. The mesh can be seen in Figures 4- 6.
Fig. 4 Rotating fluid zone mesh.
Fig. 5 The mesh around the airfoil.

10. 10
3.2 Solver Settings
Second order implicit unsteady formulation were used in Fluent. The discretization used
for all the solved equations was second order upwind scheme. SIMPLE scheme for pressure-
velocity coupling was used.
Sliding mesh was used to simulate the rotation of the turbine in which the rotating fluid
zone is sliding relative to the stationary fluid zone. The period for one turbine revolution is
𝑇 = 2𝜋/𝜔. The time step for the solver was chosen to be ∆𝑡 = 𝑇/360. This means that the time
step correspond to a one degree of turbine rotation. The specific choice of this time step comes
from time step sensitivity study in which it shows that there is no need for small time step. The
details of the time step sensitivity is given in the next section. For this simulation, 𝜔 =
12𝑟𝑎𝑑/𝑠, 𝑇 = .5236 𝑠𝑒𝑐, 𝑎𝑛𝑑 ∆𝑡 = .00145 𝑠𝑒𝑐. The maximum number of iterations for each
time step was 120 iterations. With 120 iterations the residuals for momentum and turbulent
viscosity reach 10-8
while continuity residual reaches between10-5
- 10-6
.
4. Results and Discussion
4.1 Grid and time sensitivity results and validation of the CFD simulation
4.1.1 Grid sensitivity
In order to assess the suitability of the mesh to resolve the flow field around the turbine,
grid sensitivity study has been performed. The aim of this study is to determine how the flow field
changes with mesh refinement. In other words, we need to determine how fine the mesh need to
be in order to resolve the flow field accurately with minimum truncation error.
Fig. 6 Stationary fluid zone mesh that is near to the rotating mesh.

11. 11
To assess how the flow field changes with mesh refinement, the torque coefficient is used
as the metric to identify changes in the whole flow field (Torque coefficient is defined in Eq. (6)
in next section). Four different meshes were used with each one more refined than the previous
one. The details of these for meshes are summarized in Table 1. For each of these meshes the time
step was held constant at 1o
per time step i.e. ∆𝑡 = .00145 𝑠𝑒𝑐
Figure 7 shows the variation of torque coefficient with respect to time for each mesh given
in Table 1. The figures shows large variation of meshes 1 and 2. However, meshed 3 and 4 are
very close to each other over most of the time. Figure 8 shows the average torque coefficient for
one cycle versus the number of elements. It can be seen in this figure also that the difference
between mesh 3 and 4 are not that much great, in fact the difference is about 11.6%. Although
11.6% difference seems to be a little bit on the high side which suggests that more grid refinement
need to be done in order to assess whether mesh 4 has reached grid independent solution or not, it
was decided to use mesh 3 because this mesh has a compromise between the computational time
and the accuracy. In addition, it will be shown later that validation with mesh 3 shows great match
between the validation case and the current CFD simulation using mesh 3.
Mesh # # Element on
boundary1
# Element on
boundary2
# Element on
boundary3
# Element on
boundary4
Total
Elements
1 60 60 80 100 60490
2 60 60 160 160 134616
3 60 60 240 240 262014
4 60 60 240 300 338172
Table 1. Details of mesh refinement. The number of boundary refers to Fig. 3

12. 12
Fig. 7 Torque coefficient variation with time and different mesh refinement for one
revolution.
Fig. 8 Averaged torque coefficient for one revolution for different mesh refinement.

13. 13
4.1.2 Time step sensitivity
In order to assess the validity of the time step used in the simulation (1o
per time step i.e.
∆𝑡 = .00145 𝑠𝑒𝑐), the time step was reduced by half to be .5o
per time step i.e. ∆𝑡 = .000727 𝑠𝑒𝑐
and the simulation with mesh 3 was repeated. Figure 9 shows the variation of torque coefficient
with respect to time for mesh 3 using two different times steps: 1o
per time step and .5o
per time
step. The figure shows that the two cases are indistinguishable which indicates that 1o
per time
step is sufficient to reach time step independent solution. Figure 10shows the average torque
coefficient with degrees per time step. The figure indicates that the difference between the two
cases is less than .8% which reassure what we concluded from Fig. 9.
Fig. 9 Torque coefficient variation with time and different time steps for one revolution.

14. 14
4.1.3 Validation
The current CFD simulation need to be validated to assess the degree of accuracy of the
reported results. Since the turbine geometry used in this report does not have a close experimental
study, it was decided that validation will be done using Sandia Laboratory CACTUS code which
is itself has been validated against serval experimental data [8]. CACTUS is a performance
simulation code based of a free wake vortex method. CACTUS is a 3D code in which it simulates
the whole turbine and not just a 2D slice cut. In order to make the comparison with present
simulation as close as possible, the height of the turbine in CACTUS was made to be large to
decrease the 3D flow effect as much as possible. The validation is based on the instantaneous and
averaged torque coefficient.
Figure 11 shows a comparison between CACTUS and the current CFD simulation for the
torque coefficient. The figure indicates that the CFD results capture the main pattern in the torque
compared with CACTUS. The CFD results appear to be shifted in time in comparison with
CACTUS. This shift might be attributed to the ability of the CFD code to capture the dynamic stall
and its effect on the overall performance. In terms of the averaged torque coefficient, Fig. 11 shows
that the CFD results are in very good agreement with CACTUS with error less than 9%. As can
be seen the CFD overestimate the averaged torque compared with CACTUS and this is expected
due to the 3D losses effect that are captured in CACTUS but not in the current CFD simulation.
Fig. 10 Averaged torque coefficient for one revolution for time step.

15. 15
4.2 Forces and moment on the blades
During the operation of the turbine, the air exerts unsteady forces on each blade. These
forces can be decomposed into normal and tangential directions (the definition of the direction can
be seen in Fig. 12). Normal forces are undesirable loads on the structure that need to be safely
transmitted from the blade to the structure of the turbine. Tangential forces are responsible for
generating the torque of the turbine and hence the power.
In this subsection, the instantaneous forces acting on one blade during one revolution are
presented. The forces are presented in terms of the normalized force coefficient which are:
𝐶𝐹𝑛 =
𝐹𝑛
1/2𝜌𝑈∞
2 2𝑅
(4)
𝐶𝐹𝑡 =
𝐹𝑡
1/2𝜌𝑈∞
2 2𝑅
(5)
𝐶𝑄 =
𝑄
1/2𝜌𝑈∞
2 2𝑅2
(6)
Fig. 11 Comparison between current CFD simulation and CACTUS showing the
instantaneous and averaged torque coefficient over one revolution.

16. 16
Where 𝐶𝐹𝑛, 𝐶𝐹𝑡, 𝐶𝑄 are the normal force, tangential force, and torque normalized coefficients
respectively. 𝜌 is air density, 𝑈∞ is incoming wind speed, and 𝑅 is turbine radius. In the presented
figures below, the force coefficients are presented for a single blade whereas the torque coefficient
is for the whole turbine.
Figure 13 shows the normal force coefficient for one blade during one complete revolution.
The x-axis of the plots are actually time, but it has been converted to angle for convenience. Figure
13 shows that the normal force variation with time can be divided into two regions: compression
for 𝜃 <̃ 190 𝑜
and tension for 𝜃 >̃ 190 𝑜
. The region of compression is in the upwind region while
the region of tension is in the downwind region (the upwind region is refer to the flow domain
inside the turbine circle in which the air is not affected by a blade i.e. the region that the air first
encounters when flow through the turbine). It can be also noticed from Fig. 13 that the positive
and negative force distribution is not symmetric in the upwind and downwind regions. In fact the
magnitude of the maximum normal force in the upwind region is more than 2.5 times larger than
the force in the downwind region.
The variation of the normal force with time can be understood by the fact that the forces
acting on the blade are results of combination between lift and drag on the airfoil. Since the lift
and drag on a 2D airfoil depends on the angle of attack and Reynolds number, thus variation of
these force can be understood by variation of angle of attack and Reynolds number as the blade is
rotating. Due to rotation, the blade sees, with respect to its own frame of reference, a perceived
wind that is composed of the free stream wind speed and also the liner speed of the blade itself.
Since the direction of the linear speed of the blade changes with time (due to changing of blade
location along the circle), both the perceived wind direction and magnitude is changing with time
as seen by the blade. This change in the perceived wind is the main cause for the unsteady nature
of the normal force. In addition, changes in the magnitude of the forces indicate changes in the lift
force which is linked with the circulation around the blade. This means that as a result of unsteady
forces, bound circulation around the blade is also changing with time which necessitates shedding
vortices in the wake of the blade as a consequence of Kelvin’s theorem for conservation of
circulation. More on this idea will be shown when we present the flow visualization in the next
section.
To understand the trend in the normal force magnitude shown in Fig. 13 (mainly the trend
in the upwind region in which the force magnitude increases until 𝜃 ≈ 80 𝑜
and then decreases until
it reaches zero at 𝜃 ≈ 190 𝑜
), Fig. 14 is used. Figure 14shows schematically how the angle of attack
and the perceived wind changes as a function of the azimuthal angle (notice in Fig. 14 the angle
starts at 𝜃 = −90 instead of 𝜃 = 0 as we defined it earlier in Fig. 12). Figure 14shows that in the
region 0 < 𝜃 < 90 (−90 < 𝜃 < 0 in Fig. 14 notation) the angle of attack increases in magnitude
from zero to maximum at 𝜃 = 90 and then it starts to decrease in magnitude until it reaches zero
again at 𝜃 = 180. The trend in forces is also similar to the angle of attack trend. Using this
observation from Fig. 14 it becomes now easy to explain the trend in normal force seen in Fig. 13.
The variation in normal force in Fig. 13 is a consequence of the angle of attack variation described
in Fig. 14. One should notice though that the maximum value of the normal force in Fig. 13 does
not coincide exactly within 𝜃 = 90. This is probably because of the induction field, caused by
vortices in the wake of the blade, which modifies the flow ahead of the blade. Another important
aspect that worth notice, is the difference between the upwind and downwind force distribution. It
is clearly seen in Fig. 13 that the downwind force distribution does not follow the increase decrease
trend of the upwind half, but is relatively constant. This is mainly due to the fact that in the
downwind portion, the blade operates in a wind condition that is extremely different from the

17. 17
upstream wind. The air velocity in the downwind portion is reduced because energy has been
already extracted from it and also it contains the wake of the blade. This wind condition severely
changes the angle of attack and the perceived wind on the downwind portion of the blade compared
with the upwind portion. More on this idea will be presented in the next section in which with the
help of flow visualization we will try to explain this phenomena more clearly.
Figure 15 shows the tangential force coefficient for one blade during one complete
revolution similar to Fig. 13. In Fig. 15 one notice that similar to the normal force distribution
discussed earlier, the tangential force distribution is different between the upwind and downwind
portion. The tangential force reaches maximum at about 𝜃 = 90 which is what is expected from
ideal considerations discussed earlier. Figure 15 shows that there exist small time interval in which
the tangential force is negative, this means that the torque is negative during these time interval
which leads to negative power extraction. The tangential force during the upwind portion has large
variation whereas during the downwind portion it is fairly uniform. In addition, Fig. 15 shows the
tangential force is larger during the upwind portion compared with downwind portion which
indicates that larger energy extraction is accomplished during the upwind part. It is worth to notice
that the tangential force peak is smaller in magnitude than the normal force peak by about a factor
of 3.75.
Figure 16 shows the torque coefficient for the whole turbine i.e. considering both blades.
The torque distribution with time is similar to the tangential force distribution except that there
exist two distinct peak in the distribution. This is due to the fact that we are considering both
blades. Each peak corresponds to the blade being in the upwind portion of the cycle near 𝜃 = 90.
Figure 16 shows that even for the case of two blades, there exist time intervals in which the torque
is negative indicating that the turbine exhibits negative energy extraction in some portion of the
cycle. Although that the peak to peak variation in torque is about .67, the average torque over one
complete cycle is only .22 as shown in Fig. 16. This low value of torque is a consequence of
operation in the downwind portion.
𝜃
𝑛
𝑡
Fig. 12 Definition of the angle θ and the normal and tangential directions.

18. 18
Fig. 13 The instantaneous normal force coefficient for blade 1 during one revolution.
Fig. 14 Schematic of the ideal variation of angle of attack and forces on the blade as a
function of azimuthal angle. Figure taken from [9].

19. 19
Fig. 15 The instantaneous tangential force coefficient for blade 1 during one revolution.
Fig. 16 The instantaneous and mean torque coefficient for the whole turbine during one
revolution.

20. 20
4.3 Flow Visualization
In this subsection, the flow field through the turbine is examined to gain insight on the
operation of the turbine. The results presented emphasize the description of the turbine wake and
the unsteady vortex shedding and vortex-blade interaction. Five complete turbine revolutions were
simulated and the results presented in this section are all from the fifth revolution. The reason for
choosing five revolutions is because the averaged torque coefficient over the fifth and fourth
revolutions differs by about .5%. This variation is deemed small and hence convergence is attained
in terms of reaching periodic state and there is no need to simulate more revolutions.
Figure 17 shows contour plot of velocity magnitude in region near to the turbine at different
time instances (again different times are represented in terms of angles for convenience). The free
stream velocity of the wind is from left to right. It is clear from Fig. 17 that the distinct feature at
different times is the wake of turbine that extends several diameter downstream. The velocity in
the wake is about third the free stream velocity (free stream velocity is 12 m/s). It is interesting to
notice that the wake start developing immediately downstream of the upper blade causing the flow
inside the turbine circle to be of lower velocity. The velocity of the wake gets even lower
downstream of the lower blade. The fact the wake develops immediately after the upwind (upper)
blade give rise to the implication that the downwind (lower) blade always operates at reduce flow
condition that is different from what the upwind blade operates in. Operation in two different flow
regimes imply necessary that the forces acting on the downwind blade will be of different
magnitudes that the forces acting on the upwind blade. This idea has been already mentioned in
the previous section when discussing the trend in the forces acting on the blade in the upwind and
downwind portion.
The wake velocity and shape downstream of the turbine seems not to change as the turbine
is rotating as can be seen by comparing the different snap shots of Fig. 17. However, the wake
velocity inside the turbine circle does change as the turbine is rotating. This change in wake
velocity inside the circle is probably a result of upwind blade energy extraction from the air the
flows around it.
It can be noticed that outside of the low velocity region, the velocity there is actually higher
than the free stream velocity. The reason for this increase in velocity is probably because of the
blockage effect of the whole turbine. In other words, not all the streamlines ahead of the turbine
flows into the turbine but some streamlines diverge away from the turbine and don’t get effect
directly by the blades. These streamlines need to increase in velocity during the process of
diverging and that is what is observed just outside the wake region.
There remains features in the wake that has not been explained yet. For instance, the shape
of the downstream wake is a “cloud-like” i.e. the outer boundary of the wake is not straight but
undergoes ups and downs. Also, there exists multiple discrete regions downstream of the turbine
and just outside the wake that has considerably high velocity. These feature will be examined
further when considering vorticity field.
Figure 18 shows contour plot of vorticity field near the turbine for different time instants.
The different time instances snap shots correspond to those shown in Fig. 17. Figure 18 shows that
the wake of the turbine is composed of concentrated vortices that turn either counterclockwise or
clockwise depending on their locations. The clockwise vortices are on top and the
counterclockwise ones are in bottom. All the different snap shots in Fig. 18 indicate that as the

21. 21
vortices move downstream their magnitudes get lower and they get smeared out as they diffuse
into the lower vorticity region in the rest of the wake.
The source of the concentrated vortices in the downstream wake of the turbine can be traced
to the vortex shedding by the blades. For instance, following the upper blade in Fig. 18 as it rotates,
indicate that the vorticity shed from the trailing edge of the blade moves away from the blade in a
roughly circular path and then becomes one of the concentrated vortex region at the top of the
turbine. The bottom concentrated vortices downstream of the turbine can be associated with the
separation of the large vortex on the surface of the bottom blade. This large vortex can be seen in
Fig. 18 at 𝜃 = 0 𝑜
on the bottom blade as a circle-like region that begins roughly mid chord and
extend a little bit beyond the trailing edge (this region is shown in white color due to its magnitude
being out of scale). Following this region in Fig. 18 at 𝜃 = 45 𝑜
shows that the vortex has separated
from the surface of the blade and then it moves downstream to constitute the lower part of the
vorticity region downstream of the turbine as can be seen by following Fig. 18 at 𝜃 = 90 𝑜
− 180 𝑜
.
It is interesting to notice that the top and bottom vortices downstream of the turbine originate from
two different flow situation, namely, the top vortices are results of continuous vortex shedding
from the trailing edge, whereas the bottom vortices are results of separation of large vortex region
on the surface of the whole blade. The “cloud-like” shape of the wake mentioned earlier can be
seen as a results of the complex vorticity field in the wake. In addition, the high speed multiple
discrete regions downstream of the turbine which are at the edge of the bottom wake can be seen
to be a results of the separated large vortex from the bottom blade.
Figure 19 shows contour of vorticity field for the top blade at different time instances. The
purpose of this plot is to investigate the large vortex on the surface of bottom blade that is shown
in Fig. 18 at 𝜃 = 0 𝑜
. Figure 19 shows that there is a small bubble with high vorticity at the leading
edge at 𝜃 = 150 𝑜
. This bubble grows rapidly into large vortex that moves with the blade. The
vortex grows in size until it covers the whole surface of the blade as seen in Fig. 19 at 𝜃 =
135 𝑜
, 150 𝑜
. This vortex is the same as the previously mentioned large vortex on the surface of the
bottom blade. Since this vortex originates at the leading edge, then this is vortex should be called
the leading edge vortex.

22. 22
Fig. 17 Contour of velocity magnitude at different time showing the wake of the turbine.
(Values are in m/s)
a) 𝜃 = 0 𝑜
b) 𝜃 = 45 𝑜
c) 𝜃 = 90 𝑜 d) 𝜃 = 135 𝑜
d) 𝜃 = 180 𝑜

23. 23
Fig. 18 Contour of vorticity field at different time showing the vortex shedding and
interaction. Positive values are counterclockwise. (White regions are values outside the
range of the figure scale) (Values are in 1/s)
.
a) 𝜃 = 0 𝑜
b) 𝜃 = 45 𝑜
c) 𝜃 = 90 𝑜 d) 𝜃 = 135 𝑜
d) 𝜃 = 180 𝑜

24. 24
Fig. 19 Contour of vorticity field following the top blade at differ time instances. The figure
shows the evolution of the leading edge vortex on the surface of the blade. (Values are
in 1/s)
a) 𝜃 = 105 𝑜
b) 𝜃 = 135 𝑜
c) 𝜃 = 150 𝑜 d) 𝜃 = 180 𝑜

25. 25
4.4 Results of varying incoming wind speed
In this subsection, the effect of varying the incoming wind speed on the turbine
performance is presented. Instead of making the incoming wind speed as parameter and then
evaluating the performance of the turbine at different values of the parameter, a more interesting
scenario is presented here. The wind speed is continuously varied in a transient fashion during the
simulation. Hence, at each time step the wind speed is changing with some prescribed profile in
time. The specific profile chosen in this report is shown in Fig. 20 in which the incoming velocity
is plotted against the normalized time (normalized time is the time divided by the period of turbine
rotation). As can be seen in Fig. 20 the velocity profile that will be simulated decreases linearly at
a rate of .5 m/s for each revolution. The results in this subsection have been done for three turbine
revolution, i.e. the wind speed went from 12 m/s to 10.5 m/s.
Figure 21 shows the instantaneous normal force coefficient for blade 1 during the wind
speed transient. In the same figure, the normal force coefficient for constant wind speed at 12 m/s
is plotted also for comparison. It can be seen that the case of variable wind speed is almost identical
to the constant wind speed with the exception of the valleys values of the force. The figure shows
that as the time increases, the absolute value of the most negative force value for the variable speed
wind case gets reduced compared with the constant wind speed. This behavior is due to the linear
decrease of the wind speed with time. Figure 22 shows a plot similar to Fig. 21 but for the tangential
force coefficient. Same transient behavior is noted in figure Fig. 22 as in Fig. 21for the variable
Fig. 20 Incoming wind speed variation with time for three turbine revolutions.

26. 26
speed wind case. It can be seen by comparing Fig. 21and Fig. 22 that the change in amplitude of
the tangential force coefficient is more than that of the normal force coefficient especially if we
look at the last revolution trends (720 < 𝜃 < 1080). Lastly, Fig. 23 shows the torque coefficient
for the whole turbine for both the constant and variable wind speed. Same transient behavior seen
in Fig. 21 and Fig. 22 can be noticed in Fig. 23. It is interesting to see that the lowest value of the
torque coefficient is not affected by the change in velocity and only the peak value gets changed.
It is worth to notice that the variable wind speed curve looks like a damped response version of
the constant wind speed. In terms of the averaged torque coefficient over the transient period of
the wind i.e. three revolution, Fig. 23 shows that the constant wind averaged torque coefficient is
0.213 while it is 0.198 for the variable wind. This means that the averaged torque coefficient has
reduced by about 7% compared with the constant wind speed.
Fig. 21 The instantaneous normal force coefficient for blade 1 during wind speed transient
plotted for total of three revolution.

27. 27
Fig. 22 The instantaneous tangential force coefficient for blade 1 during wind speed
transient plotted for total of three revolution.
Fig. 23 The instantaneous and mean torque coefficient during wind speed transient
plotted for total of three revolution for the whole turbine.

28. 28
5. Conclusion
CFD unsteady numerical simulation for the flow field over a 2D vertical axis wind turbine
VAWT has been presented. The simulated turbine is composed of a slice cut with plane
perpendicular to the axis of rotation. The flow field is solved using Fluent in which the Navier-
Stokes equation are discretized and solved using finite volume method in the computational
domain. The computational domain is composed of a rotating fluid region and stationary fluid
region. The overall shape of the domain is a semicircle. Velocity inlet boundary condition and
pressure outlet condition were used with sliding mesh model to simulate the unsteady rotation of
the turbine. Grid and time independent studies has been performed to insure that the CFD
simulation has reached grid and time independent solution. It is shown that a mesh with about
262K element with time step equivalent to 1 degree is sufficient to reach grid and time independent
solution. Validation of the CFD simulation has been performed with the Sandia Laboratory
CACTUS code. Validation shows the current CFD simulation capture the averaged torque
coefficient with error less than 9%. Main performance parameters of the turbine such as the forces
on the blade and the turbine torque have been reported and their trends were explained. Flow
visualization have been reported in terms of both the velocity and the vorticity field. Velocity field
visualization shows that the turbine exchanges energy by reducing the kinetic energy of the main
wind which was clear by examining the wake of the turbine. Downwind blade operates always at
reduced velocity flow from the free stream flow. This reduced velocity flow is highly turbulent
and full of vortices that are shed from the blades. The vorticity field shows mainly that the wake
of the turbine is composed of concentrated vortices with clock-wise and counter clockwise sense.
Counter clockwise vortices are results of a large separated leading edge vortex from the downwind
blade, whereas the clock wise vortices are results of continuous shedding of the vortex of the
trailing edge of the blade. Incoming wind velocity was varied to simulate unsteady wind condition.
The wind speed was varied between 12-10.5 m/s in three turbine revolution. The response of the
turbine to this change in wind condition in terms of the forces and torque on the blade showed that
the general time variation trend is still the same compared with constant wind condition. The
difference is in the value of the peak forces and torque. These peaks gets reduced as the wind
reduces its speed. The averaged torque coefficient over the period of wind transient has decreased
by about 7% compared with the constant wind speed case.

29. 29
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