TASK QUARTERLY vol. 22, No 3, 2018, pp. 195–209
VELOCITY FIELD AROUND DARRIEUS
WIND TURBINE ROTOR USING ACTUATOR
CELL MODEL AND OTHER CFD METHODS
KRZYSZTOF ROGOWSKI AND KLAUDIA ROGOWSKA
Warsaw University of Technology
The Institute of Aeronautics and Applied Mechanics
Nowowiejska 24, 00–665 Warsaw, Poland
(received: 4 May 2018; revised: 11 June 2018;
accepted: 29 June 2018; published online: 6 July 2018)
Abstract: The main purpose of this work is to analyze the usefulness of the active cell model
(ACM) developed by the author of this article to estimate the ow eld around a single-bladed
vertical-axis wind turbine (VAWT) with the Darrieus-type rotor. The obtained ow velocity elds
were compared with the experimental values taken from the literature available on the Internet.
Additionally, the ow elds around the rotor and the aerodynamic forces were determined using
the following approaches: the -RNG turbulence model, the scale-adaptive simulation (SAS) and
the laminar model. The velocity proles behind the turbine rotor obtained with all numerical
approaches are consistent with the experiment. The aerodynamic blade loads obtained using
numerical methods also appear to be satisfactory.
Keywords: CFD, vertical-axis wind turbine, aerodynamic blade loads, wake modeling
1.1. Vertical-axis wind turbines
Even though the share of the vertical-axis wind turbine (VAWT) is currently
decreasing in Poland, the number of patents related to them continues to increase.
There is also a growing interest in this topic in the world as evidenced by the
number of publications.
The archetype of the VAWT is considered to be a Persian windmill (Fi-
gure 1a) from the 2nd century BC used to drive a quern. This device resembled
a water wheel in which half of the rotor was shielded from the wind. The source
of torque in these devices was the drag dierence on the windmill blades. Such
turbines are called drag-driven devices .
196 K. Rogowski and K. Rogowska
Devices with the Darrieus rotor operate on a completely dierent princi-
ple. Darrieus wind turbines are an alternative to traditional horizontal-axis pro-
peller-type machines. This lift-driven device proposed by a French aeronautical
engineer, Georges Jean Marie Darrieus was patented by U. S. Patent Oce in
1931 . According to the inventor, the rotor should have properly curved bla-
des. The shape of this curvature should be close to a rotating rope. This shape
is called the Troposkien shape (from Greek: – turning; – rope).
A rotor blade having such a shape only transfers the tensile and compressive
stresses. The aerodynamic torque in this device comes from an aerodynamic force
generated on the rotor blades that have aerodynamic proles. The silhouette of
a typical two-bladed Darrieus wind turbine is shown in Figure 1b. There are also
modications of this rotor, Figure 1c shows, for example, a rotor with straight
blades. It is a VAWT created in 1980 in the well-known company McDonnell Do-
uglas. The French engineer’s idea was forgotten for almost half a century, then it
was discovered in the mid-1970s, mainly in the USA and Canada. At that time,
extensive research work was undertaken, resulting in the creation of wind farms in
California in Tehachapi, Altamont Pass and San Gorgonio. About 500 machines
of this type were built there with the rated powers of 150–300kW. In 1987, the
largest wind turbine of this type was built – the Canadian EOL with an installed
capacity of 3.8MW .
The Darrieus concept was again forgotten when the idea of oating wind
turbines appeared in recent years. The low centre of gravity of the rotor is the
reason why the Darrieus wind turbine is better for oshore applications than
a traditional wind turbine with a horizontal axis of rotation .
(a) (b) (c)
Figure 1. (a) Persian windmill silhouette ; (b) Darrieus wind turbine ;
(c) H-type Darrieus wind turbine 
Velocity Field around Darrieus Wind Turbine Rotor197
1.2. State of the art
Currently, the most well-known tool for analyzing the aerodynamic perfor-
mance of the Darrieus rotor is the double-multiple streamtube (DMS) model. It
is a method that uses two actuator disks working in tandem. One of these disks
represents the upwind part of the rotor and the second the downwind part. This
method takes into account one component of the ow only, therefore, it is suitable
for rotors with low solidity. The calculation of aerodynamic forces is possible due
to the combination of the actuator disc theory with the blade element theory .
However, the aerodynamic blade loads obtained with this method are not satis-
factory . Strickland et al.  proved in their report that an aerodynamic wake is
determined wrongly when these methods are used. The same authors have deve-
loped a much more accurate approach based on vortex equations. This approach
is still used, but it is more computationally expensive. An interesting approach
is the actuator cylinder model developed by Madsen . In this approach the
curvilinear surface is used instead of a at surface of the actuator disc as in the
case of the DMS approach. On this circular surface Volume forces are applied in
the radial direction.
Nowadays Computational Fluid Dynamics (CFD) methods have been de-
veloped very strongly. They solve the average Navier-Stokes equations whereas
turbulence is solved by means of additional models called turbulence models. Ho-
wever, these methods are very costly computationally because they require mode-
ling using dense grids [7, 8]. Although the rotor calculation shown in Figures 1b
or 1c is theoretically possible using CFD methods, this is not usually done. This is
because these methods incorrectly determine the critical attack angle . In ad-
dition, the use of a large number of processors is not economical. This determines
the need to look for new solutions. The presented article is an extended version of
Rogowski’s article . The article presents a comparison of speed distributions
obtained by means of various numerical approaches and the actuator cell model
(ACM) approach developed by Rogowski .
2. Wind turbine
2.1. Wind turbine rotor
The wind turbine rotor presented in this paper consists of only one straight
blade with the NACA 0012 airfoil. The chord length of the airfoil, , is 9.14cm. The
rotor radius, , was established to be 0.61 m giving the rotor solidity, ,
of 0.15. The most important parameter determining the rotor operation is the
so called tip speed ratio, TSR. This parameter is dened as a non-dimensionless
ratio of the rotor blade velocity (the tangential velocity of the blade), 𝑡, to the
undisturbed ow velocity, 0. The tip speed ratio can be written as:
where: is the angular velocity of the rotor. In the case of the examining rotor the
tip speed ratio value is 5.0. This means that the tangential velocity of the rotor is
198 K. Rogowski and K. Rogowska
ve times larger than the undisturbed ow velocity. According to Paraschivoiu ,
the optimum value of the rotor power coecient, 𝑝, for rotors that have a low
solidity value () is obtained for TSRs of 4–5. The rotor power coecient
is dened as: 𝑝power extracted by rotor
For a typical Darrieus-type rotor, the maximum value of 𝑝is around 0.4.
Therefore, in these simulations a rotor operating at the optimal TSR was tested.
For optimal TSRs, the so called, secondary eects are negligible . Moreover,
local angles of attack do not exceed static critical angles of attack, thanks to this,
the phenomenon of dynamic stall does not appear. According to Paraschivoiu 
the secondary eects are associated with e.g.: the presence of struts, the rotor
geometry, the presence of spoilers, etc. During simulations the angular velocity of
the rotor was 0.75rad/s and the undisturbed ow velocity was 0.091m/s.
The numerical experiment performed in this work corresponds to the
experiment of Strickland et al. . This experiment was performed in a water
towing tank. The selected operating uid was water due to the lower angular
velocity of the rotor at the same tip speed ratio. In addition, studies conducted
in water made it easier to visualize the ow eld. More information about the
experiment settings can be found in two reports and in a scientic article [6,
11, 12]. Figure 2 presents a sketch of a one-bladed rotor.
Figure 2. Silhouette of one-bladed vertical axis wind turbine
2.2. Aerodynamic characteristics of the rotor
The Darrieus wind turbine is a lift-driven device consisting of a number of
curved blades. A rotor with straight blades can be called a Darrieus-type rotor.
The principle of the rotor operation results from the creation of a lifting force
on its blades. The local velocity at the rotor, , is lower than the undisturbed
ow velocity. The relative velocity, , is the sum of the vector of the tangential
velocity of the blade and the velocity vector . The angle between the relative
velocity vector and the chord line is called the angle of attack, . This angle
changes with the azimuth angle, , and depends on the tip speed ratio and the
Velocity Field around Darrieus Wind Turbine Rotor199
blade Reynolds number. The azimuth angle uniquely determines the position
of the rotor and is measured as shown in Figure 3. The direction of the drag
is the same as the direction of the relative velocity , whereas the lift force is
perpendicular to this velocity. The aerodynamic force components, the lift force
and the drag, projected onto the normal and tangent directions give the normal
and tangential forces, respectively.
During the experiments of Strickland et al.  both aerodynamic blade
loads as well as velocity proles behind the rotor were measured. The velocity
proles were measured at the distance of one rotor diameter downstream behind
the rotor, as shown in Figure 3. The wake velocity component 𝑥is normalized
by the undisturbed ow velocity 0.
Figure 3. Two-dimensional rotor model; forces acting on the rotor blade and velocity vectors
In this work, numerical methods were used to assess the aerodynamic
properties of the rotor: aerodynamic forces and ow parameters in the rotor area.
The classic full CFD approaches were used as well as the method developed by
Rogowski  – the Actuator Cell Model (ACM).
3.1. Ful l CFD approach
The full CFD approach, for the purposes of this article, means the numerical
approach in which the boundary layer is modeled using a turbulence model and
an appropriate grid near the blade edges. The ow around the Darrieus rotor
(or a Darrieus-type rotor) cannot be considered as steady. Fluctuations in the
rotational torque of a one-, two-and even three-bladed rotor are so large that the
ow must be considered as unsteady. Therefore, the moving mesh technique was
utilized in all simulations. In this concept, the rotor is surrounded by a large square
stationary area, a computational domain. Moreover, there is a smaller circular
area in the vicinity of the turbine rotor which rotates during the simulation with
the same angular velocity as the rotor. The data between these two areas is
200 K. Rogowski and K. Rogowska
exchanged via the interface. Rogowski et al. proved in their previous works [13, 14]
that the ratio of the square side length to the rotor diameter should be at
least 10. Otherwise, the power coecient results may be somewhat overstated.
Interestingly, the result obtained by Rogowski et al. [13, 14] is right both for
drag-driven rotors (such as e.g. Savonius rotors) as well as for Darrieus-type
rotors. Figure 4 shows schematically the concept of the numerical approach and
the boundary conditions: the velocity inlet, the pressure outlet, the walls, and the
symmetry. The symmetry boundary condition can be used as a zero-shear stress
wall boundary condition.
Figure 4. Numerical approach and boundary conditions
In all the cases presented in this article, the rotor was modeled as a two-di-
mensional object consisting of one airfoil rotating with respect to the axis of
rotation. In the two examined cases, the two-dimensional Navier-Stokes (NS)
equations were taken into account for the RNG -turbulence model and for
the laminar model. In the case of the Scale-Adaptive Simulation approach, the
two-dimensional ow was considered using three-dimensional NS equations. The-
refore, the presented approach is essentially a 2.5D approach.
As mentioned in the previous paragraph, one of the approaches used was
the -RNG turbulence model. This model solves two variables: the turbulence
kinetic energy and the rate of its dissipation. This approach was developed using
the so called Re-Normalization Group methods, hence, the acronym RNG.
In the laminar approach, no turbulence model is considered. Two equations
of momentum and equation of continuity are solved only. Therefore, the presented
approach can be treated as a direct numerical simulation (DNS) approach.
The scale-adaptive simulation (SAS) model is an improved unsteady-ave-
raged Navier-Stokes (URANS) approach. It enables the solution of a turbulent
spectrum in an unstable ow. This is a model developed on the well-known shear
stress transport (SST) formulation.
More about the models used can be found, for example, in the ANSYS Fluent
Velocity Field around Darrieus Wind Turbine Rotor201
3.2. Actuator cell concept
The actuator cell model (ACM) is an original approach developed by
Rogowski . Thanks to the moving mesh technique, it is also a model that
allows analyzing unsteady ows. In this approach, the boundary layer around the
airfoil of the rotor blade is not modeled. The presence of the blade is taken into
account by means of momentum sources, which are introduced into the laminar
Navier-Stokes equations. In other words, in the ACM approach aerodynamic loads
do not result from the solution of uid motion equations. These loads can come, for
example, from the blade element theory. The technique devised by Rogowski 
is still being developed. In this article, the authors want to show that the proposed
method can correctly determine the velocity eld around the rotor and behind the
rotor based on the set aerodynamic force function (aerodynamic load as a function
of the azimuth). The function of aerodynamic forces used in the presented research
was created based on the experimental measurements of Strickland et al. . For
a given rotor position determined using an azimuth , the aerodynamic force
components are interpolated and entered into the NS equations as:
where: 𝑐is the volume of the mesh cell, 𝑥and 𝑦are aerodynamic force
components in the CFD solver system. The ACM model has been implemented
in the commercial solver, ANSYS Fluent, which uses the Cartesian coordinate
system. The geometrical model of the rotor tested in this work is related to the
Cartesian coordinate system as shown in Figures 2–4. In the Cartesian coordinate
system, the components of aerodynamic forces are expressed as:
The aerodynamic forces dened by means of Equations (4)–(5), presented in the
result section 4, have been normalized:
where: is the uid density.
3.3. Mesh distribution
Both the ACM approach and full CFD models require the use of calculation
grids. The grid used for the URANS approach with the -RNG turbulence model
and for the laminar model is shown in Figures 5a and 5b. This grid consists of
a structural mesh near the edge of the rotor blade and a non-structural mesh in
the remaining area. The use of the structural grid provides a better representation
of the boundary layer near the edge of the blade. During all simulations, the wall
parameter was kept less than 1. Figure 5c presents the grid used for a 3D
simulation with the SAS model. The grid distribution in the plane perpendicular
202 K. Rogowski and K. Rogowska
to the rotor axis of rotation is the same as in the case of 2D CFD simulations (the
same as presented in Figures 5a and 5b). The thickness of the three-dimensional
computational domain has a length equal to two chords of the blade. Figures 5d
to 5f present the mesh distribution for the ACM approach. Figure 5f shows the
densest resolution of the grid near the cell into which the momentum sources are
introduced. The number of items for the 3D mesh is 3.9 million, for the 2D mesh
it is 131 958 items, whereas in the case of the ACM model, the number of items is
43 707. All the grids presented in this article were thoroughly investigated due to
the cell density. A detailed description of these tests and a detailed description of
these grids can be found in the articles [7, 10].
(a) (b) (c)
(d) (e) (f)
Figure 5. Mesh for full CFD models (Figures a–c) and mesh for ACM model (Figures d–f)
4.1. Aerodynamic blade loads
This chapter presents a comparison of the aerodynamic blade loads obtained
by means of: the laminar model, the SAS approach and the -RNG turbulence
model. The results obtained are compared with the experimental results of
Strickland at al. . As mentioned in Chapter 3.2, in the case of the ACM
model, the aerodynamic blade loads were not determined. The components of
aerodynamic blade loads, in the Cartesian coordinate system, were presented
in a dimensionless form (according to Equation (6)) in Figure 6. The azimuth
changes from 0 to . However, Figure 6 shows the results for the azimuth in
the range from 0 to . It was done to show that the blade loads change the
same in each rotation of the rotor (they are repeated in every rotation). As can
Velocity Field around Darrieus Wind Turbine Rotor203
be seen in Figure 6, the results of the 𝑥force component obtained by means
of dierent numerical approaches are convergent with the experimental values.
Some discrepancies are visible in the case of the 𝑦component in the azimuth
range between 100 and . In the azimuth range up to , it seems that
the best results of the 𝑦force component are given by the SAS approach.
The results of aerodynamic blade loads presented in Figure 6 obtained with
the laminar model are amazingly good. We can see that there are considerable
oscillations of aerodynamic blade loads resulting from the vortex structures that
are formed on the blade surfaces, averaged by e.g. the RNG -model. However,
these aerodynamic loads oscillate around the experimental values or around values
calculated by other methods used.
Figure 6. Aerodynamic blade load components. Comparison between full CFD approach and
experimental data 
4.2. Velocity proles
Figure 7 presents the proles of the ow velocity component 𝑥obtained
at a distance of one rotor diameter after the rotor. The velocity components were
normalized by the undisturbed ow velocity 0. The comparison of numerical and
experimental results conrms the eectiveness of the methods used for analyzing
the aerodynamic wake downstream behind the rotor. The ACM model also gives
very good quantitative velocity results.
4.3. Static pressure distribution
As can be seen in Figure 6, the blade load component values in the azimuth
range between 180 and 360 are rather smaller compared to the values of these
forces in the remaining rotor area. In the trade literature, the rotor area in the
azimuth range between 0 and is commonly referred as the upwind part of
the rotor while the remaining part is called the downwind part of the rotor. In the
upwind part of the rotor, the ow is less disturbed than in the downwind part.
Figures 8–9 present the static pressure distributions for the upwind and
downwind parts of the rotor, respectively. The pressure dierences between these
204 K. Rogowski and K. Rogowska
Figure 7. Velocity proles downstream behind the rotor: experiment (black circles), ACM
(blue solid lines), -RNG (red dashed lines), laminar (magenta dashed lines), SAS (green
areas are expressive. Figures 8f9 also show the pressure distributions obtained
by various numerical methods. If we compare the pressure distributions for the
azimuth of , they look very similar. This is related to the components of the
aerodynamic blade loads 𝑥and 𝑦, which in the range of about –are
almost the same for all calculation methods and for the experiment (cf. Figure 6).
The ACM model indicates pressure results similar to the laminar model and SAS.
Employing the -RNG turbulence model the pressure is more smooth in the
whole azimuth range. The results obtained by means of the laminar model and
the SAS model indicate the appearance of ow instabilities near the trailing edge
of the azimuth of . This is not visible in the case of the two-equation -RNG
model. Many works indicate the possibility of ow detachment in this part of the
rotor area, e.g. . Depending on the rotor, the operating conditions (tip speed
ratio) of this phenomenon may be more or less strong. This is a surprising eect
because, according to previous observations , for the optimal tip speed ratios,
the local blade angles of attack should be small and the stall phenomenon should
4.4. Vorticity magnitude
Another method of assessing the velocity eld used in this work is to com-
pare the vorticity magnitude distributions (Figure 10). The vorticity describes the
Velocity Field around Darrieus Wind Turbine Rotor205
Figure 8. Static pressure distributions for rotor upwind part
tendency of the ow eld to rotate. In uid mechanics the vorticity pseudovector
describes the local spinning motion of the ow near some point. Mathematically,
in the Cartesian coordinate system, this vector can be dened as:
where: is the del operator, is the velocity eld. In the case of a two-dimensional
ow this vector has only a component:
Figure 10 shows contour maps of the vorticity magnitude, i.e. the square root
of the vorticity vector described by Equation (8). The history of the vorticity
magnitude shown in this gure indicates that the velocity eld in the upwind
part of the rotor is slightly disturbed. In the second part of the rotor, the
interaction between the aerodynamic wake created by the rotor blade in the
upwind part and the same blade is visible (wake-blade interaction). Comparing
the aerodynamic wakes provided by the SAS and ACM models we can observe an
amazing compatibility. It seems that it is much more reliable than in the case of
the -RNG model. The aerodynamic wake provided by the laminar model is, as
expected, heavily disturbed. This model is not equipped with the “mathematical
206 K. Rogowski and K. Rogowska
Figure 9. Static pressure distributions for rotor downwind part
mechanism” of dissipative turbulent kinetic energy as in the case of the RNG
To better understand the phenomenon of blade-wake interaction, the ad-
vantages of the SAS model, which in the case of an unstable ow behaves like
a large eddy simulation (LES) model, and the three-dimensional ow model were
used. Figure 11 shows the history of vorticity in the area of the three-dimensio-
nal Darrieus-type rotor. The gure shows vortex iso-surfaces with the same value
(0.15 1/s) but opposite turns. The blue color suggests the clockwise vorticity di-
rection and the red color the opposite direction.
A ow instability appears for the azimuth of –. This resulting ow
instability evolves and moves with the main stream. Since the velocity of the
rotor blade is larger than the main stream velocity, the blade-wake interaction is
visible for the azimuth of around –. The second but smaller blade-wake
interaction is visible near the azimuth of .
The purpose of this work was to estimate the velocity eld around a one-bla-
ded vertical-axis wind turbine using full CFD models and the author’s ACM me-
thod. The results presented in this article have shown that:
The advantage of testing a single-bladed rotor is the ability to analyze the
interaction of the aerodynamic wake with a rotor blade. The analyses carried
Velocity Field around Darrieus Wind Turbine Rotor207
Figure 10. Contour maps of vorticity magnitude for dierent numerical approaches
out by the authors of this article have shown that the aerodynamic wake behind
a vertical-axis wind turbine is very complicated compared to rotors of classic
wind turbines with a horizontal axis of rotation.
Rotors of the Darrieus-type, due to large uctuations in aerodynamic forces,
cannot be tested using the RANS approach.
The use of standard turbulence models, such as e.g. the -family, is not
sucient to study the aerodynamic phenomena of the Darrieus rotor. The SAS
model shows many more details of the ow.
The ACM model gives satisfactory results in the ow eld. They are comparable
with other CFD methods. The ACM model is further studied in order to be able
to estimate the aerodynamic forces on the basis of given prole characteristics.
The intention of the authors of the paper is also to study three-dimensional
rotors using the developed method.
In its present form, the ACM method is suitable for testing mixers of some
The laminar model surprisingly well predicts the aerodynamic blade loads and
the velocity proles downstream behind the rotor. Rogowski  has also proved
208 K. Rogowski and K. Rogowska
Figure 11. Iso-surfaces of vorticity for SAS simulations
this in the case of another single-bladed rotor. This provides the basis for testing
the transitional laminar-turbulent model.
The presented numerical computations were performed in the Interdiscipli-
nary Centre for Mathematical and Computational Modeling of the Warsaw Uni-
versity. The current work was prepared as part of the computing grant GB73–5.
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