Theory development
The actuator cylinder theory begins with the assumption that a vertical slice
of a VAWT can be modeled as a two-dimensional problem.
Figure shows a 2-D representation of the VAWT, with only
one of the blades shown for simplicity, and defines the coordinate system
used in this derivation. The VAWT produces a varying normalized radial force
per unit length q(θ) as a function of azimuthal position along the
VAWT. We define the positive direction for this force q as positive radial
outward (and thus positive radially inward for the loads the fluid produces
on the VAWT). Using the two-dimensional, steady, incompressible, Euler
equations, and (for the moment) neglecting nonlinear terms, the induced
velocities at any location in the plane can be shown to be given by the
following integrals :
u(x,y)=12π∫02πq(θ)x+sinθsinθ-y-cosθcosθx+sinθ2+y-cosθ2dθ-qcos-1yinside and wake+q-cos-1ywake onlyv(x,y)=12π∫02πq(θ)x+sinθcosθ+y-cosθsinθx+sinθ2+y-cosθ2dθ,
where the x, y position is measured from the center of a unit radius
turbine, and velocities are normalized by the freestream velocity. For
evaluation points inside the cylinder the insideandwake term applies, and for evaluation points downstream of the cylinder
both the insideandwake and wakeonly terms apply. These two terms are based on an
integration path through the cylinder, where θ=cos-1y
(Fig. ). For brevity, the derivations of the above
equations are omitted, but details are available in the above-cited papers
from Madsen.
A canonical 2-D slice of a VAWT (only one blade shown) and the
coordinate system used.
Integration path for a point inside the
cylinder.
These two equations for the induced velocities (Eq. ) are
applicable for any x, y location; however, we are primarily interested in
the induced velocities only at locations on the current turbine and on other
turbines. To facilitate computation we discretize the description of each
actuator cylinder into n panels centered at the azimuthal locations:
θi=(2i-1)πnfori=1…n,Δθ=2πn.
Furthermore, as is done in the original version, we assume piecewise constant
loading across each panel. These locations are the points of interest where
will compute the radial forces and subsequently the induced velocities.
In general, we need to compute the induced velocity at every location on a
given VAWT using contributions from all VAWTs (including itself). In the
following derivation we adopt the notation that index I is the turbine we
are evaluating the velocities at, and index i represents the azimuthal
location on turbine I where we are evaluating. Index J will refer to the
turbine producing the induced velocity, and index j will indicate the
azimuthal location on turbine J where the load is producing the induced
velocity (Fig. ).
Using the azimuthal discretization, the induced velocities at a point (x,y) are expressed as a sum of integrals over individual panels. Recall that
Eq. () is normalized based on the current VAWT radius and the
freestream velocity. Because we are now considering multiple VAWTs with
potentially different radii, we need to be more explicit in defining the
normalized quantities. The generalized definitions of the x,y evaluation
positions are
xi*=xi-xJc/rJ,yi*=yi-yJc/rJ,
where xJc is the x location of the center of turbine J. If
I=J (i.e., we are evaluating the turbine's influence on itself), then
this definition is identical to the single-turbine case where the x and y
locations are then distances from the VAWT center normalized by its radius.
The velocity used in normalizing the induced velocities and the radial
loading must be the same, and for that purpose we continue to use the
freestream velocity. We introduce the asterisk superscript on the induced
velocities for clarity (e.g., u*=u/V∞). The expressions for
induced velocity at the cylinder surface depend on whether we evaluate just
upstream of the actuator disk or just downstream. The end result is the same,
as long as we are consistent. In the following derivation we evaluate on the
upstream surfaces for both halves of the actuator disk.
ui*=12π∑jqj∫θj-Δθ/2θj+Δθ/2(xi*+sinϕ)sinϕ-(yi*-cosϕ)cosϕ(xi*+sinϕ)2+(yi*-cosϕ)2dϕ-qn+1-iI=J,i>n/2-qJk+qJn+1-k{I≠J,-1≤yi*≤1,xi*≥1}whereindexksatisfiesθJk=cos-1yi*vi*=12π∑jqj∫θj-Δθ/2θj+Δθ/2(xi*+sinϕ)cosϕ+(yi*-cosϕ)sinϕ(xi*+sinϕ)2+(yi*-cosϕ)2dϕ
In these integrals we have replaced θ in the integration with the
dummy variable ϕ in order to avoid confusion with the θ terms
appearing in the integration limits. The term -qn+1-i arises when
evaluating the influence of a turbine on itself. Because we chose to evaluate
on the upstream surfaces, the upstream half of the VAWT is considered outside
of the VAWT, but the aft half is in the inside of the cylinder. This implies
that for the aft half (i.e., i>n/2) the -q(cos-1y) term must be
added. This corresponds to the loading on the front half of the turbine with
the same y value. Based on our discretization, its location can be indexed
directly as -qn+1-i.
Influence of load at location j of turbine J onto location i
of turbine I.
The following two terms for u arise when turbine I is in the wake of
turbine J. Actuator cylinder theory only includes the wake term when an
evaluation point is directly downwind from a source point (e.g., the blue
region in Fig. ). The condition corresponds to xi*≥1
and -1≤yi*≤1 and xi*2+yi*2≥1. For this wake
area, both of the terms in Eq. are applicable. The index k
corresponds to the location where θJk=cos-1yi*. Note that
cos-1yi* will likely not line up exactly with an existing grid point
θk on turbine J, but we have assumed piecewise constant loading
across a given panel, so k will correspond to the panel that is
intersected.
This model is based on integration paths like those shown in
Fig. and thus ignores the effect of wake expansion and
viscous decay. An alternative is to ignore the wake terms and instead apply a
momentum deficit factor from some other VAWT wake model. For example, we have
developed a reduced-order wake model based on computational fluid dynamics
(CFD) simulations that predicts the velocity deficit behind a VAWT
. Rather than use the above wake terms, one could use the
separate wake model to evaluate the velocity deficit at each control point
and subtract the deficit from u*. Because the focus on this paper is
on actuator cylinder theory, we will use the simple wake model that naturally
arises within the theory itself, but this methodology provides a convenient
hook to insert any wake model.
Wake region from actuator cylinder theory highlighted in blue (and
extending downstream).
For convenience in the computation, Eq. () can be expressed as
a matrix vector multiplication where the loading q is separated from
the influence coefficients.
uI*=AxIJqJvI*=AyIJqJ
The matrix AyIJ is given by
AyIJ(i,j)=12π∫θj-Δθ/2θj+Δθ/2(xi*+sinϕ)cosϕ+(yi*-cosϕ)sinϕ(xi*+sinϕ)2+(yi*-cosϕ)2dϕ.
For the AxIJ matrix we divide the contributions between
the direct influence and the wake influence: AxIJ=DxIJ+WxIJ, where
DxIJ(i,j)=12π∫θj-Δθ/2θj+Δθ/2(xi*+sinϕ)sinϕ-(yi*-cosϕ)cosϕ(xi*+sinϕ)2+(yi*-cosϕ)2dϕ,WxIJ(i,j)=-1if-1≤yi*≤1andxi*≥0andxi*2+yi*2≥1andj=k1if-1≤yi*≤1andxi*≥0andxi*2+yi*2≥1andj=n-k+10otherwise,
where index k corresponds to the panel where θJk=cos-1yi*.
If we are evaluating the influence of a turbine on itself (e.g., I=J)
then the computations in the Ax matrix can be simplified. We can
expand using the definitions for x and y along the cylinder (xi*=-sinθi and yi*=cosθi for i=1…n). As long as
i≠j, then the integral in Eq. () evaluates to
Δθ/2. When i=j the value of the integral depends on which
side of the cylinder we evaluate on. It converges to π(-1+1/n) just
outside of the cylinder and π(1+1/n) just inside. Because we chose to
evaluate on the upstream surface on both halves of the cylinder, the
integral evaluates to π(-1+1/n) on the upstream half of the cylinder
and π(1+1/n) on the downstream half of the cylinder.
DxII(i,j)=Δθ/(4π)ifi≠j(-1+1/n)/2ifi=jandi≤n/2(1+1/n)/2ifi=jandi>n/2WxII(i,j)=-1ifi>n/2andj=n+1-i0otherwise
If a user elects to use a more sophisticated wake model the Wx
term can simply be ignored and a separate momentum deficit factor can be
applied.
Faster computation
The bulk of the computational effort is contained in computing the influence
coefficient matrices AxIJ and AyIJ.
These computations consist of a double loop iterating across all evaluation
positions i on turbine I for each source position j on turbine J
(which is itself contained in a double loop across all turbines I and J).
Fortunately, some of this computation can be simplified. The expressions in
Eqs. (), (), and () apply for the cases
where I=J, or in other words for computing the influence of the turbine
on itself. A significant benefit to this equation form is that the matrices
depend only on the discretization of the cylinder and not on the details of
the blade shape or loading. For a preselected number of azimuthal segments
(e.g., n=36), these matrices can be precomputed and stored. This is true
no matter what size radius the VAWT is.
If I≠J, some reduction in computational requirements is also possible.
For each VAWT pair (I≠J), if the two VAWTs are of equal radius, then
pairs of influence coefficients between them are exactly the same. As seen in
Fig. , the distance vector from the center of one turbine
to the evaluation point on a separate turbine is exactly equal and opposite
to a vector originating from the center of the other turbine and terminating
at an azimuthal location diametrically opposite to the first evaluation
point's azimuthal location. As long as these two VAWTs are of equal radius,
these two vectors will always be equal and opposite. This corresponds to
x* and y* switching signs in Eqs. () and
(). However, the evaluation locations are always 180∘
apart in location. This corresponds to switching the sign on all sin and
cos terms. The two sign changes cancel out and thus the two evaluation
coefficients will be exactly the same. In other words, for all pairs of VAWTs
that are of equal radii, only one set of influence coefficients need be
computed. The influence coefficients for the other VAWT can be mapped over
directly. In equation form this is given by
DxJI((i+n/2)modn,(j+n/2)modn)=DxIJ(i,j),∀i=1…n,j=1…nifrI=rJ
and similarly for Ay. Note that there is no symmetry in the wake terms
(Eq. ). If a second turbine is in the wake of the first, the
first turbine will clearly not be in the wake of the second turbine.
The influence coefficient calculations between a pair of VAWTs will
always have paired locations that have exactly equal and opposite distance
vectors if the two VAWTs are of equal radius. These two evaluation locations
result in the exact same influence coefficients, reducing the amount of
calculations that must be performed.
Finally, we can reduce the number of computations required for VAWTs that
have large separation distances. If a VAWT pair has a large separation
distance (e.g., xIc-xJc2+yIc-yJc2>10rI), then when
iterating across index i the value for positions xi and yi will
change very little. The computation can be simplified by neglecting these
very minor changes and instead using the distance between VAWT centers
(independent of i):
xi*→xIc-xJc/rJ,yi*→yIc-yJc/rJ.
With this simplification the matrices in Eqs. ()
and ()
can be computed by iterating only in j and filling an entire column per
iteration. Additionally, for these large separations the wake terms should be
negligible and can be skipped in the computation.
Body forces
With the induced velocities u* and v*, we can compute the
body forces produced by the VAWT. The volume forces produced by the VAWT are
modeled as acting along an infinitesimally small radial distance and in a
direction normal to the surface of the cylinder (the tangential component is
much smaller than the normal force and can be reasonably neglected in the
volume forces of the Euler equations). The radial volume force is
fr(θ)=Fr′rjΔθΔrLρV∞2,
where Fr′ is an azimuthally averaged radial force per unit length
in a direction pointing into the center of the cylinder, rj is the radius
of the local VAWT cross section, and rjΔθΔr is the
in-plane area across which the force acts (Fig. ). The last
term comes from the normalization of the Euler equations, where L is some
relevant length scale.
In-plane area for volume force at a given azimuthal station.
Because the force acts across an infinitesimal small radial distance, the
radial force acts as a pressure jump
q(θ)=limϵ→01L∫rj-ϵrj+ϵfr(θ)dr=limϵ→01L∫rj-ϵrj+ϵFr′rjΔθdrLρV∞2dr,=Fr′rjΔθ1ρV∞2.
Here, the 1/L is necessary to be consistent with the normalization. It does not
matter which reference length is used in normalizing q(θ) because the
length scales cancel.
Figure shows the relative components of velocity in the
frame of the airfoil. It consists of contributions from the freestream
velocity, the velocity due to rotation, and the induced velocities from
itself and other turbines.
Vj=V∞(1+uj)x^+V∞vjy^-Ωjrjt^
Relative components of velocity in the frame of the
airfoil.
Using the following coordinate transformations,
x^=-cosθjt^-sinθjn^y^=-sinθjt^+cosθjn^,
the velocity can be expressed in the n^–t^ plane as
Vj=-V∞(1+uj)sinθj+V∞vjcosθjn^+-V∞(1+uj)cosθj-V∞vjsinθj-Ωjrjt^.
These velocity components are depicted in Fig. .
Components of velocity resolved into n^-t^ plane.
If we define the magnitudes
Vnj≡V∞(1+uj)sinθ-V∞vjcosθ,Vtj≡V∞(1+uj)cosθ+V∞vjsinθ+Ωjrj,
then
Vj=-Vnjn^-Vtjt^
and the magnitude of the local relative velocity and local inflow angle
(Fig. ) are
Wj=Vnj2+Vtj2,ϕj=tan-1VnjVtj.
The angle of attack, Reynolds number, and lift and drag coefficients can then
be estimated as
αj=ϕj-βRej=ρWjcμclj=f(αj,Rej)cdj=f(αj,Rej).
This can be rotated into normal and tangential force coefficients (note that
cn is defined as positive in the opposite direction of n^ in
Fig. ).
cnj=cljcosϕj+cdjsinϕjctj=cljsinϕj-cdjcosϕj
Definition of normal and tangential force coefficients.
We can resolve these normal and tangential loads into a radial, tangential,
and vertical coordinate system. In doing so, we will account for blade
curvature, as is often used with VAWTs, an example of which is shown in
Fig. . The total force vector is resolved as
F=12ρW2(-cnn^+ctt^)Δa,
where the negative sign results from the coordinate system definition seen in
Fig. . From Fig. we see that the area of the blade
element is
Δa=cΔs=cΔzcosδ,
and the unit vector n^ can be expressed as
n^=cosδr^+sinδz^.
Thus, the force vector per unit depth (unit length in the z direction) is
F′=ρW2c2cosδ(-cncosδr^-cnsinδz^+ctt^).
We can simplify these expressions for the three instantaneous force
components:
R′=-cn12ρW2cT′=ct12ρW2ccosδZ′=-cn12ρW2ctanδ.
Note that the radial force is unaffected by blade curvature because although
the in-plane normal force varies with the cosine of the local curvature angle
δ (Fig. ), the area over which the force acts varies
inversely with the cosine of the angle. Blade sweep is also permitted;
however, it is assumed that the sweep is accomplished through shearing rather
than rotation. In other words, it assumed that the airfoils are still defined
relative to the streamwise direction as opposed to normal to the local blade
sweep. Thus, sweeping does not increase the area of the blade element.
Cross-sectional length of blade segment for small changes in height.
Blade curvature increases the area of the blade element for unit height, but
sweep has no effect on the blade element area as it is a shearing
operation.
For equating with the actuator cylinder theory, only the radial force is of
interest (but all components will be of use for computing overall power and
loads). Because the blades are rotating we need to compute an azimuthally
averaged value of the radial loading (recalling that the sign convention for
a positive radial loading is inward for loads the fluid produces on the
VAWT):
Frj′=cnj12ρWj2cBΔθ2π.
Substituting this radial loading into Eq. (), we find that the
radial volume force can be expressed as
qj=cnj12ρWj2cBΔθ2π1rjΔθ1ρV∞2.
After simplification, the radial force is
qj=Bc4πrjcnjWjV∞2.
Defining solidity as is typically done for a VAWT (σ=Bc/r), the
normalized radial force per unit length becomes
qj=14πσjcnjWjV∞2.
Correction factor
note that the induced velocities from the linear solution
fit well at low loading, and at high loading they produce good trends but
with significant error in overall magnitude. For a uniform loading across a
2-D actuator disk, this linear solution can be shown to produce the following
relationship between the thrust coefficient and the induction factor (a=-u/V∞):
CTlinear=4alinear.
We can equate this thrust coefficient prediction to that of blade element
momentum theory in order to produce a correction factor for
alinear. We extend the approach used by Madsen to consider more
than just the momentum region. The relationship between the thrust
coefficient and the induction factors varies more generally depending on the
induction factor :
CT=4a(1-a)a≤0.4(momentum)29(7a2-2a+4)0.4<a<1(empirical)4a(a-1)a>1(propellerbrake).
In order to get the same induction factor from the linear solution, as would
be predicted by blade element momentum theory, we need to multiply our
predicted induced velocities (and thus the thrust coefficient) by the
correction factor ka=CTlinear/CT.
The correction factors become
ka=1/(1-a)(momentum)(18a)/(7a2-2a+4)(empirical)1/(a-1)(propellerbrake).
In order to determine the value of a to use in the above equation we first
find the thrust coefficient. The instantaneous thrust coefficient can be
found from Eq. () using the coordinate system definition
that
X′=-R′sinθ-T′cosθ=12ρW2ccnsinθ-ctcosθcosδ.
The instantaneous thrust coefficient is
CTinst=X′12ρV∞2(2r)=WV∞2c2rcnsinθ-ctcosθcosδ,
where the other normalization dimension comes from the distributed loads,
which are a force per unit length in the z direction. To get the total thrust
coefficient we need to compute the azimuthal average:
CT=B2π∫02πCTinst(θ)dθ=σ4π∫02πWV∞2cnsinθ-ctcosθcosδdθ.
From the thrust coefficient we can compute the expected induction factor by
reversing Eq. ():
a=121-1-CTCT≤0.96(momentum)171+372CT-30.96<CT<2(empirical)121+1+CTCT>2(propellerbrake).
Finally, this induction factor allows us to compute the correction factor
from Eq. (). These factors should be multiplied against the induced
velocities, but because that is the quantity we need to solve for, we must
multiply against their predicted values.
Because this correction is derived for an isolated turbine, the correction
factors k1…kN should be precomputed for each individual turbine in
isolation rather than as part of the coupled solve of all turbines together.
Matrix assembly and solution procedure
From the proceeding discussion it should be noted that computing loads
depends on the induced velocities, but computing the induced velocities
depends on the loads. Thus, an iterative root-finding approach is required.
We can assemble the self-induction and mutual induction effects into one
large matrix composed of block matrices. We also need to apply the various
correction factors k for turbine J. To solve all induced velocities as
one large system we will concatenate the u and v velocity
vectors into one vector: w=[u;v]. In the equation
below, the symbol ⊙ represents an element-by-element multiplication.
u1u2⋮uN--v1v2⋮vN=k1k2⋮kN--k1k2⋮kN⊙AxAx12…Ax1NAx21Ax…Ax2N⋮⋮⋱⋮AxN1AxN2…Ax-----------------AyAy12…Ay1NAy21Ay…Ay2N⋮⋮⋮⋮AyN1AyN2…Ayq1q2⋮qN
We now have a matrix vector expression of the form w=Aq, but because q depends on w we must solve
for w using a root-finding method. The residual equation is
f(w)=Aq(w)-w=0.
Any good n-dimensional root finder can be used. This paper uses the
modified Powell hybrid method as contained in hybrd.f of Minpack.
Variations in height
The actuator cylinder theory computes all loads in two-dimensional
cross sections. We can use a representative section to represent the
whole turbine (which is more appropriate for an H-Darrieus geometry, ignoring
wind shear), or we can additionally discretize the turbine along the height
and compute loads at each section.
Example two-dimensional discretization in height and azimuthal
position of swept surface (side view).
For each azimuthal station of interest, the solution is projected onto the
instantaneous locations of the blade discretization as shown in
Fig. . For an unswept blade, this involves just a
straightforward transfer of forces as the blade discretization would
typically be exactly aligned with the surface discretization. However, for
swept blades, interpolation is necessary to resolve the forces along the
curved blade path. Furthermore, for a swept blade, the normal and tangential
directions change along the blade path. For swept blades, each point along
the blade is at some azimuthal offset (Δθ) from a reference
point (e.g., relative to the equatorial blade location), and the total
normal force, tangential force, and torque produced by the blade are (again
Δθ=0 for unswept blades)
Rblade(θ)=∫[R′(θ+Δθ)cos(Δθ)..-T′(θ+Δθ)sin(Δθ)]dzTblade(θ)=∫[R′(θ+Δθ)sin(Δθ)..+T′(θ+Δθ)cos(Δθ)]dzZblade(θ)=∫Z′(θ+Δθ)dzQblade(θ)=∫rT′(θ+Δθ)dz.
Now that the forces as a function of θ are known for one blade, the
forces for all B blades can be found. We let ΔΘj represent
the offset of blade j relative to the first blade:
ΔΘj=2π(j-1)/B.
The resulting forces in the inertial frame are then
Xall-blades(θ)=∑j=1B-Rblade(θ+ΔΘj)sin(θ+ΔΘj)-Tblade(θ+ΔΘj)cos(θ+ΔΘj)Yall-blades(θ)=∑j=1BRblade(θ+ΔΘj)cos(θ+ΔΘj)-Tblade(θ+ΔΘj)sin(θ+ΔΘj)Zall-blades(θ)=∑j=1BZblade(θ+ΔΘj).
In this representation the velocities at each height can be different to
account for wind shear or other wind distributions. This derivation is
provided for completeness, but because of the increased computational
expense, and to be consistent with the other comparisons we are making in
this paper, we will focus on using one 2-D slice for the entire turbine.
Power
In addition to the thrust coefficient and instantaneous loads, which have
already been defined, we are also interested in computing the power
coefficient. This is easily computed from the instantaneous tangential load
given in Eq. () (or Eq. ). The torque
(per unit length) is then
Q=rT′,
and the azimuthally averaged power is
P=ΩB2π∫02πQ(θ)dθ.
This is a periodic integral, and care should be taken in integrating near the
boundaries because of the way the discretization is defined (θ1 does
not start at 0). The power coefficient per unit length is then
CP=P12ρV∞3(2r).
Clockwise rotation
The following derivation assumed counterclockwise rotation. For clockwise
rotation a few minor changes must be made. Nothing in the influence
coefficients needs changing as those are purely based on location. The only
change for clockwise rotation is that the direction of t^ is reversed,
as is the direction of the Ωr velocity vector in
Figs. and . The consequence is that
the tangential velocity in Eq. () must be redefined as (note
the two minus signs)
Vtj≡-V∞(1+uj)cosθ-V∞vjsinθ+Ωjrj.
Additionally, the change in tangential direction affects the computation of
the thrust coefficient. In Eq. () the sign is reversed on the
second part of the equation. The consequence is that the total thrust
coefficient (Eq. ) would be computed as
CT=σ4π∫02πWV∞2cnsinθ+ctcosθcosδdθ.
For transferring loads to an inertial frame, or for computing total blade
loads with curved blades, a couple more changes are required.
Equation () replaces the plus sign in front of
Tblade with a minus sign (for both the X and Y equation) and
Eq. () is modified as
Rblade(θ)=∫[R′(θ+Δθ)cos(Δθ)..+T′(θ+Δθ)sin(Δθ)]dzTblade(θ)=∫[-R′(θ+Δθ)sin(Δθ)..+T′(θ+Δθ)cos(Δθ)]dz.
Two turbine interactions
This methodology was implemented and made open-source
(https://github.com/byuflowlab/vawt-ac) in Julia, which is a dynamic
programming language designed for scientific computing
(http://julialang.org). For single-turbine cases the implementation was
verified against Madsen's results . Our focus here is on
multiple-turbine cases, and for simplicity on two turbine cases.
The first configuration we examine is the Mariah Windspire 1.2 kW VAWT,
which has been the subject of multiple experimental and computational studies
. Specifically, we compare against
published 2-D unsteady Reynolds-averaged Navier–Stokes
(URANS) simulations of two closely interacting turbines
. This paper was one of the few interacting VAWT studies
with sufficient detail to make a good comparison. The turbine parameters for
this turbine are shown in Table . Wind tunnel data for the
lift and drag coefficients of this airfoil are available for a few different
conditions .
Mariah Windspire 1.2 kW VAWT parameters.
Diameter
0.6 m
Chord
0.128 m
Number of blades
3
Height
6.1 m
Airfoil
Du-06-W-200
Before comparing turbine pairs, we compared the performance of the isolated
turbine to experimental data. The National Renewable Energy Laboratory (NREL) conducted field studies with the same
turbine while installed at the National Wind Technology Center
. The isolated power coefficient and power, as a function
of wind speed, are shown using the actuator cylinder method as compared to
the experimental data in Fig. . The computational
method overpredicts the power somewhat, which is not surprising considering
that our 2-D simulations do not include the tower, struts, or tip losses and
predict aerodynamic rather than electrical power. The published CFD data
overpredict power by about the same percentage relative to experimental
data, for similar reasons . We were not able to compare
directly to the CFD data set, because the corresponding experimental data set
from Windward Engineering was no longer available (in particular the rotation
speed schedule was needed). Instead, we compared to the NREL data set, which
tabulated the corresponding rotation speed for each wind speed. Overall, the
agreement in the power performance trends is quite reasonable.
Power and power coefficient (separate y axes), as a function of
wind speed for the Windspire 1.2 kW turbine. Lines are from the actuator
cylinder method (AC), and circles are from the NREL experimental
data.
For a pair of closely spaced turbines we used the same conditions as in the
CFD study, namely two identical turbines separated by two diameters (center
to center), with the incoming wind direction varied around a full
360∘. The tip-speed ratio was set at 2.625 for both turbines, which
is an average of the two tip-speed ratios used in Zanforlin's study (2.55,
2.7, which produce essentially the same results). The relative power of the
pair of turbines, as compared to their power in isolation, is plotted versus
the wind direction in Fig. . Also shown for comparison
are the CFD results from the URANS study . While small
differences exist, the overall agreement is very good. Benefits on the order
of 5–10 % are observed across a relatively wide range on inflow angles.
These results are also in alignment with those observed experimentally for
this same turbine .
Normalized power as a function of wind direction on a polar plot,
with an overlay of the turbine rotation directions. Results shown for both
the actuator cylinder method (AC) and the CFD results of
.
Repeating this exercise for many different separations (e.g., two diameters
was used in the previous case) requires a different visualization approach;
otherwise, the plots overlap and become difficult to see. Instead of changing
the wind direction for a fixed turbine position, we move the turbines around
for a fixed wind position. The position of turbine 1 is fixed, and the
position of turbine 2 is swept in concentric circles ranging from 1 diameter
(no separation) to 6 diameters between turbine centers. The lower bound is of
course unrealistically close, but because some VAWT concepts employ dual
rotors with very close spacings, the smallest possible lower bound was used
to observe trends across a broader range. By moving the turbines we can show
the effect of relative separation and changing wind direction (via rotated
positions) simultaneously.
Figure shows the normalized power
(total power of the turbines relative to their total power in isolation) as a
function of the position of turbine 2 for counter-rotating turbines. Overlays
of turbine 1 (fixed) and turbine 2 (position changed) are shown on the plots.
The blank regions upstream and downstream of turbine 1 are regions where one
of the turbines is in the other's wake. In these cases, the power drops by a
large amount, well below the range of the color bars. The color bar range was
chosen to highlight the mutual interference outside of the wake, which is of
greater interest. We observe large regions of beneficial interference of
around 5–10 % for closely spaced turbines (with benefits exceeding
10 % for very close spacings).
The first turbine is fixed in the center, and the second turbine is
moved around in concentric circles with the wind incoming from the left. The
contours show the normalized power of the two turbines (sum of their power
normalized by sum of their power in isolation). The color scale is centered
on 1.0 so that regions of beneficial interference can be more clearly seen.
The color scale focuses on a small region (0.9–1.1) for the same reason. The
blank areas are waked regions, in which the power drops below the range of
the color scale and so are not
plotted.
For counter rotating turbines, two configurations are possible: the Counter
Up and the Counter Down configuration (Fig. ). In
Fig. the upper half of the figure
corresponds to the Counter Up configuration, and the lower half corresponds
to the Counter Down configuration. For the current turbine and conditions,
the Counter Down configuration shows somewhat larger benefits across a wider
range of inflow angles, consistent with the reported CFD studies. A
co-rotating configuration was also explored, which yielded very similar
regions of beneficial interference. The figures were so similar in these
conditions that a separate plot for the co-rotating case is not shown.
Two cases for a counter rotating pair of turbines. “Counter Up”
refers to the pair with a rotation direction facing upstream at their closest
interface, whereas the “Counter Down” configuration rotates downstream at
their closest interface.
The top row is for the linear actuator cylinder theory, showing
streamlines for an isolated turbine. The left panel shows the induced
velocity only, while the right panel includes the freestream. The bottom row
shows the same conditions, but with the full Euler equations
.
A second turbine and condition set came from a three-dimensional
incompressible Navier–Stokes simulation . This turbine
was a 3.5 kW H-Darrieus from Cleanfield Energy Corporation. The turbine
parameters are listed in Table . Wind tunnel data for the
airfoil were extracted from Sandia wind tunnel tests . The
lift and drag coefficients change significantly at the Reynolds number of
this turbine (275 000), and so to provide the best estimate possible the
coefficient data were interpolated between the two nearest data sets:
Re=160 000 and Re=360 000.
Cleanfield 3.5 kW VAWT parameters.
Diameter
2.5 m
Chord
0.4 m
Number of blades
3
Height
3 m
Airfoil
NACA 0015
Because three-dimensional CFD is more computationally intensive, only one
case was included in that study to compare against. That case consisted of
two turbines in the Counter Down configuration, separated (tower to tower) by
2.64 rotor radii. The turbines were operated at a tip-speed ratio of 1.5,
which was the optimal condition for the single-turbine configuration. The CFD
study reported a decrease in power for the counter-rotating pair, although
the amount of decrease was not specified. Fortunately, torque vs. azimuth
plots were presented. Extracting that data from the plot and integrating
shows that the CFD simulations predicted a 12 % decrease in power for the
counter-rotating pair as compared to the isolated turbines.
Actuator cylinder theory was used for the same turbine at the optimal
tip-speed ratio predicted using AC theory. A decrease in power for the
Counter Down configuration was also observed of about 15 %. However, the
optimal tip-speed ratio differed from the experimental data
. The nondimensional power curves were very similar, but
with a shift in tip-speed ratio. To be consistent, the optimal tip-speed
ratio was used in both cases (2.9 for the actuator cylinder case). Without
more data to compare against, the robustness of this configuration to other
conditions could not be assessed.
A third data set was examined, which came from wind tunnel tests of co- and counter-rotating VAWT pairs . This study used very high solidity rotors and
found beneficial interference for counter-rotating turbines across all wind
speeds tested. For co-rotating turbines, beneficial interference was realized
across some wind speeds, and detrimental interference at others. The turbines
used in this wind tunnel study were inefficient under the conditions tested,
with an extremely small power coefficient of 2×10-3 (corresponding
power of under 0.3 W). The percent increase in power of turbine pairs
was significant, but in absolute terms the power increase was very small.
Similar relative benefits with actuator cylinder theory were observed
depending on the assumed airfoil data. This turbine used a custom-made
airfoil section and, without corresponding wind tunnel data, detailed
comparisons to actuator cylinder theory were not possible.
In addition to comparing power output, we compared the induced velocity field
generated by the VAWT. Ultimately, this induced velocity is what changes the
incoming flow field for a second turbine. The streamlines for the induced
velocity field of the isolated VAWT and the total velocity field (with
freestream added) are shown in Fig. . For
comparison, the streamlines for a solution based on the full Euler equations
are shown below . Although the actuator cylinder theory
uses a linearization and only adds a nonlinear correction, the induced and
total velocity fields compare well to those from the Euler equations. These
induced velocities were also found to compare well to that from an unsteady
panel simulation of a VAWT (see Figs. 3.7 and 3.8 in ).
A properly positioned pair of VAWTs can produce power increases, but one must
be careful in extrapolating results to the wind farm. Even in just the two-turbine case, the
overall benefits may or may not be realized depending on the site conditions.
Although power increases of around 5–10 % may be realized across a wide
range of inflow angles, for inflow angles that create wake interference the
power loss can drop by 50 %. As an example, assuming a uniform wind rose
(wind equally probable in all directions), the data in
Fig. yield an expected value of power of around 90 %
of that in isolation (both using this method and the CFD results). In other
words, with a uniform wind rose the wake losses overwhelm the benefits of
close spacing. A more directional wind rose, on the other hand, could realize
overall power increases.
Additionally, a wind farm consists of many turbines, and accounting for wake
effects across the full wind rose is essential. VAWTs tend to produce shorter
wakes than HAWTs, which can help alleviate some of the spacing challenges.
However, if turbines are intentionally positioned close together to create
beneficial interference, then they also have the potential to create strong
power losses through wake effects.
Beneficial interference may be possible not just for pairs of turbines, but
for carefully positioned arrays of turbines, as demonstrated in 2-D URANS simulations . However, the benefits have only been
explored with one wind direction. Our past research in HAWT wind farm
optimization suggests that when optimizing turbine positioning under
uncertainty of wind direction, the optimal configurations spread out and are
not in aligned rows in order to minimize wake interference
. Further investigation is needed to better
understand how to optimize VAWT farms.