-
Vertical-Axis Wind Turbines in Oblique Flow:
Sensitivity to Rotor Geometry
Frank Scheurich Richard E. Brown
University of Glasgow, Glasgow, UK University of Strathclyde,
Glasgow, UK
E-Mail: [email protected] E-Mail:
[email protected]
Abstract
Increasing interest is being shown worldwide in theapplication
of vertical-axis wind turbines for decen-tralised electricity
generation within cities. The dis-tortion of the onset air flow by
buildings within theurban environment might however, under certain
con-ditions of wind speed or direction, cause vertical-axis wind
turbines to operate in oblique flow – inother words in conditions
in which the wind vectoris non-perpendicular to the axis of
rotation of the tur-bine. Little is known about the effect on the
oper-ation of a vertical-axis wind turbine when the windis
perturbed from supposedly optimal conditions. Inthe present study,
the Vorticity Transport Model hasbeen used to simulate the
aerodynamic performanceand wake dynamics, both in normal and in
obliqueflow, of three different vertical-axis wind turbines:
onewith a straight-bladed configuration, another with
acurved-bladed configuration and another with a heli-cally twisted
configuration. The results partly con-firm previous experimental
measurements that sug-gest that a straight-bladed vertical-axis
wind turbinethat operates in oblique flow might produce a
higherpower coefficient compared to when it is operated innormal
flow. The simulations suggest, however, thatsignificantly higher
power coefficients in oblique floware obtained only at higher tip
speed ratios, and in-deed only if the height of the turbine is not
large com-pared to its radius. Furthermore, it is shown thata
vertical-axis wind turbine with blades that are he-lically twisted
around its rotational axis produces arelatively steady power
coefficient in both normal andoblique flow when compared to that
produced by tur-bines with either a straight- or a curved-bladed
con-figuration.
Keywords: vertical-axis wind turbine; oblique flow;numerical
modelling; Vorticity Transport Model
Nomenclature
A swept areaAR aspect ratio, b/cb blade spanc blade chordCP
power coefficient, P/
1
2ρAV 3
∞
CP,Normal power coefficient in normal flowCP,Oblique power
coefficient in oblique flow∆CP unsteady component of the
power coefficient, CP − CP,MeanFt sectional force - tangential
to the chordH turbine heightP powerR radius of the rotor at blade
mid-spanRe blade Reynolds number, Re = ΩRc/νS vorticity sourceu
flow velocityub flow velocity relative to the bladeV∞ wind speedβ
oblique flow angle, i.e.
angle between wind vector and horizon
λ tip speed ratio, ΩR/V∞ν kinematic viscosityψ azimuth angleρ
densityω vorticity, ∇× uωb vorticity bound to rotor bladesΩ angular
velocity of the rotor
1. Introduction
In recent years, there has been increasing interestin the
deployment of wind turbines within cities. Incontrast to current
plans for providing electricity atthe scale of the national grid by
concentrating large-scale wind turbines in on- or offshore wind
farms, an-other potentially effective strategy for harvesting
windenergy, particularly in the urban environment, relieson using
small-scale wind turbines, with rotor diam-eters of only several
metres. These devices can bedistributed within the urban
environment and thuscontribute to a decentralised energy supply in
whichthe urban energy requirement is provided by on-site
-
energy production equipment. The design of a windturbine that
operates efficiently within a city poses asignificant challenge,
however, since the wind in thebuilt environment is characterised by
frequent and of-ten rapid changes in direction and speed. In
thesewind conditions, vertical-axis wind turbines might of-fer
several advantages over horizontal-axis wind tur-bines. This is
because vertical-axis turbines do notrequire a yaw control system,
whereas horizontal-axiswind turbines have to be rotated in order to
trackchanges in wind direction. In addition, the gearboxand the
generator of a vertical-axis turbine can be sit-uated at the base
of the turbine, thereby reducing theloads on the tower under
unsteady wind conditions,and facilitating the maintenance of the
system. Theprincipal advantage of these features of a
vertical-axisconfiguration is to enable a somewhat more
compactdesign that alleviates the material stress on the towerand
requires fewer mechanical components comparedto a horizontal-axis
turbine.
A concern though is that the local distortion to theair flow
that is caused by the presence of buildingsin urban environments
might, under certain condi-tions of wind speed and direction, cause
vertical-axiswind turbines to operate in oblique flow – in
otherwords in conditions in which the wind vector is
non-perpendicular to the axis of rotation of the turbine.Little is
known about the effect on the operation of avertical-axis wind
turbine when the wind is perturbedfrom supposedly optimal
conditions. In what appearsto be one of the only experimental
measurements ofthe behaviour of vertical-axis wind turbines in
obliqueflow, Mertens et al. [1] and Ferreira et al. [2] showedthat
their straight-bladed vertical-axis wind turbineproduced a higher
power coefficient when operated inoblique flow compared to when it
was operated in nor-mal flow – in other words in conditions in
which thewind vector was perpendicular to the axis of rotationof
the turbine.
In the present study, the Vorticity Transport Modelhas been used
to simulate the aerodynamic perfor-mance and wake dynamics, both in
normal and inoblique flow, of three different vertical-axis wind
tur-bines: one with a straight-bladed configuration, an-other with
a curved-bladed configuration and anotherwith a helically twisted
configuration. The resultspartly confirm the observations described
in Refs. [1]and [2] but also suggest that significantly higher
powercoefficients in oblique flow are obtained only at highertip
speed ratios and indeed only if the ratio betweenthe height of the
turbine and the radius of the rotoris sufficiently low. The results
that will be presentedin this paper also indicate that straight-
and curved-bladed vertical-axis wind turbines produce power
co-efficients that vary significantly within one revolution
of the rotor irrespective of whether the turbine is oper-ated in
normal flow or in oblique flow. A turbine withblades that are
helically twisted around its rotationalaxis produces a relatively
steady power coefficient, incontrast, under the same flow
conditions.
2. Computational Aerodynamics
The Vorticity Transport Model (VTM), developedby Brown [3], and
extended by Brown and Line [4],simulates the aerodynamics of wind
turbines by pro-viding a high-fidelity representation of the
dynamicsof the wake that is generated by the turbine rotor.In
contrast to more conventional CFD techniques inwhich the flow
variables are pressure, velocity and den-sity, the VTM is based on
the vorticity-velocity formof the unsteady incompressible
Navier-Stokes equation
∂
∂tω + u · ∇ω − ω · ∇u = S + ν∇2ω (1)
The advection, stretching, and diffusion terms withinEquation
(1) describe the changes in the vorticityfield, ω, with time at any
point in space, as a func-tion of the velocity field, u, and the
kinematic vis-cosity, ν. The physical condition that vorticity
mayneither be created nor destroyed within the flow, andthus may
only be created at the solid surfaces that areimmersed within the
fluid, is accounted for using thevorticity source term, S. The
vorticity source termis determined as the sum of the temporal and
spatialvariations in the bound vorticity, ωb, on the turbineblades,
so that
S = −d
dtωb + ub∇ · ωb (2)
In the Vorticity Transport Model, Equation (1) isdiscretised in
finite-volume form using a structuredCartesian mesh within a domain
that encloses the tur-bine rotor, and then advanced through time
using anoperator-splitting technique. The numerical diffusionof
vorticity within the flow field surrounding the windturbine is kept
at a very low level by using a Rie-mann problem technique based on
the Weighted Av-erage Flux method developed by Toro [5] to
advancethe vorticity convection term in Equation (1) throughtime.
This approach permits many rotor revolutionsto be captured without
significant spatial smearingof the wake structure and at a very low
computa-tional cost compared with those techniques that arebased on
the pressure-velocity-density formulation ofthe Navier-Stokes
equations. Dissipation of the wakedoes still occur, however,
through the proper physicalprocess of natural vortical instability.
The bound vor-ticity distribution on the blades of the rotor is
mod-elled using an extension of lifting-line theory. The
-
lifting-line approach has been modified appropriatelyby the use
of two-dimensional experimental data inorder to represent the real
performance of any givenaerofoil. The effect of dynamic stall on
the aero-dynamic performance of the blades is accounted forin the
VTM by using a semi-empirical dynamic stallmodel that is based on
the method that was proposedby Leishman and Beddoes [6].
Scheurich et al. [7] have validated VTM predic-tions against the
experimental measurements of theperformance of a straight-bladed
vertical-axis windturbine that were made by Strickland et al.
[8].Scheurich et al. [9] have also used the VTM to studythe
influence of blade curvature and helical blade twiston the
performance of vertical-axis wind turbines innormal flow. These
works have shown the VTM tobe an effective tool for resolving the
effects of bladegeometry, aerofoil performance and wake dynamics
onthe behaviour of vertical-axis wind turbines.
3. Turbine Model
The geometry of each of the three vertical-axis windturbines
that are investigated in this paper is illus-trated in Figure 1.
The blades of each turbine are sep-arated by 120◦ azimuth. Starting
from the straight-bladed configuration, shown in Figure 1(a), the
radiallocation of each blade section was displaced using
ahyperbolic cosine distribution in order to yield thetroposkien
shape of the curved-bladed configurationshown in Figure 1(b). The
helically twisted config-uration, shown in Figure 1(c), was then
obtained bytwisting the blades around the rotor axis.
The maximum radius, R, of each rotor is at the mid-span of the
reference blade (‘blade 1’) of the turbine,and is identical for
each of the three different config-urations. This radius is used as
the reference radiusof the rotor when presenting non-dimensional
data forthe performance of the turbine. The tip speed ratio,λ, is
defined as the ratio between the circumferentialvelocity at the
mid-span of the blade, ΩR, and thewind speed, V∞. The orientation
of the blades andspecification of the azimuth angle of the turbine
withrespect to the mid-span of blade 1 is shown in Fig-ure 2. The
key parameters of the rotor, common toall three turbines, are
summarised in Table 1.
Table 1: Rotor parameters.
Number of blades 3Aerofoil section NACA 0015Blade Reynolds
number at mid-span 800, 000Chord-to-radius ratio at mid-span
0.115
Figure 1: Geometry of the vertical-axis wind turbineswith (a)
straight, (b) curved, and (c) helically twistedblades.
Figure 2: Diagram showing the wind direction, thedefinition of
rotor azimuth and direction of positivenormal and tangential
forces, as well as the relativepositions of the rotor blades.
Figure 3 shows the definition of the oblique flowangle, β,
together with a representation of the wakethat is produced by the
turbine with helically twistedblades in oblique flow, as visualised
by plotting oneof the surfaces within the flow field surrounding
therotor on which the vorticity has a uniform magnitude.
The influence of the ratio between the height of theturbine and
the radius of the rotor, henceforth calledsimply the
height-to-radius ratio, on the performanceof the straight-bladed
turbine both in normal and inoblique flow will be discussed in the
following section,whereas an analysis of the influence of blade
curva-ture and helical blade twist on turbine performance
ispresented in section 5.
-
Figure 3: VTM-predicted flow field of the vertical-axiswind
turbine with helically twisted blades at a tip speedratio λ = 3.5
and at an oblique flow angle β = 20◦,visualised by plotting an
isosurface of vorticity.
4. Height-to-Radius Ratio
The power coefficient produced by the straight-bladed
vertical-axis wind turbine for three differentheight-to-radius
ratios is shown in Figure 4 whenthe turbine is operated at three
different tip speedratios in normal flow. The rotor radius was
keptconstant for all the turbine configurations that
wereinvestigated. Thus, a lower turbine height-to-radiusratio
results in a lower aspect ratio of the blades,and consequently, in
a slightly lower aerodynamicefficiency of the turbine. This is the
principal reasonfor the reduction in performance with
decreasingheight-to-radius ratio seen in Figure 4. This figure
isfundamental in understanding the behaviour of theturbine in
oblique flow.
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
Tip speed ratio, λ
Pow
er c
oeffi
cien
t, C
P, N
orm
al
H/R=3.0, AR=26H/R=2.1, AR=18H/R=1.2, AR=10
Figure 4: Influence of turbine height-to-radius ratio(H/R) on
the VTM-predicted power coefficient of astraight-bladed
vertical-axis wind turbine in normalflow at tip speed ratios 2.0,
3.5 and 5.0.
The simplest theory for the performance of avertical-axis wind
turbine in oblique flow would con-sider the rotor from the
perspective of swept-wing the-ory [10]. According to this theory,
the component ofvelocity parallel to the turbine axis would have no
ef-fect on the aerodynamics of the turbine and thus, forthe same
wind speed V∞, the performance of the tur-bine in oblique flow
would be the same as that of thesame turbine in normal flow when
operated at windspeed V∞ cosβ and thus at an increased effective
tipspeed ratio λ/ cosβ. In other words
Cp(λ, β) = Cp(λ/ cosβ, 0) cos3 β (3)
According to this simple theory, whether the powerproduced by
the turbine would increase or decreaseas the flow became more
oblique would depend onthe gradient of the CP vs. λ curve at the
particularoperating point of the turbine: for operating pointsto
the right of the maximum in the CP − λ curvethe effective increase
in tip speed ratio in oblique flowshould result in a tendency for
the power coefficient todecrease, and vice versa. In all cases
though the powercoefficient would tend to reduce at higher oblique
flowangles as a result of its dependency on cos3 β.
Figure 5 shows the power coefficient produced bythe
straight-bladed vertical-axis wind turbine, as pre-dicted by the
VTM for the same three different height-to-radius ratios at the
three different tip speed ratiosas shown in Figure 4, but when the
turbine is operatedwith various angles of oblique flow. The
predictionsof the simulation are consistent with the simple
the-ory, to the extent that the power produced by thethree turbines
under most of the simulated operatingconditions remains relatively
constant or even reducessomewhat as the flow becomes increasingly
oblique.At the lowest tip speed ratio, the performance of allthe
turbines is predicted to be marginally enhancedin oblique flow, but
this is to be expected from thesimple theory given that the
operating points of theturbines under the simulated conditions all
lie in theregime where the gradient of the CP − λ curve is
pos-itive (see Figure 4). Interestingly, though, Figure 5shows a
marked anomaly in the predicted performanceof the turbine with the
smallest height-to-radius ratiowhen compared to the predictions of
the simple the-ory. Indeed, this turbine is predicted to develop
asignificantly higher power coefficient in oblique flowthan in
normal flow – at least when operated at hightip-speed ratio. These
results are consistent with theexperimental study documented in
Refs. [1] and [2].When Figure 5 is taken in its entirety, however,
itshows that the increase in the power output of theturbine in
oblique flow that was observed in the ex-periments should by no
means be taken as evidence ofa general characteristic of
vertical-axis wind turbines.
-
0 1 2 3 4 5 60.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Tip speed ratio, λ
CP
,Obl
ique
/ C
P,N
orm
al
β = 10°
β = 20°
β = 30°
(a) H/R = 3.0
0 1 2 3 4 5 60.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Tip speed ratio, λ
CP
,Obl
ique
/ C
P,N
orm
al
β = 10°
β = 20°
β = 30°
(b) H/R = 2.1
0 1 2 3 4 5 60.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
Tip speed ratio, λ
CP
,Obl
ique
/ C
P,N
orm
al
β = 10°
β = 20°
β = 30°
(c) H/R = 1.2
Figure 5: Influence of turbine height-to-radius ratioon the
VTM-predicted power coefficient of a straight-bladed vertical-axis
wind turbine at three differentoblique flow angles.
The reason why vertical-axis wind turbines withcertain
characteristics might produce higher power co-efficients in oblique
flow than in normal flow is re-vealed in Figure 6. This figure
shows the VTM-predicted vorticity distribution on a plane that is
im-mersed within the wake that is produced by the tur-bine with the
smallest height-to-radius ratio that wassimulated. This plane is
aligned with the wind direc-tion and contains the axis of rotation
of the turbine.The vorticity distribution is depicted at the
instant
when blade 1 is located at 270◦ azimuth. The flowfield is
represented using contours of the componentof vorticity
perpendicular to the plane, thereby em-phasising the vorticity that
is trailed by the blades.The dark rendering corresponds to
vorticity with aclockwise sense of rotation, and the light
rendering tovorticity with a counter-clockwise sense of
rotation.
As the turbine rotates, its blades encounter thewake of the
turbine as they traverse the downwindpart of their azimuthal cycle
(i.e. between ψ = 180◦
and ψ = 360◦). In normal flow, the entire length ofeach blade of
the rotor is immersed in the wake of theturbine during the downwind
portion of its cycle – asshown in Figure 6(a). In oblique flow,
however, theconvection of the wake is skewed, as shown in Fig-ure
6(b), thereby allowing a portion of each blade tooperate, over its
entire azimuth, in a flow regime inwhich the influence of the wake
is reduced significantlycompared to the situation in normal flow.
The resul-tant effect on the tangential force, and, consequently,on
the power that is produced by the turbine, is il-lustrated in
Figure 7. In normal flow, the interactionbetween the wake and the
blades acts effectively tosuppress the loading on each blade during
the down-wind portion of its cycle – see Figure 7(b) – and hencethe
tangential force that is produced during the up-wind segment of the
blade cycle is the major contrib-utor to the power that is produced
by the turbine. Inoblique flow, however – see Figure 7(d) – the
contri-bution to the tangential force from the loading that
isproduced on the undisturbed portion of the blade dur-ing the
downwind portion of its cycle contributes to anoverall increase in
the power that is produced by theturbine. This is despite the fact
that the peak loadingon the blades is reduced somewhat by the sweep
effectalluded to earlier, in other words as a result of the
re-duction of the component of the wind vector that isperpendicular
to the blade.
A simple geometric argument suggests then that thelarger the
height-to-radius ratio of the turbine, thesmaller the proportion of
the downwind blade that isexposed to relatively undisturbed flow by
the effectsof wake skew – and hence the smaller the proportionof
the blade that develops significant tangential forceduring the
downwind portion of its trajectory. Thisobservation suggests thus
that turbines with largerheight-to-radius ratio should be less
susceptible to theeffects of wake skew than those with smaller.
Whenthe wake skew effect is considered together with thebasic
effects of oblique flow that are captured by thesimple sweep-based
theory, it can then be understoodwhy only those turbines with
smaller height relative totheir radius (in other words those that
are more sus-ceptible to the effects of wake sweep) should
exhibitan increase in power output as the oncoming windbecomes
increasingly oblique.
-
(a) Normal flow (β = 0◦)
(b) Oblique flow at β = 20◦
Figure 6: Computed vorticity field surrounding
thestraight-bladed vertical-axis wind turbine with H/R =1.2 at λ =
3.5, represented using contours of the vor-ticity component that is
perpendicular to a verticalplane containing the axis of rotation of
the turbine.Blade 1 is located at 270◦ azimuth.
The results presented in Figure 5 suggest also thatthe effect of
wake skew on the power produced by therotor becomes increasingly
important the higher thetip speed ratio of the turbine. This can be
explainedin terms of the detailed mechanics of the vorticity
dis-tribution in the wake of the turbine. A previous studyby the
present authors [9] has shown the subsequentmutual interaction
between the vortex filaments, oncecreated behind the blades, to
result in a significant co-alescence of vorticity in the immediate
vicinity of theblades during the downwind part of their cycle.
Theeffect of this coalescence appears to be to enhance theinfluence
of the wake in ameliorating the load pro-duced by the blades during
their passage downwind ofthe rotational axis of the turbine. At
higher tip speedratios, the vortical structures in the turbine wake
con-vect more slowly downstream relative to the motion ofthe blades
than at low tip speed ratios and the coales-cence of the vorticity
is more pronounced. Conversely,at lower tip speed ratios the wake
is swept away fromthe rotational trajectory of the blades before
signifi-
0 4590 135
180 225270 315
360
−1
0
1
2
3
Azimuth, ψ [deg]
Blade span
Ft /
(0.
5 ρ
c V
∞2)
(a) Normal flow (β = 0◦)
(b) Normal flow (β = 0◦)
0 4590 135
180 225270 315
360
−1
0
1
2
3
Azimuth, ψ [deg]
Blade span
Ft /
(0.
5 ρ
c V
∞2)
(c) Oblique flow (β = 20◦)
(d) Oblique flow (β = 20◦)
Figure 7: VTM-predicted variation with azimuth ofthe
non-dimensional tangential force along the bladespan of the
straight-bladed vertical-axis wind turbinewith H/R = 1.2 when
operated at λ = 3.5 in normalflow (β = 0◦) and in oblique flow with
β = 20◦.
-
cant coalescence can take place, and consequently theinfluence
of the wake on the loads produced on theblades is enhanced to a
lesser extent by this mecha-nism. Note too that the mutual
interaction betweenthe vortex filaments seems to be responsible for
a non-linearity in the trajectory of the wake that precludesovert
use of a simple geometric model to account forthe effects of wake
skew on the performance of theturbine. This nonlinearity can be
seen quite clearlyin the trajectory of the vortices originating
from thebottom of the rotor that are shown in Figure 6(b).
0 45 90 135 180 225 270 315 360−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
Azimuth, ψ [deg]
∆ C
P
β = 0° β = 20° β = 30°
(a) Straight − bladed
0 45 90 135 180 225 270 315 360−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
Azimuth, ψ [deg]
∆ C
P
β = 0° β = 20° β = 30°
(b) Curved − bladed
0 45 90 135 180 225 270 315 360−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
Azimuth, ψ [deg]
∆ C
P
β = 0° β = 20° β = 30°
(c) Helically twisted
Figure 8: Influence of rotor geometry on the unsteadycomponent
of the VTM-predicted power coefficient ofthree different turbine
configurations with H/R = 3.0in normal flow and in oblique flow at
λ = 3.5.
5. Rotor Geometry
In normal flow, the power coefficients that are pro-duced by
both a straight-bladed and a curved-bladedvertical-axis turbine as
well as the forces that act onthe rotor vary cyclically within one
revolution of theturbine, as shown by Scheurich et al. [9]. These
cyclicloads cause vibrations that can generate noise andthat can
also lead to material fatigue of the turbinestructure. The
variation with azimuth of the power co-efficient is reduced
slightly in oblique flow conditions,as shown in Figures 8(a) and
8(b). This is becausethe larger second local peak in tangential
force that isgenerated between 180◦ and 360◦ azimuth in
obliqueflow, as shown in Figure 7, results in a greater symme-try
in the forces between the blades on the advancingand on the
retreating sides of the rotor. The powercoefficient that is
developed by the helically twistedconfiguration is relatively
steady over the entire az-imuth, in comparison, irrespective of
whether the tur-bine is operated in normal or oblique flow, as
depictedin Figure 8(c).
6. Conclusions
A vertical-axis wind turbine can generate a higherpower
coefficient in oblique flow, compared to thatdeveloped in normal
flow, if its height-to-radius ra-tio is sufficiently small. This is
because oblique flowskews the convection of the wake, thereby
allowing asignificant portion of the blade to operate, over
itsentire azimuth, in a flow region in which the influ-ence of the
wake is significantly reduced comparedto the situation in normal
flow. This effect becomesmore dominant at higher tip speed ratios
since obliqueflow causes the vortical structures in the turbine
wake,which otherwise convect relatively slowly downstream,to be
swept away more efficiently from the rotationaltrajectory of the
blades than at lower tip speed ratios.The variation of the power
coefficient within one rev-olution of the rotor that is observed
both for straight-and for curved-bladed vertical-axis wind turbines
innormal flow is shown to be reduced, to some extent,in oblique
flow. The power coefficient that is pro-duced by a vertical-axis
wind turbine with helicallytwisted blades is fairly steady over the
entire azimuth,by comparison, irrespective of whether the turbine
isoperated in normal or in oblique flow. As such, theresults
presented here show that, in contrast to infer-ences drawn from
limited previous experimental data,the behaviour of a vertical-axis
wind turbine in obliqueflow conditions is influenced by its tip
speed ratio aswell as its geometry - more specifically the
configura-tion of its blades and its height-to-radius ratio.
-
References
1. Mertens, S., van Kuik, G., and van Bussel, G.Performance of
an H-Darrieus in the skewed flow on aroof. Journal of Solar Energy
Engineering , Vol. 125,pp. 433–440, 2003.
2. Ferreira, C., Dixon, K., Hofemann, C., van Kuik,G., and van
Bussel, G. The VAWT in skew: Stereo-PIV and vortex modelling.
AIAA-2009-1219. In Pro-ceedings of the 47th AIAA Aerospace Sciences
Meet-ing, Orlando, USA, 2009.
3. Brown, R. E. Rotor wake modelling for flightdynamic
simulation of helicopters. AIAA Journal ,Vol. 38, No. 1, pp. 57-63,
2000.
4. Brown, R. E., and Line, A. J. Efficient high-resolution wake
modelling using the vorticity trans-port equation. AIAA Journal ,
Vol. 43, No. 7,pp. 1434–1443, 2005.
5. Toro, E. A weighted average flux method forhyperbolic
conservation laws. In Proceedings of theRoyal Society of London,
Series A: Mathematical andPhysical Sciences, Vol. 423, No. 1865,
pp. 401–418,1989.
6. Leishman, J. G., and Beddoes, T. S. A semi-empirical model
for dynamic stall. Journal of theAmerican Helicopter Society , Vol.
34, No. 3, pp. 3–17, 1989.
7. Scheurich, F., Fletcher, T. M., and Brown R. E.Simulating the
aerodynamic performance and wakedynamics of a vertical-axis wind
turbine. Wind En-ergy , in press. Published online in advance of
print(4 June 2010). DOI: 10.1002/we.409
8. Strickland, J. H., Smith, T., and Sun, K. Avortex model of
the Darrieus turbine: an analyti-cal and experimental study. Sandia
Technical ReportSAND81-7017 , 1981.
9. Scheurich, F., Fletcher, T. M., and Brown R. E.The effect of
blade geometry on the loads producedby vertical-axis wind turbines.
Proceedings of the In-stitution of Mechanical Engineers, Part A,
Journal ofPower and Energy, in press. Accepted for publicationin
October 2010.
10. Jones, R. T. Wing Theory , Princeton UniversityPress,
1990.
