Environmental Engineering Reference
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circulation constant. At the azimuthal positions of y = 90 deg and y = 270 deg, each blade is
“flipped” (
i.e.
, its pitch angle changes sign) to produce a circulation of equal magnitude but
opposite sign. This change of sign in circulation is necessary to maintain a positive force in the
tangential direction. The wake consists of concentrated vortices moving downwind in vortex-
street fashion. As
B
® ¥ and
c
® 0 while the product
Bc
is held constant, the discontinuous
vortex streets become continuous
vortex sheets
, and the entire flow field becomes steady.
The velocity field of the rotor consists of three parts that can be superimposed to ob-
tain the flow field: the free-stream velocity,
U
; the wake-vortex sheet velocity,
v
w
; and the
bound-vortex sheet velocity,
v
b
. Analysis of this vortex sheet model of the flow field about
a giromill shows that the power and thrust coefficients of a VAWT have the same limits as
those of a HAWT, given by Equations (5-15a) and (5-30) [Wilson 1978]. In addition, this
analysis also yields a
shear
(
crosswind
)
force coefficient
,
C
S
, as follows:
0.5 r
U
2
R H
= -
p
S
l
a
2
(5-47)
C
S
=
(1 -
a
)
where
H
= height of the giromill rotor (m)
Fixed-Wake Streamtube Analysis
In order to analyze the aerodynamic behavior of a VAWT in accordance with the stream-
tube approach, first consider an airfoil traversing the idealized path shown in Figure 5-33(a).
This airfoil is assumed to be symmetrical in cross-section and have negligible drag. Between
points
A
and
B
the airfoil moves parallel to the free stream and generates no force.
At point
B
, the airfoil changes direction and sheds vorticity. It then moves across the
wind for a distance
w
, generating both lift and circulation. Again at point
C
, the direction of
motion is changed and vorticity is shed, opposite in sign to that shed at
B
. From point
C
to
point
D
no force is generated. Finally, on the path from
D
to
A
, the airfoil once again gener-
ates lift and circulation, the latter opposite in direction to the circulation along path
BC
. The
wake system that is generated is illustrated in Figure 5-33(b), in which the local circulation
direction is indicated by the arrows and where
G
f
, G
r
= magnitudes of front and rear circulation, respectively (m
2
/s)
Axial Induction of a VAWT Rotor
We may take certain liberties with the idealized path illustrated in Figure 5-32(a) without
changing the wake pattern. In Figure 5-33(c), the width
w
has been reduced relative to the
distance
AB
, and the paths
BC
and
DA
have been made to conform to the path of a Darrieus
rotor blade. Thus, we have a streamtube of a Darrieus rotor. Considering the flow in other
streamtubes to be similar, we arrive at several significant observations:
-- Since flow along
BC
is influenced only by flow inside the vortex street shed by
BC
,
forces on an airfoil traversing
BC
are not influenced by adjacent streamtubes.
-- As the streamtube gets smaller in width, the wake from
BC
appears as a semi-infinite
vortex street. The resulting induced velocity at the front is due only to the semi-in-
finite wake of the front caused by G
f
.
-- An airfoil on the rear path
DA
“sees” an infinite wake caused by the front circulation,
G
f
, and a semi-infinite wake caused by the rear circulation, G
r
. Thus, the rear in-
duced velocity depends on both the front and rear circulations.
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