Environmental Engineering Reference
In-Depth Information
streamtubes of rectangular cross-section are then followed from the upstream
undisturbed fl ow to the upstream half of the cylinder where the fl ow in a particular
streamtube is retarded to
U
u
such that:
( 1)
A local pressure drop from
p
+
to
p
−
occurs across the upstream face of the actua-
tor cylinder. Following the streamtube downstream the pressure recovers so that
at a given point somewhere between the upstream and downstream faces of the
cylinder the static pressure returns to the freestream value,
p
∞
(Fig. 10). Here the
local air velocity,
U
a
, is assumed to be
UU
∞
=
(1
−
a
)
u
u
(2 )
UU
=
(1
−
2
a
)
a
∞
u
There is a further pressure drop at the downstream interaction with the actuator
cylinder and the pressure then again recovers to the freestream datum some dis-
tance downstream in the wake which fl ows at velocity
U
w
:
(3 )
UU a
=
(1
−
)
and
U U
=
(1
−
2
a
)
d
a
d
w
a
d
The retardation of the fl ow through the domain of interest leads to an expansion in
the cross-sectional areas of the streamtube, from
A
u
to
A
d
, through the rotor. The
effect of this expansion is accounted for explicitly in the calculation of the magni-
tude and direction of the resultant wind direction at the turbine blades as illustrated
schematically in Fig. 11.
In the case of VAWTs it is generally useful to resolve the lift and drag forces
acting on the blades into components acting normally (radially) and tangen-
tially (chordwise),
F
n
and
F
t
, respectively. In the case of a VAWT with blades
of fi xed pitch, the chord of the blades is generally held perpendicular to the
radius from the rotor axis. The lift and drag coeffi cients,
C
L
and
C
D
, for a given
aerofoil section may be manipulated to provide the non-dimensional normal
A
d
A
u
db
d
b
Figure 11: Illustration of streamtube expansion and nomenclature, after Sharpe [23].
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