Environmental Engineering Reference
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through the disk has been infinitely diluted, or when the wake has grown to an infinite
diameter with a negligible velocity deficit. It is important to establish an
invariant
for this
process, noting that neither mass nor energy flux is conserved in the wake.
If one considers the situation of a single turbine in a uniform flow on flat terrain, it can
be seen that the thrust of the rotor must be manifest in the wake, assuming that the turbu-
lent skin friction on the ground plane is not greatly affected by the wake flow. The thrust
is equal to the rate of change of the momentum deficit in the wake, or
R
W
é
¥
T
=
d
dt
ò
ò
(6-8)
ê
2pr
(
U
-
V
)
dx r dr
= 2pr
(
U
-
V
)
V r dr
0
ë
0
where
T
= rotor thrust force (N)
r
,
x
=
radial and downwind coordinates measured from the rotor center (m)
U
=
free stream wind speed at the elevation of the rotor center (m/s)
V
= local velocity in the wake; function of the radial coordinate
r
(m/s)
R
W
=
effective radius of the wake (m)
Equation (6-8) can be recognized as the general form of Equation (5-9). Any effect of the
ground in creating a non-axisymmetric wake has been neglected here and will be accounted
for later. Equation (6-8) provides the basic conservation law for the wake development.
We now require a “shape” law to define the velocity profile
V = V(r)
and a “scale” law that
will define the effective wake radius,
R
W
.
Wake Geometry Models
It has been observed that wind turbine wakes develop according to several fairly-well
defined regimes at different downwind distances, and these can be idealized as shown in
Figure 6-16. First, there is the flow emerging from the rotor disk itself at section
A
. For
well-designed blades, air loading produces an almost uniform velocity deficit, removing the
same energy at all radii. The wake velocity is assumed to be constant across this section
and equal to (
1 - a
)
U
,
where, as before,
a
is the axial induction factor. The initial wake is
considered to behave like an inviscid slug of uniformly-reduced velocity submerged in an
outer flow, forming a so-called
co-flowing jet
or
potential core.
Proceeding downwind from the rotor to section
B
,
the velocity deficit profile is “top
hat” shaped, and an intense shear layer is developed, attenuating the velocity discontinuity.
It is assumed that this shear layer extends itself outward and inward until the inviscid core
region is eliminated at section
C
. The wake from
A
to
C
is defined as being in the
initial
potential core regime.
Farther downwind at section
D
,
it is assumed that the wake has adopted its asymptotic
“bell-shaped” profile, like the radial velocity distribution far downstream of a body of revo-
lution. This is the beginning of the
fully-developed wake regime.
The profile of the wake
at
C
, where the shear layer has penetrated to the flow center line, is not the same as the far-
wake profile at
D
,
so it is necessary to assume that there is a
transitional regime
of a
certain streamwise extent that joins these two sections. It now remains to establish simple
analytical models for intermediate wake profiles, between the rotor and far downwind, and
the transitions from one to another [Lissaman 1976, 1979].
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