Wind Turbine Airfoils and Rotor Wakes - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 403

Two examples will illustrate the investigation of this sensitivity to spacing. In the first,
turbines are placed in a single windwise row, with the constraint that a given number of
units has to be placed on a strip of a given length. This is sufficiently simple that a vari-
ational process can be used to determine an ideal spacing. The optimal distribution is
symmetrical about the center of the strip, with the spacing somewhat larger for the units
near the center [Lissaman et al. 1982]. This implies that increasing the spacing in the
center serves to provide a region of re-energization for the flow, a conclusion supported by
Kaminsky et al. [1987]. The second example is a square array with a constant wind occur-
ring with equal frequency from all directions. An optimal-spacing study of this case also
indicates that it is slightly advantageous to have larger spacing in the interior region, with
a heavier concentration of the turbines near the periphery of the square.
The most important conclusion from these optimal spacing studies of simple arrays is
that uniform spacing gives an array energy output that is only insignificantly less than that
for an optimum spacing, for normal spacings in the range of 6 to 10 diameters. Moreover,
this energy difference is smaller than the implied accuracy associated with the approxima-
tions in the mathematical models. This provides the valuable practical result that a uniform
spacing, at least in the prevailing wind direction, is an excellent initial arrangement which
can then be optimized to account for the actual site features and wind characteristics.

Complex Terrain

In the case of a non-uniform wind field caused by complex terrain it is not possible to
use the basic invariant of the momentum deficit, which is connected to the thrust of the
turbine. In complex terrain the thrust in a plane downwind of the turbine is manifested in
pressure perturbations on the sloping ground as well as in the velocity profile in the wake.
Thus, the momentum deficit in the wake is not conserved for non-uniform flows. To
handle this case we will use the concept of invariance of the total energy in the flow after
dissipative losses due to entrainment have been taken into account [Lissaman et al. 1986].
If there were no losses, as the flow moves into areas of different pressure the wake
speed would change in accordance with the Bernoulli equation connecting fluid speed and
pressure. However, entrainment causes dissipation losses in total head, so that speed cannot
be connected to pressure alone. As a first approximation, it can be assumed that dissipation
(which is a function of shear gradients and flow distortion) is the same for non-uniform and
uniform flows. Thus, our modeling approach for varying ambient pressure is to take the
wake field which would exist over flat terrain, calculate the dissipation for this constant-
pressure case, and then apply this dissipation to the wake flow speed at each turbine
location on complex terrain, the varying pressure case.
For example, consider identical turbines at the two sites shown in Figure 6-17, where
site (a) is flat and site (b) has complex terrain. The wake velocities can be expressed as

(6-10a)

V f = U f (1 - d f )

V c = U c (1 - d c ) (6-10b)

where
V f = wake speed at a given location over flat terrain (m/s)
V c = wake speed at the same location over complex terrain (m/s)
U f , U c = ambient wind speeds for flat and complex terrain, respectively (m/s)
d f = known wake deficit factor, calculated using flat terrain model
d c = unknown wake deficit factor for complex terrain