Wind Turbine Airfoils and Rotor Wakes - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 401

d R W
dt

= k W (s 0 + s G ) (6-9a)

d R W
dx

s 0 + s G
U

= k W (6-9b)

where k W = empirical constant

This model produces a turbulent energy level which increases from free stream to a
higher level upon passage through the rotor disk, is reinforced by the shear layers
immediately downwind of the disk, and then decays to return to the ambient level. Details
are given in [Lissaman 1976].

Transitional Regime

The transition law describing flow from the inviscid core to the far wake is based
simply on establishing a geometric relationship that will provide a smooth transition from
the profile at section C to that at section D in Figure 6-16.

Ground Effects

A refinement of Equations (6-9) takes into account differences between vertical and
lateral ambient turbulence near the ground and accounts for differences in vertical and
lateral growth rates. This causes the wake to adopt an elliptical cross-section downwind of
its original circular shape. Another ground effect is taken into account by the standard
method of reflection , so that the actual wake perturbation at a given downwind station now
becomes the sum of that in the direct flow and that in the image flow. If the perturbations
are linearized, then the thrust invariant is now associated with the thrust of the rotor plus
its image. Thus, conservation of the integral for a single wake automatically takes into
account the ground effect by the presence of the image perturbation. In other words, the
momentum defect “removed” by the ground is “restored” by the addition to the actual wake
of the ground perturbation flow of the image.

Integration of Wakes for Array Effect

In most models it is assumed that the wakes of an array of turbines may be directly
superimposed. This is a linearizing assumption, based on the physical fact that the wake
perturbations caused by a single turbine are relatively small. Normally, velocity deficits are
less than 5 percent of the free-stream velocity by a distance of five diameters downwind of a
rotor. Thus it is a good approximation to disregard any interaction between wakes.
The normal procedure for calculating the wake interference for a given array of turbines
is therefore straightforward and as follows: For the given wind azimuth the most-upwind
unit is selected, and its wake geometry and velocity deficits are calculated for a specified
wind speed and turbulence intensity, progressing downwind through the array. The deficit
at each turbine is tabulated. Turbine control parameters (such as cut-in and rated wind
speeds) may be introduced into the model, as well as different rotor areas and elevations.
Then the most-upwind of the remaining turbines is selected and its inflow velocity deter-
mined. In general, this will be the vector sum of the free-stream flow and the wake