Environmental Engineering Reference
In-Depth Information
Two mathematical models or “laws” are commonly used to quantify the vertical profile
of wind speed over regions of homogeneous, flat terrain (
e.g.
,
fields, deserts, and prairies).
These are the
logarithmic/linear law
and the
power law.
The former can be derived
theoretically from basic principles of fluid mechanics. It is valid over large ranges of
altitude and incorporates the phenomenon of
atmospheric stability.
By contrast, the power
law is empirical, and its validity is generally limited to the lower elevations of the
atmosphere. Because of its simplicity, however, the power law is the engineering model
most commonly used to describe wind speed variations with elevation above ground.
Logarithmic/Linear Law for Vertical Profiles of Wind Speed
There are many detailed derivations and discussions of this model in the literature [
e.g.
Plate 1971, Haugen 1973, Panofsky and Dutton 1984]. Therefore, only a brief description
of the principle features of the logarithmic/linear law is given here. The basic equation is
U
=
U
*
(8-10a)
k
[ln(
z
/
z
0
) + y
s
(
z
/
L
s
)],
z
>>
z
0
where
U
*
= friction velocity (m/s)
k = von Karman constant, approximately equal to 0.4
z
= elevation above ground level (m)
z
0
=
empirical surface roughness length (m)
y
s
( ) = atmospheric stability function dependent on
z/L
s
(m/s)
L
s
=
Monin-Obukhov stability length [see,
e.g.
,
Mikhail 1984] (m)
The surface roughness length,
z
0
,
is an empirical parameter which characterizes the
influence of surface irregularities on the vertical wind speed profile. The rougher the terrain
(
i.e.
,
the larger the surface obstructions that oppose the flow of the wind) the thicker will
be the affected layer of air and the more gradual will be the increase of velocity with
height. In the absence of experimental data,
z
0
must be selected on the basis of visual
inspection of the terrain upwind of the turbine and reference to tables such as Table 8-3
[
e.g.
,
Counihan 1975, Kaufman 1977, Frost
et al.
1978]. The parameter
U
*
,
the friction
velocity at the ground, is a function of the surface shear stress and the air density. Since
U
*
is not easily evaluated, the ratio
U
*
/k
.
is generally calculated using a reference wind
speed at a specified reference elevation. Thus
U
*
k
U
r
ln(
z
r
/
z
0
) + y
s
(
z
r
/
L
s
)
(8-10b)
=
,
z
>>
z
0
where
z
r
= reference elevation = 10 m
U
r
=
reference steady wind speed at the reference elevation (m/s)
Atmospheric Stability and Wind Shear
The stability of the atmosphere is governed by the vertical temperature distribution
resulting from radiative heating or cooling of the earth's surface and the subsequent
convective mixing of the air adjacent to the surface. Atmospheric stability states are
classified as
stable, neutrally stable
,
or
unstable.
These states are important to our
modeling of the vertical profile of the steady wind speed because of the different amounts
Search WWH ::
Custom Search