Geoscience Reference
In-Depth Information
3.1.2 Power Law
Sometimes, instead of the logarithmic profile laws (
3.6
)or(
3.16
), which have been
derived from physical and dimensional arguments, an empirical power law is used
to describe the vertical wind profile:
a
z
z
r
u
ð
z
Þ¼
u
ð
z
r
Þ
ð
3
:
22
Þ
where z
r
is a reference height and a is the power law exponent (sometimes called
the ''Hellmann exponent''). The exponent a depends on surface roughness and the
thermal stability of the Prandtl layer. The analysis of the relationship between
the logarithmic law (
3.6
)or(
3.16
) and the power law (
3.22
) is not easy, because
thermal stability is described quite differently in both formulations. The following
section shows how (
3.6
)or(
3.16
) and (
3.22
) are related to each other and whether
they can be used really interchangeably.
3.1.3 Comparison Between Logarithmic and Power Law
The choice of a suitable way of describing the wind profile is often made by
practical arguments. Although today computer resources set nearly no limits any
more to the rapid integration of complex equations, the power law (
3.22
) is often
chosen due to its mathematical simplicity. It is often claimed that both descriptions
lead more or less to the same results. A comparison of the parameters of the two
profile laws for neutral stratification is given in Table
3.1
.
The following analysis shows theoretically how closely the logarithmic profiles
(
3.6
)or(
3.16
) can be described by a power law (Emeis
2005
). This is not a new
issue as Sedefian (
1980
) has derived theoretically how the power law exponent n
depends on z/z
0
and z/L
*
by equating the slopes of a logarithmic profile and a
power law. As long as the height range over which the two profiles should match is
small the solution given by Sedefian (
1980
) is practical and sufficient. One will
always find a power law with an exponent n that fits to a given logarithmic profile
at a given height.
However, today's tasks in wind engineering (the construction of large wind
turbines and the design of high buildings) often require the extrapolation of the
wind profile over considerable height intervals. For these purposes the two
descriptions are only equivalent if it is possible to find a power law that fits to the
logarithmic profile not only in slope but also in curvature over the respective
range. The following investigation will demonstrate that this is possible only for
certain combinations of surface roughness and atmospheric stability in a stably
stratified boundary-layer flow. We start the analysis for the sake of simplicity with
neutral stratification.
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