For negative n values, there is discontinuity in the so-called excluded region,
which serves as a separation between spectra under stability and instability conditions and is associated with an abrupt transition in characteristic lengths of eddies
(Fig. 3.7). For the spectral density curves for components u, v and w, the proposed
empirical functions (Kaimal et al. 1972) are as follows:
f S u ðf Þ
u 2
Ã
¼
102n
ð1 þ 33nÞ
5=3
ð3:152Þ
f S v ðf Þ
u 2
Ã
¼
17n
ð1 þ 9:5nÞ
5=3
ð3:153Þ
f S w ðf Þ
u 2
Ã
¼
2:1n
ð1 þ 5:3n 5=3 Þ
ð3:154Þ
where n is the dimensionless frequency in the form fz=u. To perform dimensional
analysis, the u i component is obtained from Kolmogorov’s Law for the inertial
sublayer (Stull 1994):
S u i ðKÞ ¼ a k e
2=3 K
À5=3
ð3:155Þ
where K is the wavenumber module and a k is the dimensionless Kolmogorov
constant, with a value of 0.55.
The following equation shows the relationship between the spectral power
function and the absolute frequencies and wavenumber module (Kaimal and
Finnigan 1994):
KS u i ðKÞ ¼ fS u i ðf Þ
ð 3:156Þ
After some development, and considering that:
u 02
i ¼ u
2
Ã
ð3:157Þ
Equation (3.156) is transformed to the following expression:
fS u i ðf Þ
u 2
Ã
¼
a k
ð2pkÞ
2=3
/
ð2=3Þ
e
n
ðÀ2=3Þ
ð3:158Þ
If k (von Karman constant) and a k is assigned the values 0.4 and 0.55,
respectively, Eq. (3.158) becomes:
nS u i ðnÞ
u 2
à /
2=3
e
¼
nS u i ðnÞ
ðkzeÞ
2=3
¼ 0:3n
ðÀ2=3Þ
ð3:159Þ
76
3 Characterization of Turbulent Flow in the Surface Boundary Layer