Geoscience Reference
In-Depth Information
Figure 15.4 The approach to the isotropic (4/3) ratio of
F
33
and
F
11
in the inertial
subrange in the 1968 Kansas experiment.
z
is distance from the surface,
L
is
the Monin-Obukhov length, and
κ
1
is streamwise wavenumber. From
Wyngaard
(
1973
).
Again, different normalizations of one-dimensional spectra are used. Sometimes
a form is used wherein the integral over the half-line is the variance, in which case
the factor 9/55 becomes 18/55.
An approach to isotropy in the inertial subrange in a turbulent shear flow should
be manifested also in a stress cospectrum that falls faster than the energy spectrum.
Lumley
(
1967
) has argued that in the
ine
rtial range
Co
13
(κ)
, the spherically aver-
aged cospectrum of kinematic stress
uw
, depends on the mean shear
U
,
,and
κ
.
Since the dependence on
U
should be linear, on dimensional grounds we then have
1
/
3
U
κ
−
7
/
3
f
U
(κ
2
)
1
/
3
2
.
Co
13
(κ)
∼
(15.86)
The argument of the function
f
here can be interpreted as
(U
/u
)
2
,where
u
is the
strain rate of an eddy of size 1
/κ
(Problem 15.23)
. When this argument is small the
turbulent strain rate dominates,
f
→
constant, and this becomes
1
/
3
U
κ
−
7
/
3
.
Co
13
(κ)
∼
(15.87)
This inertial-range behavior has been observed for the one-dimensional cospectrum
1994
).
