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In-Depth Information
Figure 14.2 The flatness factor
F
of the streamwise derivative of temperature mea-
sured in various laboratory and atmospheric flows. From
Sreenivasan and Antonia
(
1997
). Reprinted, with permission, from
Annual Review of Fluid Mechanics
,
29
,
One can show that
(Problem 7.16)
the skewness
S
and flatness factor
F
of a
stochastic variable are related by
F
+
1
|
S
|≤
.
(14.37)
2
This suggests a different criterion for local isotropy - that
S
be much less than this
upper limit:
F
+
1
|
S
|
.
(14.38)
2
Figure 14.2
shows a compilation by
Sreenivasan and Antonia
(
1997
)of
F
for the
temperature derivative in laboratory and atmospheric flows; values are as large
as 100. Thus at large
R
λ
one might argue that
S
∼
1 satisfies the constraint of
Eq. (14.38)
- i.e., that
S
1 is small. But the skewness of
∂u/∂x
plays a dynam-
ically important role in turbulence
(Problem 14.22)
, and it is also
∼
∼
1. From that
point of view
S
∼
1 is not small.
14.5.5 Local isotropy in turbulence models
Given the contrary evidence, what can we say about modeling turbulence as
locally isotropic?
In applications to acoustic and electromagnetic signal propagation, for exam-
ple, the power spectral densities (spectra) of velocity, temperature, and water-vapor
fields
(Chapter 15)
are often modeled as isotropic at large wavenumbers. There
are two aspects of such spectra: their behavior in the inertial range, which lies
