Geoscience Reference
In-Depth Information
Figure 14.1 Measurements of the skewness of the streamwise derivative of
fluctuating temperature as a function of
R
λ
. These were made in a large num-
ber of laboratory and geophysical flows with both mean shear and a mean
temperature gradient. From
Sreenivasan and Antonia
(
1997
). Reprinted, with per-
mission, from
Annual Review of Fluid Mechanics
,
29
, ©1997 by Annual Reviews,
In grid turbulence having a lateral (
y
) gradient of mean temperature,
Tong and
Warhaf t
(
1994
) found even larger skewness (
1
.
8) for
∂θ/∂y
.
Sreenivasan
(
1991
) attributes the skewness of the temperature derivative in shear
flows with a mean temperature gradient to elongated eddy structures with a pre-
ferred orientation. Their signature is the “ramp-cliff” nature of temperature signals
(Figure 7.2)
observed in heated jets and in the surface layer on a sunny day. In this
model the skewness is due largely to the “cliffs” - strong, one-signed temperature
gradients. Thus it appears that scalar-derivative skewness is tied to the large-scale
structure of the flow, in conflict with the ideas underlying the hypothesis of local
isotropy.
∼
14.5.3 The maintenance of local anisotropy
We can gain some insight into local anisotropy of a scalar field
†
through the con-
servation equation for scalar derivative skewness. In the process we'll see that the
conservation equations for moments of derivative fields, and their scaling rules,
differ in important ways from those for the energy-containing-range.
We'll begin at the second-moment level. Taking the gradient of the conserved
scalar
equation (1.31)
and using our tilde notation for full fields yields an equation
for
g
i
=
˜
∂
c/∂x
i
,
˜
∂
2
˜
∂
∂t
+
u
j
∂
g
i
˜
g
i
˜
g
j
∂
u
j
g
i
˜
∂x
j
=−˜
∂x
i
+
γ
∂x
j
∂x
j
.
(14.29)
†
The author is indebted to Chenning Tong for discussions on this topic.
