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∂u
1
∂x
1
2
∂u
2
∂x
1
2
∂u
3
∂x
1
2
1
2
1
2
=
=
,
(14.26)
were approximately verified across the turbulent wake of a cylinder. But since then
numerous deviations from the isotropic forms for derivative covariances,
Eq. (14.18)
for velocity and
Eq. (14.23)
for a conserved scalar, have been documented in low
and moderate Reynolds number shear flows. For example, in a plane jet
Antonia
u
1
,
2
u
1
,
2
2
u
1
,
1
u
1
,
1
=
u
1
,
1
u
1
,
2
(u
1
,
1
u
1
,
1
)
1
/
2
(u
1
,
2
u
1
,
2
)
1
/
2
1
.
8
,
=
0
.
23
,
(14.27)
while under local isotropy these values are 1.0 and 0, respectively. In a nearly
homogeneous turbulent shear flow
Tavoularis and Corrsin
(
1981
) measured values
of 2.2 and
0
.
44, respectively.
Antonia
et al
.
(
1991
) found similar evidence of local
anisotropy in the dissipation-rate tensor in direct numerical simulation of turbulent
duct flow.
Shen and Warhaft
(
2000
) studied the fine structure of the velocity field in grid
turbulence having an unusually large Reynolds number -
R
λ
∼
−
10
4
.
Second-order s
tatistics (
s
pectra, stru
cture functions) showed local isotropy, but at
fifth order, for
(∂u/∂y)
5
/
10
3
,
R
t
∼
6
×
5
/
2
, local anisotropy had emerged.
Mydlarski andWarhaft
(
1998
) found the temperature field in grid turbulence with
a lateral gradient of mean temperature to have strong, Reynolds and Péclet number
independent local anisotropy. The anisotropy was evident at second order, in devi-
ations from
Eq. (14.23)
for derivative variances, and in higher, odd-order structure
functions. They found no Reynolds or Péclet number dependence of temperature
spectra and no qualitative change in the nature of the temperature fluctuations over
the range 30
<R
λ
<
700, in strong contrast to the behavior of the turbulent veloc-
ity field in the same flow (
Mydlarski and Warhaft
,
1996
). They commented that
scalar turbulence shows high-
R
t
behavior even when the turbulent velocity field
has low
R
t
.
As we discussed in
Chapter 7
, local anisotropy of temperature derivative fields
is prominent at third order (
Warhaf t
,
2000
).
Figure 14.1
is a summary of laboratory
and atmospheric observations of the skewness of
∂θ/∂x
versus
R
λ
in flows with
both mean shear and a mean temperature gradient. The sign of this skewness is
given by (
Sreenivasan
,
1991
)
[
(∂u/∂y)
2
]
sgn
dU
dz
sgn
d
dz
.
sgn
(S
∂θ/∂x
)
=−
(14.28)
