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in all turbulent flows. His second hypothesis is that for spatial scales in the inertial
subrange, statistics depend only on
and not on
ν
.
Batchelor
(
1960
) named these
hypotheses the
universal equilibrium theory
.
7.3.1 Local isotropy
Kolmogorov's hypothesis of local isotropy has in the past been considered to be
generally consistent with observations, but the more recent, persistent evidence
of anisotropy in the fine structure of turbulent scalar fields has caused a shift of
opinion. The most often cited evidence is the nonzero skewness of the streamwise
derivative of fluctuating temperature
θ
,x
in shear flows
(Problem 7.20)
,
θ
,x
θ
,x
θ
,x
S
=
θ
,x
θ
,x
3
/
2
.
(7.26)
A typical
θ(t)
signal measured with a fine-wire temperature sensor is shown
in
Figure 7.3
.
Its streamwise derivative is typically found through “Taylor's
hypothesis” (
Taylor
,
1938
) in the form
θ
,t
−
Uθ
,x
,
(7.27)
with
U
the mean streamwise velocity. With adequate bandwidth
θ
,t
and, with
Eq. (7.27)
, its streamwise derivative
θ
,x
are easily measured. Since as we saw
in
Chapter 5
their principal contributions come from eddies beyond the inertial
subrange, deriv
ative s
tatistics are candidates for the local-isotropy assumption.
One such is
(θ
,x
)
3
, which changes sign under reversal of the
x
-direction and,
hence, must vanish in a locally isotropic field. But according to
Warhaf t
(
2000
),
measurements in the laboratory and the atmosphere show that flows with mean
shear and a mean temperature gradient have temperature-derivative skewness of
magnitude
1
.
Figure 7.2
shows a turbulent temperature signal measured in a heated jet. Its
“ramp-cliff” structure, which has been observed in many other flows, shows domi-
nant, one-signed contributions to the temperature derivative, the sign being dictated
by the imposed large-scale temperature gradient. This suggests a direct connec-
tion between the statistics of very large and very small structures in the scalar
field.
This evidence of local anisotropy startled the turbulence community. In a turbu-
lent flow the eddy size range
/η
R
3
/
t
can be very large; it was plausible that
eddies of such disparate scales are only weakly connected through the long cascades
of energy and scalar variance and do not directly interact. Thus the notion of local
∼
