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so its ensemble average is
.
If the averaging scale
r
is small compared to
,so
that any spatial inhomogeneity has a negligible effect, the ensemble average of the
locally averaged dissipation rate
r
is also
.
Obukhov
(
1962
) discussed the application of this newer scaling hypothesis. The
velocity scale of eddies of spatial scale
r
, which was originally taken as
(r)
1
/
3
,
now depends on
r
and so needs a different symbol:
(
r
r)
1
/
3
,
r
fixed
.
u
r
(r)
=
(7.33)
(
r
r)
1
/
3
is hypothe-
sized to be the velocity scale for the subset of measurements with the specified
r
value. Similarity expressions made from
u
r
(r)
must then undergo a final
averaging over all values of
r
. This causes the predictions of the original and
revised
Kolm
ogo
rov hypotheses to differ; for a similarity expression
f(
r
,r)
, say,
f(
r
,
r)
A new notion here is
conditional scaling
.Thatis,
u
r
(r)
=
=
f(
r
,r)
=
f(,r).
Only if
f
is linear in
r
, which is not often the case,
does
f(
r
,r)
f(
r
,r)
.
The original Kolmogorov hypothesis holds that velocity derivativemoments scale
with
and
ν
so that, for example,
∂u
∂x
=
2
∂u
∂x
3
∂u
∂x
4
ν
3
/
2
ν
2
ν
,
∼
∼
,
∼
.
(7.34)
The skewness and flatness factor of the velocity derivative were thus originally
predicted to be constants,
∂u
∂x
3
∂u
∂x
4
S
∂u/∂x
=
=
constant
,F
∂u/∂x
=
=
constant
,
(7.35)
∂u
∂x
2
3
/
2
∂u
∂x
2
2
which is not observed
(
Figure 7.5
)
. Under the revised hypothesis these moments
become
∂u
∂x
2
∂u
∂x
3
∂u
∂x
4
3
/
2
r
r
ν
,
r
ν
∼
ν
=
∼
,
∼
ν
,
(7.36)
so this skewness and flatness factor are predicted to be
3
/
2
r
r
3
/
2
,
S
∂u/∂x
=
∂u/∂x
=
2
.
(7.37)
As discussed in
Part III
, variables that have high intermittency (such as derivatives
of turbulent velocity and temperature at large
R
t
) are observed to have probability
