Geoscience Reference
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Figure 7.5 Observations of the skewness and flatness factors of the streamwise
velocity derivative in a variety of turbulent flows as a function of
R
λ
. The lines
are the prediction of
Eq. (7.43)
,
(F
∂u/∂x
)
3
/
8
, that is based on the
revised Kolmogorov hypothesis. From
Sreenivasan and Antonia
,
1997
. Reprinted,
with permission, from
Annual Review of Fluid Mechanics
,
29
, ©1997 by Annual
−
S
∂u/∂x
∼
∂u/∂x
∼
u/η,
so the predicted mean value of the fourth power of the derivative is
of order
∂u
∂x
4
u
4
η
4
×
u
4
η
4
×
η
2
λ
2
u
4
η
2
λ
2
.
∼
volume fraction
∼
∼
(7.29)
By the definition of the Taylor microscale
λ
the derivative variance is
(∂u/∂x)
2
∼
u
2
/λ
2
,
so the flatness factor of the velocity derivative goes as
∂x
4
∂u
∂u
λ
η
2
R
1
/
2
F
∂u/∂x
=
∼
∼
∼
R
λ
.
(7.30)
∂x
2
2
t
increase with
R
t
, this simple model considerably overpredicts that rate of increase.