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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
Reviews, www.annualreviews.org .
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 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
Observations in a wide variety of flows ( Figure 7.5 ) indicate that while F ∂u/∂x does
increase with R t , this simple model considerably overpredicts that rate of increase.
 
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