Geoscience Reference
In-Depth Information
• The average of an average is the average:
a
˜
= ˜
a A
=
A).
(2.6)
• The average of a fluctuation is zero:
(
a
˜
−˜
a)
=
0
(a
=
0 ).
(2.7)
• The average of a derivative is the derivative of the average (the commutative property):
˜
˜
˜
˜
a
∂x i =
a
∂x i ;
a
∂t =
a
∂t .
(2.8)
These can be easily proved (Problem 2.1) from the definition (2.3) of the ensemble
average.
It follows from the distributive property that the ensemble average of a product is
a b
˜
=
(A
+
a)(B
+
b)
=
AB
+
Ab
+
aB
+
ab.
(2.9)
From the definition of the ensemble average, Eq. (2.3) , the cross terms vanish
because the mean is a constant in the averaging process (Problem 2.1) :
Ab
=
aB
=
0 .
(2.10)
Thus, using the decomposition
c the Reynolds stress
τ ij in the ensemble-averaged Navier-Stokes equation (1.40) and the corresponding
turbulent scalar flux f i in (1.41) can be written as
u i
˜
=
U i +
u i ,
c
˜
=
C
+
τ ij
ρ = ˜
u i ˜
u j − ˜
u i
u j
˜
=
u i u j ,
i = ˜
c
u i − ˜
˜
c
u i =
˜
cu i .
(2.11)
These fluxes produc ed by ensemble averaging are covariances of the turbulence
field. If a covariance ab is nonzero, a and b are said to be correlated . We'll see in
Chapter 4 that any two turbulent variables tend to be correlated unless mean-flow
symmetry requires otherwise.
2.2.4 A simple example of ensemble averaging
We can demonstrate ensemble averaging with realizations of a random sine wave
in one dimension,
u(x, α) =
sin (κx + φ α ),
α =
1 , 2 ,...,N,
(2.12)
 
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