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)