Geoscience Reference
In-Depth Information
5.2 The fluctuation equations
For two turbulent variables
a
and
b
b
, the sum of e
nse
mble-mean
and fluctuating parts, we can derive the equation for their covariance
ab
as follows.
Since ensemble averaging and differentiation commute, we can write
a
˜
=
A
+
=
B
+
∂ab
∂t
=
a
∂b
∂t
+
b
∂a
∂t
.
(5.1)
Thus we form the conservation equation for
ab
by multiplying the
b
equation by
a
, multiplying the
a
equation by
b
, adding, and ensemble averaging.
To derive the conservation equation for a fluctuating variable we:
• Introduce the mean-fluctuating decomposition into the conservation equation for the full
variable; we call this the
full
equation.
• Ensemble average the full equation to produce the mean equation.
• Subtract the mean equation from the full equation to find the fluctuating equation.
We'll take a conserved scalar through this process. With
u
i
=
˜
U
i
+
u
i
,
c
˜
=
C
+
c,
Eq. (1.31)
for the full variable is
γ
∂
2
(C
∂(C
+
c)
u
j
)
∂(C
+
c)
+
c)
+
(U
j
+
=
.
(5.2)
∂t
∂x
j
∂x
j
∂x
j
Ensemble averaging and the averaging rules in
Chapter 2
yield the mean equation:
∂
2
C
∂x
j
∂x
j
.
∂u
j
c
∂C
∂t
+
U
j
∂C
∂x
j
+
∂x
j
=
γ
(5.3)
Subtracting the mean
equation (5.3)
from the full
equation (5.2)
produces the
equation for the fluctuation:
∂x
j
u
j
c
u
j
c
=
∂
2
c
∂x
j
∂x
j
.
∂c
∂t
+
∂c
∂x
j
+
u
j
∂C
∂
U
j
∂x
j
+
−
γ
(5.4)
The full variable
c(
x
,t)
could be a constituent density, in which case it must be
positive. But its fluctuation has no such restriction:
c
˜
=˜
c
−
C
is negative when the
full variable
c(
x
,t)
is less than its ensemble mean
C(
x
,t)
.
The same process yields the equation for the fluctuating velocity field:
˜
∂x
j
u
i
u
j
−
u
i
u
j
=−
∂
2
u
i
∂x
j
∂x
j
.
∂u
i
∂t
+
U
j
∂u
i
u
j
∂U
i
∂
1
ρ
∂p
∂x
i
+
∂x
j
+
∂x
j
+
ν
(5.5)
