Geoscience Reference
In-Depth Information
Because of the stretching term on the right side, the scalar gradient, like vorticity,
is not a conserved variable. But their dot product
g
i
ω
i
is conserved
(Problem 1.12)
.
If we decompose the variables in
Eq. (14.29)
into mean and fluctuating parts,
g
i
=
˜
G
i
+
g
i
,
u
i
=
˜
U
i
+
u
i
,
(14.30)
the equation for the fluctuating gradient is
∂g
i
∂t
+
U
j
∂g
i
u
j
∂G
i
u
j
∂g
i
u
j
∂g
i
∂x
j
∂x
j
+
∂x
j
+
∂x
j
−
∂
2
g
i
∂x
j
∂x
j
.
G
j
∂u
j
g
j
∂U
j
g
j
∂u
j
g
j
∂u
j
=−
∂x
i
−
∂x
i
−
∂x
i
+
∂x
i
+
γ
(14.31)
Multiplying
Eq. (14.31)
by 2
g
i
, averaging, and rewriting the molecular term and
dropping its diffusion part yields the evolution equation for the gradient variance:
2
u
j
g
i
∂G
i
∂g
i
g
i
∂t
+
U
j
∂g
i
g
i
∂x
j
∂x
j
+
g
i
g
j
∂U
j
∂x
i
+
g
i
∂u
j
+
∂x
i
G
j
∂u
j
g
i
g
i
∂x
j
2
∂u
j
2
γ
∂g
i
∂x
j
∂g
i
∂x
j
.
+
+
∂x
i
g
i
g
j
=−
(14.32)
The first four terms (counting the bracketed terms as one) on the left side and the
term on the right side of
Eq. (14.32)
are of types that also appear in the equations for
second moments of energy-containing range variables - in order, local time change,
mean advection, mean-gradient production, turbulent transport, and (on the right
side) molecular destruction. The fifth term on the left side is a new type that we'll
call
turbulent production
.
We'll scale the terms in
Eq. (14.32)
as in
Chapter 5
, reintroducing the scalar
intensity scale
s
and the Taylor microscale
λ
:
=
ν
∂u
i
∂x
j
ν
u
2
λ
2
u
3
.
∂u
i
∂x
j
s
=
(c
2
)
1
/
2
,
∼
∼
(14.33)
The fluctuating gradients of the scalar and velocity scale as
s/λ
and
u/λ
, respec-
tively, and their mean gradients scale as
s/
and
u/.
We'll take the time scale of
the time-change term as that of the evolution of the turbulent flow structure, the
large-eddy time scale
/u
. We'll assume that
γ
∼
ν
.
Since the far right side of
Eq. (14.33)
implies that
/λ
∼
uλ/ν
≡
R
λ
, the Taylor-
microscale turbulence Reynolds number, the relative importance of the terms on
