Geoscience Reference
In-Depth Information
This prompts two final questions. The first concerns the substantial-derivative
form of a conservation equation, which we often interpret as the equation
following
a parcel
. How can we follow parcels in a turbulent flow if they are rapidly distorted
and annihilated?
The answer is that we need “follow” a parcel only long enough to define this
time derivative. Since the derivative involves the limit process
t
→
0, this is a
vanishingly small time.
A second question concerns mixing: if an advected constituent in a flow is
conserved, how can it mix?
We said that a scalar constituent following
Eq. (1.31)
changes only
through the
effects of molecular diffusion
. Molecular diffusion is the final stage of turbulent
mixing. Without molecular diffusion a scalar constituent cannot truly mix; it will
ultimately be finely but nonuniformly distributed by the turbulence, the spatial scale
of the nonuniformities being that of the smallest turbulent eddies
(
Figure 3.4
b
).
3.4 Space-averaged equations
We stated the “Reynolds averaging rules” in
Eqs. (2.5)
-
(2.8)
. Any linear average
has the distributive property
(2.5)
,and
(2.7)
is a consequence of
(2.6)
.Rules
(2.6)
and
(2.8)
are
• The average of an average is the average:
a
˜
= ˜
a.
(
2
.
6
)
• 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
)
The ensemble average satisfies both of these rules. Now let's consider their
applicability to two other commonly used averages.
The “record average,” the average of a recorded measurement of
f(x,t)
, say,
over an
x
-record of length
L
,is
L
1
L
f
rec
=
f(x,t)dx.
(3.26)
0
Since
f
rec
does not depend on
x
, averaging it over the record yields
f
rec
L
L
f
rec
L
rec
1
L
f
rec
dx
f
rec
,
=
=
dx
=
(3.27)
0
0
so the record average satisfies rule
(2.6)
.
