Geoscience Reference
In-Depth Information
Figure 4.10
Fluxes of a scalar
property due to flow in the
x direction through a small control
volume.
z
∂
F
F
+
δ
x
∂
x
F
δ
z
x
δ
y
δ
x
y
Consider first how the turbulent fluctuations contribute to the flux of a scalar
property s through the small control volume of
Fig. 4.10
. The instantaneous flux
(rate of transport per unit area) of the scalar in the x direction across the left face of
the cuboid is just the product of velocity and scalar concentration:
u
0
s
0
F
¼
ð
U
þ
Þ
ð
S
þ
Þ
ð
4
:
31
Þ
which, since u
0
¼
s
0
¼
0, has an average value of:
F
u
0
s
0
:
Þ
The first term US is the transport due to advection in the mean flow, and u
0
s
0
is
an additional turbulent flux due the covariance
2
of the fluctuations of u and s.
Consideration of the scalar transport in the y and z
dire
c
tion
s leads to analogous
advective and turbulent flux components VS, WS and v
0
s
0
;
¼
US
þ
ð
4
:
32
w
0
s
0
.
4.3.3
The advection-diffusion equation
Combining all the above fluxes, we can write a conservation statement for the scalar s
in the control volume by comparing transports across opposing faces of the cuboid. In
Fig. 4.10
you can see that the net input due to flow in the x direction is the change in F
between the left and right faces multiplied by the area of the face dydz, which is just:
dydz
þ
@
F
@
¼
@
F
@
dxdydz
¼
@
@
F
F
x
dx
x
dxdydz
x
US
þ
u
0
s
0
:
ð
4
:
33
Þ
Adding the equivalent contributions from flow through the other four faces of the
cuboid, we can write the conservation statement for s as:
þ
@
@
þ
@
@
@
@
þ ¼
@
S
þ
u
0
s
0
þ
v
0
s
0
þ
w
0
s
0
ð
:
Þ
x
US
y
VS
z
WS
4
34
@
t
where
@
s/
@
t is the rate of change of the mean concentration S and we have introduced
S
to represent any non-conservative processes by which the scalar may be produced
or decay.
2
The covariance of two quantities is the time average of their product. Unrelated quantities exhibit
zero covariance.
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