Geoscience Reference
In-Depth Information
4.3 Mass flux in scalar diffusion
In
Chapter 3
we discussed molecular and turbulent diffusion of an effluent released
from a point source into a flow. We showed that the turbulent problem, which
involves turbulent mixing, molecular diffusion, and ensemble averaging, has some
“virtual” aspects. Here, after discussing the solution for the laminar case, we will
present G. I. Taylor's treatment of the turbulent problem.
4.3.1 Laminar flow
Figure 4.2
shows a simple problem in molecular diffusion. A line source of strength
Q
(units of mass emission rate per unit length), located at
x
0and
oriented in the
y
-direction, discharges a conserved effluent into a uniform laminar
flow of constant velocity
u
i
=
=
0
,z
=
(u,
0
,
0
)
. Under steady conditions the conservation
equation (1.31)
for the effluent reduces to a balance between streamwise advection
and molecular diffusion in the streamwise and vertical directions:
u
∂c
γ
∂
2
c
.
∂
2
c
∂z
2
∂x
=
∂x
2
+
(4.6)
We will make the “thin-plume” approximation that the streamwise length scale
of the plume is much larger than the plume thickness. This allows us to neglect
streamwise diffusion, and
Eq. (4.6)
reduces to
γ
∂
2
c
u
∂c
∂x
=
∂z
2
,
(4.7)
Figure 4.2 An artist's sketch, exaggerated in the vertical (more so for the left panel),
of diffusion of a conserved scalar from a crosswind (
y
-direction) line source in a
uniform flow. Left: The profiles of concentration
c
and lateral flux
γ∂c/∂z
in
laminar flow. The plume width grows
as
x
1
/
2
,
Eq. (4.9)
. Right: Profiles of mean
concentration
C
, lateral turbulent flux
wc
, and eddy diffusivity
K
in the long-time
limit in homogeneous turbulent flow. Here the plume width grows initially as
x
and then as
x
1
/
2
,
Eq. (4.29)
.
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