Geoscience Reference
In-Depth Information
Figure 3.3 Instantaneous (top left) and ensemble-averaged (top right) plumes
downstream of a continuous point source in turbulent flow. Bottom: concentrations
observed along various paths in those plumes. On a path within the instantaneous
plume the concentration
c
changes only through molecular diffusion, which can
be slow (
Section 3.3.1
). On path A along the centerline of the ensemble-averaged
plume, the downstream broadening of that plume causes
C
to decrease; on path
B, away from the centerline, this broadening causes
C
to initially increase as the
path becomes closer (in plume widths) to the centerline.
˜
In the steady case
Eq. (3.13)
for the mean concentration reduces to
U
∂C
∂cu
i
∂x
+
∂x
i
=
0
,
(3.18)
the effects of molecular diffusion on
C
being negligible. We can write this as
D
m
C
Dt
m
U
∂C
∂u
1
c
∂x
1
−
∂u
2
c
∂x
2
−
∂u
3
c
∂x
3
=
∂x
=−
,
(3.19)
where
D
m
/Dt
m
is the time derivative following the mean motion.
Equation (3.19)
says that on this “virtual mean trajectory”
C
changes solely due to the divergence
of the turbulent flux of
c
-stuff.
If the mean plume is thin compared with distance from the source, the turbulent
flux divergence in
Eq. (3.19)
is dominated by its cross-plume contributions. Taking