Geoscience Reference
In-Depth Information
We can diagnose the behavior of
wc
, the vertical component of the turbulent flux
of effluent. In the upper half of the plume we expect upward motion (positive
w
)
to be associated with concentrated effluent (positive
c
) that originates nearer the
centerplane, and downward motion (negative
w
) associated with diluted effluent
(negative
c
) that originates farther from the center
plan
e. (
c
cannot be negative, but
its fluctu
atin
g part
c
can be.) Each leads to positive
wc
. In the same way, we expect
negative
wc
in the bo
ttom
half of the plume. Since the mean flow is symmetric
about the centerplane,
wc
must vanish there, just as the molecular flux does in the
laminar problem. Final
ly,
the vanishing of
c
in the effluent-free flow far away from
the centerplane makes
wc
˜
0there.
These deduced
C
and
wc
profiles in the turbulent-diffusion problem are also
sketched in
Figure 4.2
.
Their similarity to those in molecular diffusion suggests
a similarity between the concentration
equation (4.7)
in laminar flow and the
mean concentration
equation (4.12)
in homogeneous turbulent flow. If there were a
z
-independent eddy diffusivity
K(x)
such that
=
K(x)
∂C
wc(x, z)
=−
∂z
,
(4.13)
then at any
x
the
C
equation (4.12)
would have the same form as
Eq. (4.7)
for
laminar flow,
K(x)
∂
2
C
U
∂C
∂x
=
∂z
2
,
(4.14)
and the mean concentration
C(z)
at any
x
would also be Gaussian, as observed:
e
−
z
2
Q
√
2
πUσ
t
2
σ
t
C(x,z)
=
.
(4.15)
This is the molecular diffusion form
(4.8)
but with the mean velocity
U
and the
width parameter of the mean concentration distribution,
σ
t
(x)
.
The behavior of the eddy diffusivity
K(x)
in this problemcan be deduced through
a Lagrangian analysis as follows.
†
If we substitute the Gaussian solution
(4.15)
into
Eq. (4.14)
for the mean concentration and solve for
K(x)
we find
dσ
t
U
2
K(x)
=
dx
.
(4.16)
At any
x
Ut
the mean concentration
C(x,z)
is proportional to the probability
that
z
p
(t)
, the vertical displacement of an effluent particle
at
time
t
after release, is
=
z
. Thus we can interpret
σ
t
as the Lagrangian quantity
z
p
(t)
, the mean-squared
†
This analysis of G. I. Taylor's 1921 solution is adapted from
Csanady
(
1973
).
