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whose solution
(Problem 4.5)
is the
Gaussian plume
,
Q
√
2
πuσ
e
−
z
2
c(x, z)
=
2
σ
2
.
(4.8)
The “plume width”
σ(x)
is
2
γx
u
1
/
2
2
γ
ux
1
/
2
σ
x
=
σ
=
,
.
(4.9)
Equation (4.9)
shows that
σ
grows more slowly than streamwise distance
x
.If
u
=5ms
−
1
and
γ
10
−
5
m
2
s
−
1
, at 4 cm downstream of the source
σ
is only
1% of that. The thin-plume approximation only improves farther downstream; at a
point 4 m downstream, for example, it is 0.1%.
The vertical flux of
c
here is
=
−
γ∂c/∂z.
From
Eqs. (4.8)
and
(4.9)
this is
γ
∂c
γz
σ
2
c(z).
−
∂z
=
(4.10)
It is sketched on
Figure 4.2
along with the
c
profile. Because of the symmetry of
the problem the
c
flux vanishes on the centerplane; elsewhere the flux is directed
away from the
c
-rich fluid there.
4.3.2 Turbulent flow
Now let's consider the same problem in a homogeneous turbulent flow. An instanta-
neous turbulent plume has an irregular shape; the ensemble-averaged (mean) plume
(the result of averaging many instantaneous, sharp-edged plumes, each different in
detail) is smooth
(
Figure 2.3
)
. The mean concentration profile across it (which has
been measured in homogeneous laboratory turbulent flows) follows the Gaussian
solution
(4.8)
closely.
The ensemble-averaged concentration equation in the homogeneous-turbulence
case is
U
∂C
∂x
∂uc
∂x
∂wc
∂z
=−
−
,
(4.11)
where the mean velocity
U
i
=
(U,
0
,
0
)
and
U
is constant. As in the laminar case
(4.6)
this is a balance between streamwise advection and turbulent diffusion in the
streamwise and vertical directions. The molecular diffusion terms are negligible
everywhere, there being no solid boundaries in this problem.
If
we make the
thin-plume
assumption, which implies here that
∂uc/∂x
∂wc/∂z
,
Eq. (4.11)
reduces to
U
∂C
∂wc
∂z
∂x
=−
.
(4.12)
