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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
the width of the mean plume (Figure 3.3) as L p , the turbulent velocity scale as
 
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