Environmental Engineering Reference
In-Depth Information
ut
u
upper boundary
y'
u(y)
y
u'(y)
W
P
P'
x
x'
tracer at
t
= 0
tracer at
t
velocity distribution
lower boundary
Figure 3.25.
Longitudinal dispersion.
mixing of the tracer within the fluid is described by the
advection-diffusion equation given by
first major assumption (to be subsequently validated)
that the effect of longitudinal dispersion on the attenu-
ation of the tracer concentration is much greater than
the effect of longitudinal turbulent diffusion, and so
longitudinal turbulent diffusion can be neglected in this
analysis. Combining Equations (3.230) and (3.231) and
taking
ε
x
= 0 yields
∂
∂
c
t
∂
∂
c
x
∂
∂
∂
∂
c
x
∂
∂
∂
∂
c
y
+
(3.227)
+
u
=
ε
ε
x
y
x
y
where
c
is the tracer concentration,
t
is time, and
ε
x
and
ε
y
are the longitudinal and transverse turbulent diffu-
sion coefficients, respectively. The tracer concentration,
c
, and the fluid velocity,
u
, can be expressed in terms of
their transverse-averaged values such that
∂
∂
c
t
+
∂ ′
∂
c
t
+ ′
∂
∂
′
+ ′
∂ ′
c
x
c
x
∂
∂
∂ ′
∂
c
y
(3.232)
u
u
∂
′
=
ε
y
Taking the transverse average of each term in Equation
(3.232), and noting that the transverse average of the
primed quantities
u
′ and
c
′ are equal to zero, yields
c x y
( ,
)
=
c x
( )
+ ′
c x y
( ,
)
and
u y
( )
= + ′
u u y
( )
(3.228)
where
∂
∂
c
t
+
′
∂ ′
c
x
0
(3.233)
1
1
u
∂ ′
=
W
W
∫
∫
( )
=
(
)
( )
c x
c x y dy
,
and
u
=
u y dy
W
W
0
0
(3.229)
Subtracting Equation (3.233) from Equation (3.232)
yields
w
here the overbar indicates
a
transverse average, and
u
is independent of
x
, while
c
is a function of
x
. Com-
bining Equations (3.227) and (3.228) gives
=
∂ ′
∂
c
t
+
′
∂
c
x
′
∂ ′
∂ ′
−
′
∂ ′
c
x
c
x
∂
∂
∂ ′
∂
c
y
u
∂ ′
+
u
u
ε
(3.234)
∂ ′
y
∂
(
c
+ ′
∂
c
)
∂
(
c
+ ′
∂
c
)
∂
∂
∂
(
c
+ ′
∂
c
)
∂
∂
∂ ′
∂
c
y
+
+
u
=
ε
ε
x
y
t
x
x
x
y
Further simplification of Equation (3.234) can b
e
achieved by making the additional assumptions that
c
and
c
′ are slowly varying with time and that
(3.230)
.
These assumptions correspond to ∂
c
′/∂
t
≈ 0 and the
terms in square brackets being much smaller than the
second term in Equation (3.234). The physical justifica-
tion for these assumptions is that that as the tracer is
dispersed in the longitudinal direction, transverse gra-
dients in the tracer concentration enhance transverse
diffusion, thereby reducing the tr
a
nsver
s
e variations in
concentration,
c
′(
y
), causing
c c
′
Because of the symmetry that is usually achieved by
viewing a tracer concentration distribution relative to
coordinate axes moving with the mean flow, the follow-
ing transformation can be made
x ut
(3.231)
x
′ = −
where
x
′ is the longitudinal coordinate measured
re
la-
tive to an origin moving at the mean fluid velocity,
u
, as
illustrated in Figure 3.25. At this point, we will make our
, and
c
to vary slowly.
These assumptions lead to the following reduced form
of Equation (3.234)
c c
′
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