Contaminant and sediment transport by advection and diffusion 201
a concurrent advection and diffusion process. The relative importance of
the two processes is quantified through the Peclet number, P
UL
D
e =
, where
L is a characteristic spatial scale of the phenomenon. P e < 1 indicates that
the diffusion prevails, while P e > 1 indicates that the advection is the predominant process.
8.2 NUMERICAL SOLUTIONS
OF THE TRANSPORT MODEL
The effects of numerical diffusion and dispersion of hyperbolic equations on the accuracy of the computational results have been discussed
in Chapter 3. However, the numerical solution of the general transport
model, governed by a mixed hyperbolic–parabolic equation, poses a unique
computational challenge. For instance, if a ‘diffusive’ numerical scheme
is selected, depending on the Peclet number, the numerical diffusion may
distort the effects of the naturally occurring diffusion.
An explicit finite differences scheme can be obtained by using, for the
advection term, forward differences for the time derivative, and backward
differences for the space derivative (FTBS, forward in time, backward
in space); and, for the diffusion term, a second-order centred differences
scheme:
C
C
t
U
C C
x
D
C
C C
x
i
n
i
n
i
i
n
i
n
i
n
i
n
i
n
+
−
+
−
−
+
−
=
−
+
1
1
1
1
2
∆
∆
∆
( )
2 2
(8.11)
or
C
C U
t
x
C C
D
t
x
C
C
i
n
i
n
i
i
n
i
n
i
n
i
n
+
−
+
=
−
−
(
)+
−
+
1
1
2
1
2
∆
∆
∆
∆
( )
C C i
n
−
(
)
1
(8.12)
It should be noted that for negative U, the second term in Equation 8.11
becomes U
C
C
x
i
i
n
i
n
+
+ −
1
1
∆
and changes accordingly in Equation 8.12. The preceding numerical scheme is stable but diffusive. For small Peclet numbers
the real diffusion is high, and the numerical diffusion becomes insignificant. On the contrary, for Pe > 1, selection of a proper numerical scheme
is very important, since the numerical diffusion may be larger than the
physical one. In those cases it is recommended to apply either the Fromm
scheme or total variation diminishing (TVD) scheme (see Chapter 3). More
specifically, in the TVD scheme the backward diffusive difference used
for the advection is combined with additive diffusive terms involving a
a concurrent advection and diffusion process. The relative importance of
the two processes is quantified through the Peclet number, P
UL
D
e =
, where
L is a characteristic spatial scale of the phenomenon. P e < 1 indicates that
the diffusion prevails, while P e > 1 indicates that the advection is the predominant process.
8.2 NUMERICAL SOLUTIONS
OF THE TRANSPORT MODEL
The effects of numerical diffusion and dispersion of hyperbolic equations on the accuracy of the computational results have been discussed
in Chapter 3. However, the numerical solution of the general transport
model, governed by a mixed hyperbolic–parabolic equation, poses a unique
computational challenge. For instance, if a ‘diffusive’ numerical scheme
is selected, depending on the Peclet number, the numerical diffusion may
distort the effects of the naturally occurring diffusion.
An explicit finite differences scheme can be obtained by using, for the
advection term, forward differences for the time derivative, and backward
differences for the space derivative (FTBS, forward in time, backward
in space); and, for the diffusion term, a second-order centred differences
scheme:
C
C
t
U
C C
x
D
C
C C
x
i
n
i
n
i
i
n
i
n
i
n
i
n
i
n
+
−
+
−
−
+
−
=
−
+
1
1
1
1
2
∆
∆
∆
( )
2 2
(8.11)
or
C
C U
t
x
C C
D
t
x
C
C
i
n
i
n
i
i
n
i
n
i
n
i
n
+
−
+
=
−
−
(
)+
−
+
1
1
2
1
2
∆
∆
∆
∆
( )
C C i
n
−
(
)
1
(8.12)
It should be noted that for negative U, the second term in Equation 8.11
becomes U
C
C
x
i
i
n
i
n
+
+ −
1
1
∆
and changes accordingly in Equation 8.12. The preceding numerical scheme is stable but diffusive. For small Peclet numbers
the real diffusion is high, and the numerical diffusion becomes insignificant. On the contrary, for Pe > 1, selection of a proper numerical scheme
is very important, since the numerical diffusion may be larger than the
physical one. In those cases it is recommended to apply either the Fromm
scheme or total variation diminishing (TVD) scheme (see Chapter 3). More
specifically, in the TVD scheme the backward diffusive difference used
for the advection is combined with additive diffusive terms involving a