Geoscience Reference
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where A 0 is the cross-sectional area of still water zones, and A is the cross-sectional
area of main flow. The momentum equation (5.2) or (5.54) is not changed, in which
only the main flow area A is used.
5.3 1-D CALCULATION OF SEDIMENT TRANSPORT
5.3.1 1-D equilibrium sediment transport model
The sediment continuity equation (5.38) is used to determine bed change in the equi-
librium sediment transport model. For uniform sediment in a rectangular channel, it
becomes
p m )
z b
+
q t
(
1
=
0
(5.123)
t
x
where z b is the bed elevation, and q t is the sediment transport rate per unit channel
width. As demonstrated in Eq. (5.37), q t is determined using a sediment transport
capacity formula.
Many numerical schemes have been used to discretize Eq. (5.123). Saiedi (1997)
summarized some of them. For example, applying the Preissmann scheme expressed
as Eqs. (4.35) and (4.36) to Eq. (5.123) yields
p m ) ψ
z b , i + 1 + (
ψ)
1
z b , i
1
q n + 1
t , i
q n + 1
t , i
(
1
+
x [ θ(
)
+
1
t
q t , i + 1
q t , i ) ]=
+ (
1
θ)(
0
(5.124)
where
z b , i is the change in bed elevation at cross-section i in time step
t , i.e.,
z n + 1
b , i
z b , i . The spatial and temporal weighting factors in Eq. (5.124) were
given various values, e.g.,
z b , i
=
0.5 by Cunge and Perdreau (1973).
De Vries (1981) adopted a Lax-type scheme for the bed change term and an explicit
central difference scheme for the gradient of sediment discharge:
ψ =
z n + 1
b , i
1
z b , i + ψ
b
2 (
p m )
z b , i 1 +
z b , i + 1 )
(
1
(
1
ψ
)
b
t
1
q t , i + 1
q t , i 1 ) =
+
x (
0
(5.125)
2
where
b is a weighting factor, which can enhance numerical stability but may
introduce numerical diffusion. A small value should be used for
ψ
ψ b .
Gessler (1971) and Thomas (1982) used the forward difference scheme for the bed
change term and the central difference scheme for the gradient of sediment discharge:
p m )
z b , i
1
q t , i + 1
q t , i 1 ) =
(
1
+
x (
0
(5.126)
t
2
 
 
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