Geoscience Reference
In-Depth Information
and then
A
m
,
i
+
1
(
p
bk
,
i
+
1
−
p
∗
bk
,
i
+
1
)
f
1
e
0
k
+
f
0
k
+
A
n
+
1
m
,
i
A
b
,
i
+
1
≤
+
+
1
p
∗
bk
,
i
+
1
Q
∗
n
+
1
tk
,
i
p
bk
,
i
+
1
p
∗
bk
,
i
+
1
θ(
1
−
e
k
)(
1
−
θ
)
1
t
p
+
−
(5.159)
L
n
+
1
t
,
i
p
m
)
(
1
−
+
1
Because the last term on the right-hand side of inequality (5.159) is negative but
vanishes when
1, the implicit scheme allows for larger time steps than the
explicit scheme. After considering the stability conditions of the Preissmann scheme
for sediment transport equation, condition (5.159) for the implicit scheme can be
easily satisfied. One of the safest treatments is to impose
θ
=
p
A
n
+
m
and
θ
=
1,
|
A
b
|≤
A
n
+
1
m
A
m
, which is a sufficient but not necessary condition.
≈
Sensitivity of bed-material gradation to mixing layer thickness
Assuming
A
n
+
1
m
,
i
A
m
,
i
+
1
in Eq. (5.152) and differentiating
p
n
+
1
bk
,
i
A
m
,
i
+
1
=
=
1
with
+
1
+
respect to
A
m
,
i
+
1
yields
∂
∂
2
p
n
+
1
bk
,
i
p
n
+
1
bk
,
i
A
m
,
i
+
1
+
1
+
1
0
=
≤
1
Q
∗
n
+
1
tk
,
i
∂
A
m
,
i
+
1
∂
A
m
,
i
+
1
A
m
,
i
+
1
+
f
1
(
1
−
e
k
)
+
1
θ
p
=
1
θ
p
=
(5.160)
p
n
+
1
bk
,
i
A
m
,
i
+
1
represents the change in bed-material gradation per
unit change in mixing layer thickness. Eq. (5.160) shows that the implicit scheme has
smaller
The gradient
∂
1
/∂
+
p
n
+
1
bk
,
i
∂
/∂
A
m
,
i
+
1
and is thus less sensitive to
A
m
,
i
+
1
than the explicit scheme.
+
1
5.3.4 Treatments for sediment transport in
channel networks
If a channel network is concerned, the sediment transport at channel confluences
and splits, as well as hydraulic structures, needs to be treated specially. At hydraulic
structures such as culverts, drop structures, weirs, and measuring flumes, erosion is not
allowed; thus, the beds are fixed and the sediment discharges are constant through
them. For bridge crossings, 1-D models are able to simulate the bed change due to
channel contraction, but not the local scour due to 3-D flow features. However, the
maximum local scour depth and volume can be estimated using empirical functions.
In analogy to the flow calculation described in Section 5.2.1.3, the sediment trans-
port at a channel confluence or split can generally be computed by applying Eqs.
(5.130) and (5.131) or Eq. (5.137). For this computation, the downstream cross-
section at the confluence or the upstream cross-section at the split needs to be divided
into two parts. This approach was successfully used by Wu (1991) in a quasi-steady
model. However, a simpler approach, which is described below, may be used if the
three cross-sections at the confluence or split are located very close together.
