Geoscience Reference
In-Depth Information
9.2.1 Governing equations
Hydrodynamic equations
To account for the interactions among flow, sediment transport, and bed change, the
generalized 1-D shallow water equations (5.177) and (5.178) should be used, which
are written as follows by setting
β
=
1 and neglecting side flows:
∂(ρ
A
)
+
∂(ρ
Q
)
+
ρ
b
∂
A
b
∂
=
0
(9.59)
∂
t
∂
x
t
Q
2
A
g
n
2
Q
∂
∂
)
+
∂
∂
ρ
gA
∂
z
s
1
2
gAh
p
∂ρ
|
Q
|
t
(ρ
Q
+
ρ
x
+
x
+
ρ
=
0
(9.60)
∂
∂
AR
4
/
3
x
The effect of alluvial bed roughness can be accounted for through the dependence
of Manning
n
on flow and sediment conditions, but constant Manning
n
values are
adopted here for simplicity. The sensitivity analysis performed by Wu and Wang
(2007) shows that the Manning
n
affects very little on the model results, and constant
Manning
n
values in a dam-break event can be justified.
Sediment transport equations
As described in Section 2.1.2.3, the total-load sediment in natural rivers may be divided
into bed load and suspended load as per sediment transport mode or into bed-material
load and wash load as per sediment source. The former approach is adopted here and,
thus, the total-load transport rate,
Q
t
, is computed by
Q
t
=
QC
t
=
QC
+
Q
b
(9.61)
where
C
is the suspended-load volumetric concentration averaged over the cross-
section, and
Q
b
is the bed-load transport rate. The 1-D suspended-load transport
equation (2.108) is written as
∂
∂
)
+
∂
∂
t
(
AC
x
(
QC
)
=
B
(
E
b
−
D
b
)
(9.62)
where
D
b
and
E
b
are the sediment deposition and entrainment rates at the inter-
face between the bed-load and suspended-load layers, defined as
D
b
=
ω
sm
c
b
and
E
b
sm
being the settling velocity of sediment particles in turbid
water. To consider the effect of sediment concentration,
=
ω
sm
c
b
∗
with
ω
sm
is determined using the
Richardson-Zaki (1954) formula (3.19), which is written as
ω
n
ω
=
(
1
−
C
t
)
ω
(9.63)
sm
s
where
s
is the settling velocity of single particles in clear water, determined using
the Zhang formula (3.12) if no measurement data is provided; and
n
is an empirical
exponent of about 4.0.
ω
