Geoscience Reference
In-Depth Information
The near-bed concentration
c
b
is related to the average concentration
C
by
c
b
=
α
c
C
,
as described in Section 2.5.1. Following Cao
et al
. (2004),
α
c
is determined by
p
m
)/
α
=
min
[
α
0
,
(
1
−
C
]
(9.64)
c
where
α
0
is given a value of 2.0. Note that Eq. (9.64) limits
c
b
up to a maximum value
equal to 1
p
m
.
The 1-D bed-load transport equation is
−
Q
b
U
b
∂
∂
+
∂
Q
b
∂
1
L
(
=
Q
b
∗
−
Q
b
)
(9.65)
t
x
where
Q
b
∗
is the equilibrium (capacity) bed-load transport rate;
U
b
is the bed-load
velocity, which can be evaluated using the van Rijn (1984a) formula (3.136) or
Fig. (3.27) but is set to be the flow velocity here for simplicity; and
L
is the adaptation
length of sediment, determined using Eq. (2.155).
The bed change can be determined by
p
m
)
∂
A
b
∂
1
L
(
(
1
−
=
B
(
D
b
−
E
b
)
+
Q
b
−
Q
b
∗
)
(9.66)
t
Empirical formulas
To close the aforementioned sediment transport model, additional empirical formulas
are required to determine the equilibrium bed-load transport rate
Q
b
∗
and suspended-
load near-bed concentration
c
b
∗
. The van Rijn (1984a & b) formulas (3.70) and (3.95)
herein are used. However, these formulas, which were calibrated in the lower flow
regime, should be modified for extension to the situation of dam-break flow.
According to the observation of Fraccarollo and Capart (2002), the sediment
concentration in the lower layer near the bed under dam-break flow conditions is very
high (nearly the same as that of bed material). Therefore, Wu and Wang (2007) intro-
duced a correction factor for the transport stage number in the van Rijn (1984a & b)
formulas by replacing the water density with the mixture density near the bed:
U
2
∗
U
2
∗
τ
b
b
b
c
=
gd
=
k
t
(9.67)
τ
θ
(ρ
/ρ
mb
−
1
)
θ
(ρ
/ρ
f
−
1
)
gd
c
s
c
s
where
c
is the critical Shields number for sediment incipient motion, determined using
the Shields diagram;
θ
ρ
mb
is the density of the water and sediment mixture near the
bed; and
k
t
is the correction factor, expressed as
k
t
, with
c
a
being a concentration of sediment near the bed. In principle,
c
a
can be related to the
average sediment concentration in the bed-load layer or the depth-averaged suspended-
load concentration, but this usually requires iteration in the sediment module. On the
other hand, the dynamic pressure in the dam-break wave front might affect sediment
entrainment and transport significantly, but it is very difficult to consider this effect
=
1
+
c
a
ρ
/
[
(
1
−
c
a
)ρ
f
]
s