Geoscience Reference
In-Depth Information
Using the finite volume method in Section 9.1, one can derive the discretized
equation (9.10), with the intercell fluxes determined using the first-order upwind
scheme as
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
Q
i
Q
i
2
Q
i
+
1
Q
i
+
1
2
A
i
A
i
+
1
Q
i
+
1
C
i
+
1
Q
b
,
i
+
1
/
/
F
i
+
1
/
2
=
F
i
+
1
/
2
=
,
if
Q
≥
0;
,
if
Q
<
0
Q
i
C
i
Q
b
,
i
(9.71)
The friction term and sediment exchange terms on the right-hand sides of Eqs. (9.68)
and (9.69) are evaluated in a pointwise manner using the values at cell center
i
, and
the flow density gradient term in Eq. (9.69) is discretized using the central difference
scheme at time level
n
. The water surface gradient term in Eq. (9.69) is discretized
using the scheme in Eq. (9.55) proposed by Ying
et al
. (2004).
The bed change equation (9.66) is discretized in time as
B
i
(
t
1
L
(
D
b
,
i
−
E
b
,
i
)
+
Q
b
,
i
−
Q
b
∗
,
i
)
A
b
,
i
=
(9.72)
1
−
p
m
The numerical solution is obtained through the following steps: (a) Solve the
continuity equation (9.68) to obtain the flow area and, in turn, the water level; (b) Solve
the momentum equation (9.69) to obtain the flow discharge and, in turn, the veloc-
ity; (c) Calculate the equilibrium suspended-load near-bed concentration and bed-load
transport rate; (d) Solve the sediment transport equations (9.62) and (9.65) to obtain
the actual suspended-load concentration and bed-load transport rate; (e) Calculate the
bed change using Eq. (9.72); and (f) Continue steps (a)-(e) for the next time step until
the entire time period is finished.
Because the aforementioned solution procedure is explicit, the computational time
step should be limited by the Courant-Friedrichs-Lewy (CFL) condition for flow com-
putation and additional numerical stability conditions for sediment transport and bed
change computations.
9.2.3 Examples
The movable-bed dam-break flow model described above was verified by Wu and
Wang (2007) using two sets of laboratory experiments performed in Taipei (University
of Taiwan) and Louvain (Université Catholique de Louvain) (Capart and Young, 1998;
Fraccarollo and Capart, 2002). Both experiments concerned small-scale dam-break
waves over movable beds in prismatic channels with rectangular cross-sections. They
differed primarily in the used sediment materials. In the Taipei test, the sediment
particles were artificial spherical pearls covered with a shiny white coating, having
a diameter of 6.1mm, a density of 1048 kg
m
−
3
, and a settling velocity of about
·
s
−
1
. In the Louvain test, the sediment particles were cylindrical PVC pellets
7.6 cm
·
