Geoscience Reference
In-Depth Information
in the framework of a 1-D model. As an approximation, this effect is lumped into the
correction factor
k
t
in Eq. (9.67). By trial and error, it is found that
k
t
/ρ
f
is
adequate for the two test cases presented in Section 9.2.3. This value of
k
t
is equivalent
to a value of 0.6 for
c
a
, which qualitatively agrees with the observation of Fraccarollo
and Capart (2002). Because the parameter
c
a
is not included explicitly in this relation
of
k
t
, the solution procedure in the sediment module is simplified significantly.
Note that the reference level for near-bed suspended-load concentration in the orig-
inal van Rijn formula is defined at the equivalent roughness height or half the dune
height, which is not investigated well in the case of dam-break flow. Because the cor-
rection factor
k
t
is a lumped parameter in the modified sediment transport capacity
formulas, this reference level should also be interpreted as an empirical parameter.
For simplicity, the reference level herein is set at max
≈
1
+
1.5
ρ
s
, in which
d
is the
sediment size. In addition, in the range of high shear stress, the near-bed equilib-
rium concentration of suspended load determined using the van Rijn (1984b) formula
may be larger than 1
(
2
d
, 0.005
h
)
p
m
; this is not physically reasonable and is eliminated in the
simulation by imposing an upper bound of 1
−
p
m
to
c
b
∗
.
−
9.2.2 Numerical methods
Eqs. (9.59), (9.60), (9.62), (9.65), and (9.66) constitute a hyperbolic system, which can
be solved numerically using the shock-capturing schemes introduced in Section 9.1.
As an example, the upwind flux scheme is used here.
To establish an explicit algorithm, Eqs. (9.59) and (9.60) are reformulated by
eliminating the flow density on their left-hand sides using the relation
ρ
=
ρ
f
(
1
−
C
t
)
+
ρ
s
C
t
and Eqs. (9.62), (9.65), and (9.66). The derived continuity and momentum
equations are
B
∂
A
∂
+
∂
Q
1
1
L
(
=
(
E
b
−
D
b
)
+
Q
b
∗
−
Q
b
)
(9.68)
p
m
t
∂
x
1
−
Q
2
A
g
n
2
Q
∂
Q
∂
+
∂
∂
gA
∂
z
s
1
2
gAh
p
1
∂ρ
∂
|
Q
|
=−
x
−
x
−
AR
4
/
3
t
x
∂
ρ
U
1
B
−
ρ
−
ρ
f
ρ
C
t
1
L
(
s
−
(
E
b
−
D
b
)
+
Q
b
∗
−
Q
b
)
p
m
1
−
(9.69)
Eqs. (9.68), (9.69), (9.62), and (9.65) are written in conservative form as Eq. (9.1),
in which
and
F
(
)
represent the vectors of unknown variables and fluxes:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
A
Q
AC
Q
b
/
Q
Q
2
A
QC
Q
b
/
=
,
F
(
)
=
(9.70)
U
b
and
S
(
)
includes the remaining terms in each equation.
