Geoscience Reference
In-Depth Information
both helical flow and gravity. Parker (1984) simplified the result of Kikkawa
et al
.as
q
bn
q
bs
=
1
+
α
µ
p
c
c
tan
δ
b
+
tan
ϕ
(6.96)
λ
µ
s
c
where
q
bs
and
q
bn
are the bed-load transport rates along the longitudinal and transverse
directions, respectively;
ϕ
is the lateral inclination of the bed;
is the Shields number;
c
is the critical Shields number
=
0.04;
µ
c
is the dynamic coefficient of Coulomb
friction;
α
p
is the ratio of lift coefficient to drag coefficient; and
λ
s
is a sheltering
coefficient. Kikkawa
et al
. (1976) evaluated
µ
c
,
α
p
, and
λ
s
as 0.43, 0.85, and 0.59,
respectively.
Struiksma
et al
. (1985) and Sekine and Parker (1992) also proposed a similar
relation as
q
bn
q
bs
=
δ
b
−
β
b
∂
z
b
tan
(6.97)
∂
n
U
2
where
β
b
is a coefficient. Struiksma
et al
. (1985) suggested
β
=
1
/
[
χ
(
∗
)
]
with
χ
b
0
0
4
.
By applying Eq. (3.132) to the
x
and
y
directions, Wu (2004) derived a method that
replaces the bed-load transport direction cosines
1
/
varying between 1 and 2, while Sekine and Parker (1992) gave
β
b
=
0.75
(
/)
c
α
bx
and
α
by
by
α
bx
,
e
and
α
by
,
e
:
α
by
,
e
=
τ
b
α
bx
+
λ
0
τ
c
sin
α
bx
,
e
ϕ
x
/
sin
φ
r
(6.98)
τ
b
α
by
+
λ
τ
c
sin
ϕ
/
sin
φ
0
y
r
where
y
are the bed angles with the horizontal along
x
- and
y
-directions (with
positive values denoting downslope beds), respectively. Eq. (6.98) can also be written
as Eq. (6.97) with
ϕ
x
and
ϕ
/τ
b
)/
τ
b
in Eq. (6.98) may be replaced
β
=
λ
(τ
sin
φ
r
. Note that
0
c
b
by the total bed shear stress
τ
b
if the bed-load transport capacity is determined by
formulas that consider
τ
b
as the tractive force for bed-load transport.
The dispersion fluxes in the suspended-load transport equation and the adjustment
of the bed-load transport angle tend to move sediment from the outer bank toward
the inner bank. Therefore, with such enhancements, the depth-averaged 2-D model
can reasonably predict erosion along the outer bank and deposition along the inner
bank. This is demonstrated in the following example.
Wu and Wang (2004a) simulated the sediment transport and morphological change
in an 180
◦
bend under unsteady flow conditions, which were experimentally inves-
tigated by Yen and Lee (1995). The width of the flume was 1m, the radius of
curvature at the centerline was 4m, and the initial bed slope was 0.002. The flow
hydrograph was triangular. The base flow discharge was 0.02m
3
s
−
1
, and the base
flow depth,
h
o
, was 0.0544m. In the simulated case (Run 4), the peak flow dis-
charge was 0.053m
3
s
−
1
, and the duration was 240min. The peak of the hydrograph
was set at the first third of its duration. The sediment was non-uniform and had a
median diameter of 1.0mm and a standard deviation of 2.5. The Manning rough-
ness coefficient was given as
d
1
/
6
50
/
20 in the simulation, with
d
50
being the median
