Geoscience Reference
In-Depth Information
Wu's method
Many sediment transport formulas, such as those of van Rijn (1984a & b) and Wu
et al
. (2000b), can be expressed as
q
b
(τ
b
/τ
. Two approaches may
be used to consider the effect of bed slope in this group of formulas. One is to correct
the critical shear stress
=
f
)
or
f
(τ
b
/τ
)
c
c
c
using the method of Brooks (1963) or van Rijn (1989).
A disadvantage of this approach is that when the bed slope angle is close to the repose
angle, the corrected critical shear stress usually goes to zero and, thus, the calculated
sediment transport rate perhaps tends to be infinite. This situation should be limited.
The other approach is to add the streamwise component of gravity to the grain shear
stress
τ
τ
b
or the bed shear stress
c
so that the situation of zero
critical shear stress can be avoided. The effective tractive force
τ
b
without modifying
τ
τ
be
(Wu, 2004) is thus
determined by
a
6
(ρ
τ
be
=
τ
b
+
λ
−
ρ)
gd
sin
ϕ
(3.130)
s
s
where
a
is a coefficient related to the shape and location of sediment particles, and
λ
s
τ
b
may be replaced by
is a friction factor. Note that
τ
b
in Eq. (3.130), depending on
the formula considered.
Because the friction factor
s
is difficult to determine, Eq. (3.130) is not ready for
use. In the case where the bed slope angle
λ
ϕ
is equal to the repose angle
φ
r
, sediment
τ
b
=
particles will start moving (
τ
be
=
τ
c
) even without any hydraulic action (
0). Using
a
6
this condition, one can derive
λ
=
τ
/
[
(ρ
−
ρ)
gd
sin
φ
]
. Inserting this relation into
s
c
s
r
Eq. (3.130) yields
τ
be
=
τ
b
+
τ
c
sin
ϕ/
sin
φ
(3.131)
r
The coefficients
λ
s
and
a
are replaced by the critical shear stress
τ
c
and the repose
angle
r
, which are easier to evaluate. However, the test performed byWu (2004) using
the experimental data of Damgaard
et al
. (1997) shows that Eq. (3.131) is adequate
for negative (up) slopes, but for positive (down) slopes the following modification is
needed:
φ
τ
be
=
τ
b
+
λ
τ
c
sin
ϕ/
sin
φ
r
(3.132)
0
where
c
on horizontal, upslope,
and downslope beds; it is related to flow and sediment conditions as well as bed
slope. When the above correction is applied to the Wu
et al
. (2000b) bed-load
and suspended-load transport formulas (3.80) and (3.102),
λ
0
is a coefficient.
λ
0
may consider the difference in
τ
λ
0
has the following
form (Wu, 2004):
1
ϕ
≤
0
λ
=
(3.133)
0
(τ
b
/τ
0.15
e
2sin
ϕ/
sin
φ
r
1
+
0.22
)
ϕ>
0
c
