Geoscience Reference
In-Depth Information
(
−
1.75
0.331
0.045
)
c
b
∗
=
(3.96)
(
−
1.75
1
+
0.72
0.045
)
=
U
2
where
.
Two issues regarding the aforementioned formulas of near-bed suspended-load con-
centration should be pointed out. One is that the suspended-load concentration near
the channel bed is very difficult to measure at the present time and has to be extrapo-
lated from those measured in the upper flow layer with the aid of an assumed vertical
distribution of sediment concentration. The accuracy and reliability of this analysis
highly depend on the used distribution function of sediment concentration near the
bed. The often used Rouse distribution is not reliable near the bed, and some later
modifications introduced in Section 3.5.1 do not improve much indeed. Therefore, the
calibration of these formulas using direct measurement data near the bed should be
carried out in the future.
The other issue is that the near-bed concentration is defined at different reference
levels in different formulas. Each formula should be applied only at the height where
the near-bed concentration is defined. This makes comparison of these formulas very
difficult. For sediment transport modeling, it is more convenient to set the reference
level at the interface between the bed-load and suspended-load layers.
∗
/
[
(ρ
/ρ
−
1
)
gd
]
s
3.5.3 Suspended-load transport rate
Einstein's method
Einstein's (1950) method determines the suspended-load transport rate by integrating
the product of local sediment concentration
c
k
and flow velocity
u
over the suspended-
load zone from
δ(
=
2
d
)
to
h
:
h
q
s
∗
k
=
c
k
udz
(3.97)
δ
where
q
s
∗
k
is the transport rate of the
k
th size class of suspended load.
Using the Rouse distribution of sediment concentration (
σ
=
1 ) and the logarithmic
s
distribution of flow velocity in Eq. (3.30) (replacing
U
∗
by
U
∗
) yields
c
b
∗
k
h
ω
sk
κ
log
30.2
z
dz
h
/
z
−
1
U
∗
5.75
U
∗
q
s
∗
k
=
∗
h
/δ
−
1
δ
s
2.303 log
30.2
h
I
2
k
11.6
U
∗
=
c
b
∗
k
δ
∗
I
1
k
+
(3.98)
s
r
k
1
r
k
1
r
k
−
1
r
k
−
1
0.216
ζ
0.216
ζ
1
−
ζ
ζ
)
1
−
ζ
ζ
)
r
k
d
r
k
ln
where
I
1
k
=
ζ
b
(
ζ
, and
I
2
k
=
ζ
b
(
ζ
d
ζ
,
b
b
(
1
−
ζ
b
)
(
1
−
ζ
b
)
with
ζ
=
z
/
h
,
ζ
b
=
δ/
h
,
r
k
=
ω
sk
/(κ
U
∗
)
, and
s
is defined in Eq. (3.71).
