Geoscience Reference
In-Depth Information
The derived distribution function is
exp
ω
c
c
b
∗
s
=
U
∗
[
f
(η)
−
f
(η
)
]
(3.86)
b
κ
where
η
b
=
1
−
δ/
h
, and
ln
η
+
√
2
a
√
2
a
3
/
2
+
√
η
ln
1
2 arctan
√
η
+
η
+
a
f
(η)
=
−
√
η
+
a
2
1
+
η
2
arctan
1
arctan
1
2
2
a
a
+
+
−
−
=
with
a
1.526.
The Zhang distribution is shown in Fig. 3.19 as dashed lines. It improves the sedi-
ment concentration near the water surface, but the formulation is more complicated
and inconvenient to use.
Lane and Kalinske (1941) assumed
σ
=
1 and averaged the sediment diffusivity in
s
Eq. (3.83) over the flow depth as
=
κ
6
U
∗
h
¯
ε
(3.87)
s
and then introduced this value into Eq. (3.82) and derived
exp
z
c
c
b
∗
6
ω
−
δ
h
s
=
−
(3.88)
κ
U
∗
Van Rijn (1984b) also derived a vertical distribution of suspended-load concentra-
tion using the following two-layer relation of sediment diffusivity:
U
∗
1
h
z
κ
−
z
/
/σ
z
/
h
<
0.5
s
ε
=
(3.89)
s
0.25
κ
U
∗
h
/σ
s
z
/
h
≥
0.5
In the case of small concentration (
c
<
c
b
∗
<
0.001), the van Rijn distribution is
[
(
r
c
c
b
∗
h
/
z
−
1
)/(
h
/δ
−
1
)
]
z
/
h
<
0.5
=
(3.90)
)
−
r
exp
(
h
/δ
−
1
[−
4
r
(
z
/
h
−
0.5
)
]
z
/
h
≥
0.5
where
r
=
ω
/(κ
U
∗
)
.
s
ω
s
/(κ
U
∗
)
is called the suspension or Rouse number. Physically, the
Rouse number represents the effect of gravity (
The parameter
ω
s
) against the effect of turbulent diffu-
sion (
U
∗
). When the Rouse number is larger, the effect of gravity is stronger and the
distribution of sediment concentration along the flow depth is less uniform. When the
κ
