Geoscience Reference
In-Depth Information
the dispersion flux in the
x
-direction:
h
h
2
z
1
f
1
h
12
b
s
IC
1
h
D
sx
=
U
x
C
−
u
x
cdz
≈−
α
h
−
(
z
)
dz
(6.90)
δ
δ
Similarly, the dispersion flux in the
y
-direction can be derived as
h
h
2
z
1
f
1
h
22
b
s
IC
1
h
D
sy
=
U
y
C
−
u
y
cdz
≈−
α
h
−
(
z
)
dz
(6.91)
δ
δ
Using the Lane-Kalinske distribution, the integral in Eqs. (6.90) and (6.91) is
evaluated as
1
h
2
z
1
f
1
−
1
U
∗
h
m
2
U
∗
15
e
−
15
ω
s
/
U
∗
−
e
−
15
ω
s
/
U
∗
)
h
−
(
z
)
dz
=−
+
s
(
1
−
m
+
15
ω
ω
s
δ
(6.92)
As alternatives, one may also use the Rouse distribution for the suspended load and
the logarithmic law for the streamwise flow velocity in the above derivation. This is
left to the interested reader.
6.3.4 Bed-load transport in curved channels
The bed-load movement direction deviates from the main flow direction due to the
helical flow effect. Engelund (1974), Zimmermann and Kennedy (1978), and Odgaard
(1986) proposed empirical formulas for evaluating this deviation. The Engelund
formula is
7
h
r
tan
δ
b
=
(6.93)
where
b
is the angle between the bed-load movement and main flow directions.
In Odgaard's method, the direction of bed-load movement is calculated by
δ
u
by
u
bx
tan
δ
=
(6.94)
bs
which is equivalent to
u
bx
u
bx
+
u
by
u
bx
+
u
by
,
u
by
α
bx
=
α
by
=
(6.95)
where
u
bx
and
u
by
are the
x
- and
y
-components of bed-load velocity or the flow velocity
near the bed, which can be converted from
u
bs
and
u
bn
determined using Eqs. (6.83)
and (6.84).
The effect of gravity on bed-load transport in a sloped bend is also important.
Kikkawa
et al
. (1976) derived analytically the lateral bed-load transport affected by