Geoscience Reference
In-Depth Information
Wu-Wang formula
The Manning roughness coefficient
n
for a movable bed is often related to the bed
sediment size
d
by
d
1
/
6
A
n
n
=
(3.62)
where
A
n
is a roughness parameter related to bed-material size composition, particle
shape, bed forms, flow conditions, etc.
For a stationary flat bed covered with uniform sediment particles, Strickler (1923)
suggested
A
n
m
−
1
/
3
and m, respectively.
For a stationary flat bed with non-uniform sediment particles,
d
is usually set as the
median size
d
50
, and
A
n
is about 20 (Li and Liu, 1963; Zhang and Xie, 1993; Wu and
Wang, 1999). If the sediment particles with slightly irregular shapes are tightly placed
on the bed,
A
n
may have a larger value up to 24 (i.e., lower resistance to flow). If the
sediment particles with rather irregular shapes are loosely placed on the bed,
A
n
has a
smaller value between 17 and 20. In addition, if
d
is set as
d
65
or
d
90
rather than
d
50
,
A
n
has a value of 24 (Patel and Ranga Raju, 1996) or 26 (Meyer-Peter and Mueller,
1948), respectively.
For amovable bedwith sandwaves, the effect of bed forms on
A
n
should be included.
Li and Liu (1963) proposed a relation of
A
n
=
21.1. Here, the units of
n
and
d
are s
·
∼
U
/
U
c
for natural rivers:
20
)
−
3
/
2
(
U
/
U
c
1
<
U
/
U
c
≤
2.13
A
n
=
(3.63)
/
2
3
3.9
(
U
/
U
c
)
U
/
U
c
>
2.13
However, Eq. (3.63) does not agree with most of the flume and field data used in the
test performed by Wu and Wang (1999). To improve this shortcoming, Wu and Wang
established a relation between
A
n
g
1
/
2
Fr
1
/
3
τ
b
/τ
/(
)
and
c
50
, as shown in Fig. 3.13. Here,
/
gh
. The values of
A
n
/(
g
1
/
2
Fr
1
/
3
)
Fr
is the Froude number
U
decrease, and then,
τ
b
/τ
increase as
c
50
increases. Physically, this trend represents the fact that sand ripples
and dunes are formed first, and then, washed away gradually. For the convenience of
users, the relation between
A
n
/(
g
1
/
2
Fr
1
/
3
τ
b
/τ
c
50
in the range of 1
≤
τ
b
/τ
c
50
≤
)
and
55
is approximated by
(τ
b
/τ
c
50
)
1.25
8
[
1
+
0.0235
]
A
n
g
1
/
2
Fr
1
/
3
=
(3.64)
(τ
b
/τ
c
50
)
1
/
3
c
50
in Eq. (3.64) is calculated using the Shields curve mod-
ified by Chien and Wan (1983), and the grain shear stress
The critical shear stress
τ
τ
b
is calculated using
d
1
/
6
50
Eq. (3.51), with
n
calculated by
n
=
τ
b
by Eq. (3.49). The bed hydraulic
radius
R
b
is determined using Williams' (1970) method:
R
b
=
/
20 and
B
2
h
/(
1
+
0.055
h
/
)
,in
which
B
is the channel width.
