Geoscience Reference
In-Depth Information
Ashida and Michiue (1971) modified the Egiazaroff formula as
[
2
log 19
/
log
(
19
d
k
/
d
m
)
]
d
k
/
d
m
≥
0.4
ck
c
=
(3.36)
d
m
/
d
k
d
k
/
d
m
<
0.4
and Hayashi
et al
. (1980) proposed a similar modification:
[
2
ck
log 8
/
log
(
8
d
k
/
d
m
)
]
d
k
/
d
m
≥
1
c
=
(3.37)
d
m
/
d
k
d
k
/
d
m
<
1
The formulas proposed by Parker
et al
. (1982) and others can be written as
c
50
d
k
d
50
−
m
=
(3.38)
ck
where
c
50
is the critical Shields number corresponding to the medium size
d
50
of bed
material, and
m
is an empirical coefficient between 0.5-1.0.
Method of Wu et al.
Consider a mixture of sediment particles with various diameters on the bed, as shown
in Fig. 3.8. For simplicity, the sediment particles are assumed to be spheres. The drag
and lift forces acting on a particle depend on how it is resting on the bed, i.e., whether
it is hidden by other particles or exposed to flow. Its position on the bed can be
represented by its exposure height
e
, which is defined as the difference between the
apex elevations of it and the upstream particle. If
>
0, the particle is considered to
e
be at an exposed state; if
0, it is at a hidden state. For a particle with diameter
d
k
in the bed surface layer, the value of
<
e
d
j
and
d
k
. Here,
d
j
is the diameter of the upstream particle. Because the sediment particles randomly
distribute on the bed,
e
is in the range between
−
e
is a random variable.
e
is herein assumed to have a uniform
probability distribution function:
1
/(
d
k
+
d
j
)
,
−
d
j
≤
≤
d
k
e
f
=
(3.39)
0,
otherwise
The probability of particles
d
j
staying in front of particles
d
k
is assumed to be the
fraction,
p
bj
, of particles
d
j
in bed material. Therefore, the probabilities of particles
d
k
hidden and exposed due to particles
d
j
are obtained from Eq. (3.39) as follows:
d
j
d
k
+
p
hk
,
j
=
p
bj
(3.40)
d
j
d
k
d
k
+
p
ek
,
j
=
p
bj
(3.41)
d
j
