Geoscience Reference
In-Depth Information
Figure 3.8
Definition of exposure height of bed material.
The total hidden and exposed probabilities,
p
hk
and
p
ek
, of particles
d
k
are then
obtained by summing Eqs. (3.40) and (3.41) over all size classes, respectively:
N
d
j
p
hk
=
p
bj
(3.42)
d
k
+
d
j
j
=
1
N
d
k
d
k
+
p
ek
=
p
bj
(3.43)
d
j
j
=
1
where
N
is the total number of particle size classes in the non-uniform sediment
mixture.
A relation
p
hk
+
=
=
=
0.5,
which means the hidden and exposed probabilities are equal. In a non-uniform sed-
iment mixture,
p
ek
p
ek
1 exists. For uniform sediment particles,
p
hk
p
ek
p
hk
for fine particles.
This can be demonstrated with a simple example. For a sediment mixture with two
size classes
d
1
≥
p
hk
for coarse particles, and
p
ek
≤
=
1 mm,
p
b
1
=
0.4 and
d
2
=
5 mm,
p
b
2
=
0.6, one can obtain
p
h
1
=
0.6333. It is shown that more coarse
particles are exposed and more fine particles are hidden.
By using the hidden and exposed probabilities, a hiding and exposure correction
factor is defined as (Wu
et al
., 2000b)
0.7
>
p
e
1
=
0.3,
p
h
2
=
0.3667
<
p
e
2
=
p
ek
p
hk
−
m
η
=
(3.44)
k
where
m
is an empirical parameter. The criterion for sediment incipient motion
proposed by Shields (1936) is then modified as
c
p
ek
p
hk
−
m
τ
ck
d
k
=
(3.45)
(γ
−
γ)
s
where
=
0.03 and
m
=
0.6, which are calibrated using laboratory and field data, as
c