Geoscience Reference
In-Depth Information
gD
3
and
Considering spherical particles, particle submerged weight
F
G
¼
(
p
/6)
Dr
u
m
2
D
2
and lift (shear and Magnus lift)
hydrodynamic forces like drag
F
D
¼
C
D
(
p
/8)
r
C
L
ru
m
D
2
(
z
)
0.5
[
u
0.5
z
)
0.5
] were taken into account.
F
L
¼
u
/
þ
0.5
f
(
R
)
D
(
u
/
∂
∂
∂
∂
Here,
D
diameter of solitary particle resting over a horizontal bed formed by
the sediments of size
d
;
u
¼
¼
flow velocity at elevation
z
;
u
m
¼
mean flow velocity
received by the frontal area of solitary particle;
f
(
R
)
¼
1for
R
3; and
f
(
R
)
¼
0
0.289
D
(3
D
2
for
R
<
3. The lever-arms are
X
¼
0.433
Dd
/(
D
þ
d
)and
Z
¼
þ
d
2
)
0.5
/(
D
6
Dd
d
). Taking moment at the pivot
M
, the equation for threshold
condition was given by Dey as:
þ
p
p
2
p d
2
=ð
1
þ
a
1
cos
cÞ
Y
c
¼
0
:
5
d
2
0
:
5
6
d
6
C
L
d
R
= d
p
C
D
^
u
2
m
ð
3
þ
Þ
þ
u
m
ð@
^
u
=@
^
z
Þf
2
½ð
Þ@
^
u
=@
^
z
þ
f
ð
R
Þg
(18)
^
d
where
¼
d
=
D
;
^
u
¼
u
=
u
c
;
u
m
¼
^
u
m
=
u
c
;
z
^
¼
z
=
D
;
p
¼
probability of occurring
sweep event;
c ¼
instantaneous shear stress.
Using
u
for different flow regimes, Dey put forward a diagram for entrainment
threshold as
Y
c
versus
D
for different
sweep angle;
a ¼ t
t
/
t
c
;
and
t
t
¼
'
(Fig.
4
). Unlike Shields diagram, it can be
used directly for the determination of
t
c
or
u
c
.
Besides, James (
1990
) presented a generalized model of the threshold of sedi-
ment entrainment based on the analysis of forces acting on a particle, taking into
account the particle geometry, packing arrangements and variations of near-bed
flow velocity, drag, and lift. Ling (
1995
) studied the equilibrium of a solitary
particle on a sediment bed, considering spinning motion of particles. He proposed
two modes for limiting equilibrium, namely, rolling and lifting. McEwan and Heald
(
2001
) analyzed the stability of randomly deposited bed particles using a discrete
particle model. The threshold boundary shear stress could be adequately repre-
sented by a distribution of values. A Shields parameter of 0.06 for gravels found to
correspond to the distribution for which 1.4% (by weight) of particles is on motion.
1
40°
45°
36°
32°
28°
0.1
25°
ϕ = 20°
0.01
0.1
1
10
100
1000
10000
D
*
Fig. 4 Dependency of
Y
c
on
D
for different
'
