Geoscience Reference
In-Depth Information
Table 1 Explicit empirical equations for the Shields diagram
Researchers
Equation
Brownlie (
1981
)
Y
c
¼
0
:
22
R
0
:
6
b
þ
0
:
06 exp
ð
17
:
77
R
0
:
6
b
Þ
where
R
d
¼ d
(
Dgd
)
0.5
/
u
D
b
van Rijn (
1984
)
Y
c
ð
4
Þ¼
0
:
24
=
D
D
b
D
0
:
64
Y
c
ð
4
<
10
Þ¼
0
:
14
=
D
b
D
0
:
1
Y
c
ð
10
<
20
Þ¼
0
:
04
=
D
b
013
D
0
:
29
Y
c
ð
20
<
150
Þ¼
0
:
Y
c
ðD
>
150
Þ¼
0
:
055
where
D
*
2
)
1/3
¼
particle parameter, that is
d
(
Dg
/
u
Soulsby and Whitehouse
(
1997
)
0
24
D
þ
:
Y
c
¼
0
:
055
½
1
exp
ð
0
:
02
D
Þ
Paphitis (
2001
)
10
2
10
4
0
:
273
Y
c
ð
<
R
<
Þ¼
2
D
þ
0
:
046
½
1
0
:
57 exp
ð
0
:
02
D
Þ
1
þ
1
:
It is the mean curve of Paphitis (
2001
)
variables in the diagram. Consequently,
t
c
or
u
c
remains implicit. Thus, attempts
are made to derive explicit equations for the Shields diagram (Table
1
).
In another study, White (
1940
) classified a high-speed case (
R
3.5) and a
low-speed case (
R
<
3.5). High flow velocity is capable of moving larger parti-
cles. Therefore, the drag due to skin friction is insignificant as compared to the drag
due to pressure difference. The packing coefficient
p
f
was defined by
Nd
2
, where
N
is the number of particles per unit area. The shear drag per particle (i.e.,
t
c
/
N
)is
t
c
d
2
/
p
f
. At the threshold condition, the shear drag equals the product of the
submerged weight of the particle and the frictional coefficient tan
'
. Introducing
a factor, termed
turbulence factor T
f
, he obtained:
Y
c
¼
6
p
f
T
f
tan
for
R
r
'
3
:
5
(10)
4 for fully developed turbulent flow. On the
other hand, low flow velocity is capable of moving smaller particles. In this case,
the drag due to pressure difference acting on the particle is insignificant as com-
pared to the viscous force. However, the upper portion of the particle is exposed to
the shear drag that acts above the center of gravity of the particle. This effect is
taken into account introducing a coefficient
He proposed
p
f
¼
0.4 and
T
f
¼
a
f
. He proposed:
Y
c
¼
6
p
f
a
f
tan
'
for
R
<
3
:
5
(11)
0.34 as an average value.
Kurihara (
1948
) extended the work of White (
1940
) obtaining an expression for
T
f
in terms of
R
, turbulence intensity and the probability of boundary shear stress
increment. The theoretical equations were quite complex. So he proposed simpler
empirical equations of threshold boundary shear stress as
He suggested
p
f
a
f
¼
