Geoscience Reference
In-Depth Information
where
50
is the dimensionless shear stress defined in Eq. (3.72) corresponding to the
subpavement size
d
50
. This formula was verified using field data with sediment sizes
ranging from 18 to 28 mm.
Considering the fact that the bed-load transport in gravel-bed rivers is accomplished
by means of mobilization of grains exposed on the bed surface rather than substrate
particles, Parker (1990) transformed Eq. (3.73) into a surface-based relation. The
details can be found in his paper.
θ
Hsu and Holly's method
The method proposed by Hsu and Holly (1992) first determines the size distribution
of the transported sediment and then the total transport rate. The fraction of each size
class in the transportedmaterial is postulated to be proportional to the joint probability
of two factors: (1) its mobility under the prevailing hydraulic conditions, and (2) its
availability on the bed surface (active layer).
If the fluctuation of flow velocity is assumed to have the Gaussian probability
distribution, the mobility of size class
k
is derived as
exp
dx
∞
x
2
2
1
σ
√
2
P
mo
,
k
=
−
(3.74)
2
σ
π
U
ck
/
U
−
1
where
U
is the mean velocity of flow;
U
ck
is the incipient velocity of size class
k
, deter-
mined using the Qin (1980) formula (3.34) modified by recalibrating the coefficient
0.7
86
as 1.5; and
σ
is the standard deviation of the normalized fluctuating velocity
U
/
U
and has a value of about 0.2.
The availability of size class
k
is equivalent to its fractional representation on the
bed surface (active layer),
p
bk
. Thus, the fraction of size class
k
in the transported
material is
P
mo
,
k
p
bk
d
max
p
k
=
(3.75)
d
min
P
mo
,
k
p
bk
After the size distribution of the transported material is obtained, the mean size
d
mt
and mean incipient velocity
U
ct
are calculated. The total bed-load transport rate can
then be evaluated using any appropriate predictor. The Shamov formula was suggested
and modified as
U
c
min
U
U
ct
3
d
mt
h
1
/
4
12.5
d
mt
U
q
b
∗
=
−
(3.76)
m
−
1
s
−
1
),
where
q
b
∗
is the total transport rate of bed load per unit channel width (kg
·
s
−
1
).
and
U
c
min
is the incipient velocity of the smallest size class (m
·
Methods of Ranga Raju and his co-workers
Ranga Raju and his co-workers (Misri
et al
., 1984; Samaga
et al
., 1986a; Patel
and Ranga Raju, 1996) extended the Paintal (1971) uniform bed-load formula to
