Geoscience Reference
In-Depth Information
The drag and lift forces acting on the particle are usually determined by
u
b
2
C
D
a
2
d
2
F
D
=
ρ
(3.23)
u
b
2
C
L
a
3
d
2
F
L
=
ρ
(3.24)
where
u
b
is the bottom flow velocity acting on the particle;
a
2
d
2
and
a
3
d
2
are the
projected areas of the particle on the planes normal to the flow direction and the
vertical direction, respectively; and
C
D
and
C
L
are the drag and lift coefficients, related
to particle shape, position on the bed, etc.
Inserting Eqs. (3.1), (3.23), and (3.24) into Eq. (3.22) yields the critical bottom
velocity for sediment incipient motion:
2
k
1
a
1
k
2
a
2
C
D
gd
1
/
2
k
3
a
3
C
L
ρ
−
ρ
ρ
s
u
bc
=
(3.25)
+
3.2.2 Incipient motion criteria for a group
of sediment particles
Eq. (3.25) is a criterion for the incipient motion of an individual particle on the bed. For
a group of sediment particles, there are two approaches to determine the threshold
criterion of incipient motion: stochastic and deterministic. The stochastic approach
considers the sediment incipient motion as a random phenomenon due to the stochas-
tic properties of turbulent flow and sediment transport. This approach usually does
not adopt a threshold value of sediment transport rate as the criterion at which the sed-
iment particles start moving. The pioneer using the stochastic approach for sediment
transport is Einstein (1942, 1950).
The deterministic approach usually adopts a certain amount of sediment particles in
motion as the incipient motion criterion. Theoretically, a zero bed-load transport rate
should be used, but this is not meaningful in practice. Numerous experiments have
shown that even when the flow strength is much weaker than the critical condition
proposed by Shields (1936), there are still some sediment particles moving on the bed.
Kramer (1935) defined three types of motion of bed material: weak movement (only
a few particles are in motion on the bed), medium movement (the grains of mean
diameter begin to move), and general movement (all the mixture is in motion). How-
ever, his criterion is only qualitative and difficult to use. Therefore, several low levels
of bed-load transport rate were suggested as the quantitative critical condition for
incipient motion — for instance,
q
b
∗
=
14 cm
3
m
−
1
min
−
1
by Waterways Experiment
Station, U.S. Army Corps of Engineers, and
q
b
∗
/(ρ
0.000317 by Han and
He (1984). Yalin (1972) also proposed a quantitative criterion related to the number
of particles moving on the bed. For a non-uniform sediment mixture, the threshold
criterion for incipient motion is more complex because of interactions among different
size classes. Parker
et al
. (1982) suggested the following threshold condition for the
s
d
ω
)
=
s
