Geoscience Reference
In-Depth Information
layer. Therefore, one may expect that for erosion, C
/
c b
C
/
c b
and
α
α
; for
c
c
deposition, C
/
c b
C /
c b
and
α
α
.
c
c
α c is often assumed to be negligible, for
simplicity. Thus, the net exchange flux can be determined by (Han, 1980; Wu, 1991)
However, the difference between
α c and
D b
E b = αω
(
C
C
)
(2.132)
s
where
is a new adaptation coefficient.
Equating Eqs. (2.131) and (2.132) leads to
α
αω
(
C
C ) = α
ω
s C
α
ω
s C and then
s
c
c
C
α = α
+
α
)
(2.133)
c
c
c
C
C
C
α = α c + c α c )
(2.134)
C
C
When erosion occurs,
α
α
and C
<
C ; when deposition occurs,
α
α
and
c
c
c
c
C
>
C
. Substituting these relations into Eqs. (2.133) and (2.134) results in
α α
c
and
α α
. Therefore, the coefficient
α
in Eq. (2.132) is usually less than the two
c
coefficients
in Eq. (2.131) (Wu, 1991).
Galappatti and Vreugdenhil (1985) derived a function for
α
c and
α
c
through an approxi-
mate analytical integration of the pure vertical 2-D convection-diffusion equation of
suspended load. They used the “concentration” boundary condition (2.74), which
assumes equilibrium sediment transport near the bed. Armanini and di Silvio (1986)
argued that the “concentration” boundary condition may result in large errors for
fine sediments. They derived a different function for
α
through the integration of
Galappatti and Vreugdenhil by specifying the “gradient” boundary condition (2.75).
In addition, Armanini and de Silvio performed a sensitivity analysis of the approxi-
mate solutions by applying the procedure of Galappatti and Vreugdenhil directly to
the transport ( cu ) instead of to the concentration ( c ). Armanini and de Silvio's function
can be approximated as
α
h exp
1
1.5 a
h 1 / 6
1
α =
a
h +
a
ω
s
U
(2.135)
where a is th e thickness of the bottom layer, defined as a
=
33 z 0
=
33 h
/
exp
C h / g
(
, in which z 0 is the zero-velocity distance in the logarithmic velocity
distribution, and C h is the Chezy resistance coefficient of the channel. The thickness
of the bottom layer has the order of magnitude of the grain diameter when the bed is
flat, and the order of magnitude of the bed form height in the presence of bed forms.
Zhou and Lin (1998) also established a formula for
1
+ κ
)
using the analytical solutions of
the pure vertical 2-D convection-diffusion equation of suspended load with constant
diffusivity in steady, uniform flow. They adopted the analytical solution with the
“concentration” boundary condition for erosion case, and that with the “gradient”
boundary condition for deposition case. The coefficient
α
α
is determined by
2
1
R
R
4 + σ
α =
(2.136)
 
Search WWH ::




Custom Search