Geoscience Reference
In-Depth Information
The bed load is simulated using the equilibrium transport model (Wang and Adeff,
1986; van Rijn, 1987; Spasojevic and Holly, 1993; Olsen, 2003) or the non-
equilibrium transport model (Wu
et al
., 2000a). As described in Section 2.6, the
non-equilibrium transport model is more adequate. Because the bed-load layer is very
thin, the bed-load transport equation in the 3-D model has the same formulation as
the horizontal 2-D model equation (2.158):
+
∂(α
by
q
bk
)
∂(
q
bk
/
u
bk
)
+
∂(α
bx
q
bk
)
1
L
(
=
q
b
∗
k
−
q
bk
)
(7.46)
∂
t
∂
x
∂
y
with slight difference in determining the direction cosines
α
by
of bed-load
transport. Because secondary flows, such as the helical flow in curved channels, can
be simulated somewhat, their effects on sediment transport are automatically taken
into account in the 3-D model when
α
bx
and
α
by
are set as the direction cosines
of the calculated bed shear stress. However, if the bed slope is steep, the effect of
gravity should be considered by adjusting
α
bx
and
α
bx
and
α
by
using the methods introduced
in Section 6.3.4.
The bed change is determined by
∂
k
=
z
b
∂
1
L
(
p
m
)
(
−
D
bk
−
E
bk
+
q
bk
−
q
b
∗
k
)
1
(7.47)
t
or by the overall sediment balance equation:
∂
q
bk
u
bk
+
z
s
c
k
dz
x
+
∂
q
tky
∂
z
b
∂
k
+
∂
+
∂
q
tkx
∂
p
m
)
(
−
=
1
0
(7.48)
t
∂
t
y
z
b
+
δ
where
q
tkx
and
q
tky
are the specific fluxes of total load in the
x
- and
y
-directions:
u
x
c
k
−
ε
dz
z
s
s
∂
c
k
q
tkx
=
α
bx
q
bk
+
∂
x
z
b
+
δ
u
y
c
k
dz
z
s
s
∂
c
k
∂
q
tky
=
α
by
q
bk
+
−
ε
(7.49)
y
z
b
+
δ
As compared with Eq. (7.47), Eq. (7.48) more easily ensures mass conservation but
is more complex.
The equilibrium near-bed suspended-load concentration and bed-load transport rate
need to be determined using the empirical relations introduced in Sections 3.4 and 3.5.
In general, these formulas can be written as
p
bk
c
bk
,
p
bk
q
bk
c
b
∗
k
=
q
b
∗
k
=
(7.50)
where
p
bk
is the fraction of size class
k
in the mixing layer of bed material,
c
bk
is
the potential equilibrium concentration of the
k
th size class of suspended load at the
