Geoscience Reference
In-Depth Information
The bed-load transport and bed change of size class
k
are governed by Eqs. (2.158)
and (2.159), which are written below for convenience:
+
∂(α
by
q
bk
)
∂
∂(
q
bk
/
u
bk
)
+
∂(α
bx
q
bk
)
∂
1
L
(
=
q
b
∗
k
−
q
bk
)
(6.54)
∂
t
x
y
∂
z
b
∂
1
L
(
p
m
)
(
1
−
k
=
αω
sk
(
C
k
−
C
)
+
q
bk
−
q
b
∗
k
)
(6.55)
∗
k
t
and the total rate of change in bed elevation is determined by Eq. (2.160). Note that
the bed load is assumed to move along the direction of bed shear stress if the effect
of bed slope in ignored; thus, the direction cosines of bed-load movement i
n a nearly
straight channel are given by
U
x
+
U
y
,
according to Eq. (6.4). The effect of bed slope on bed-load transport is discussed in
Section 6.3.4.
To close the set of equations (6.53)-(6.55), the equilibrium suspended-load concen-
tration
C
∗
k
and the bed-load transport rate
q
b
∗
k
need to be determined using empirical
formulas, which can generally be written as
α
bx
=
U
x
/
U
and
α
by
=
U
y
/
U
with
U
=
p
bk
C
k
,
p
bk
q
bk
C
=
q
b
∗
k
=
(6.56)
∗
k
where
C
k
is the potential equilibrium concentration of the
k
th size class of suspended
load,
q
bk
is the potential equilibrium transport rate of the
k
th size class of bed load,
and
p
bk
is the fraction of size class
k
in the mixing layer of the bed material.
The multiple-layer bed material sorting model introduced in Section 2.7.2 is applied
here. For example, the bed-material gradation in the mixing layer is determined by
∂
k
+
p
bk
∂δ
∂(δ
m
p
bk
)
∂
z
b
∂
−
∂
z
b
∂
m
=
(6.57)
t
t
∂
t
t
Bed-material load transport model
The bed-material (total) load transport equation can be obtained by summing
Eqs. (6.53) and (6.54) and using Eq. (2.149) for the sediment exchange at the bed.
The resulting equation is written as
hC
tk
β
tk
E
s
,
x
h
∂(
+
∂(
hU
y
C
tk
)
∂
∂
∂
+
∂(
hU
x
C
tk
)
=
∂
∂
r
sk
C
tk
)
∂
∂
t
x
y
x
x
E
s
,
y
h
∂(
+
∂
∂
r
sk
C
tk
)
∂
+
α
ω
(
C
t
∗
k
−
C
tk
)(
k
=
1, 2,
...
,
N
)
(6.58)
t
sk
y
y
where
C
tk
and
C
t
∗
k
are the actual and equilibrium (capacity) depth-averaged
concentrations of the
k
th size class of bed-material load, respectively;
β
tk
is the correc-
tion factor determined using Eq. (2.92);
α
t
is the adaptation coefficient of bed-material
load, defined as
α
t
=
(
Uh
)/(
L
t
ω
s
)
with
L
t
being the adaptation length; and
r
sk
is the
