Geoscience Reference
In-Depth Information
coefficient to represent the diffusion and dispersion effects together. In curved chan-
nels, the dispersion fluxes become important and need to be modeled, as discussed in
Section 6.3.
Integrating the three-dimensional sediment transport equation (2.72) over the bed-
load zone leads to the bed-load mass balance equation:
+
∂(α
by
q
b
)
∂
p
m
)
∂
z
b
∂
+
∂(δ
c
δ
)
+
∂(α
bx
q
b
)
∂
(
1
−
=
D
b
−
E
b
(2.89)
t
∂
t
x
y
where
p
m
is the porosity of bed material at the bed surface,
c
δ
is the average volumetric
concentration of sediment at the bed-load zone,
q
b
is the bed-load transport rate by
volume per unit time and width (m
2
s
−
1
), and
α
by
are the direction cosines of
bed-load movement. The bed load is usually assumed to move along the direction
of bed shear stress but may be affected by secondary flows in curved channels and
gravity in channels with steep bed and bank slopes.
The first term on the left-hand side of Eq. (2.89) represents the bed change, which
results from the exchange betweenmoving sediment and bedmaterial. The second term
accounts for the storage effect. In general, the average bed-load concentration
c
δ
is
related to the bed-load transport rate
q
b
and velocity
u
b
by
c
δ
=
α
bx
and
q
b
/(δ
u
b
)
, thus yielding
q
b
u
b
+
∂(α
by
q
b
)
∂
p
m
)
∂
z
b
∂
+
∂
∂
+
∂(α
bx
q
b
)
∂
(
1
−
=
D
b
−
E
b
(2.90)
t
t
x
y
Because the bed-load velocity
u
b
is usually slower than the flow velocity, Eq. (2.90)
accounts for the temporal lag between flow and bed-load transport.
Summing Eqs. (2.86) and (2.90) leads to the overall sediment balance equation:
hC
t
β
x
+
∂
q
ty
∂
p
m
)
∂
z
b
∂
t
+
∂
+
∂
q
tx
∂
(
−
y
=
1
0
(2.91)
∂
t
t
where
C
t
is the depth-averaged concentration of total load;
q
tx
and
q
ty
are the total-
load fluxes:
q
tx
=
α
bx
q
b
+
hU
x
C
−
ε
s
h
∂
C
/∂
x
−
hD
sx
and
q
ty
=
α
by
q
b
+
hU
y
C
−
ε
s
h
∂
C
/∂
y
−
hD
sy
; and
β
t
is a correction factor for total load, related to
β
s
and
u
b
by
hC
t
1
β
=
u
b
=
(2.92)
t
hC
/β
+
q
b
/
r
s
/β
+
(
1
−
r
s
)
U
/
u
b
s
s
where
r
s
is the ratio of suspended load to total load.
2.4.2 Width-averaged 2-D model equations
Fig. 2.7 shows the configuration of a cross-section. The width-averaged quantity
of
a three-dimensional variable
φ
is defined by
b
2
b
1
φ
1
b
=
dy
(2.93)
