Geoscience Reference
In-Depth Information
Moreover, the bed shear stress is usually determined using the wall-function
approach in the 3-D model. Thus, condition (8.11) may not be satisfied at the inter-
face between 2-D (1-D) and 3-D reaches. Correction is often needed to satisfy this
condition exactly.
Sediment discharge
The suspended-load and bed-load discharges at the interfaces should satisfy the mass
balance conditions:
B
B
Q
b
,1
d
=
q
b
,2
d
dy
=
q
b
,3
d
dy
(8.13)
0
0
B
Q
1
d
C
1
d
=
U
2
d
h
2
d
C
2
d
dy
=
u
3
d
c
3
d
dydz
(8.14)
0
where
Q
b
,1
d
,
q
b
,2
d
, and
q
b
,3
d
are the total and unit bed-load transport rates; and
C
1
d
,
C
2
d
, and
c
3
d
are the section-averaged, depth-averaged, and local suspended-load
concentrations in the 1-D, 2-D, and 3-D subdomains, respectively.
Sediment transport capacity
Analogously to Eqs. (8.13) and (8.14), the bed-load and suspended-load transport
capacities at the interfaces have relations:
B
B
Q
b
∗
,1
d
=
q
b
∗
,2
d
dy
=
q
b
∗
,3
d
dy
(8.15)
0
0
B
B
1
α
Q
1
d
C
=
U
2
d
h
2
d
C
,2
d
dy
=
U
3
d
h
3
d
c
b
∗
,3
d
dy
(8.16)
∗
,1
d
∗
0
0
c
∗
where
Q
b
∗
,1
d
,
q
b
∗
,2
d
, and
q
b
∗
,3
d
are the total and unit equilibrium bed-load transport
rates;
C
∗
,1
d
,
C
∗
,2
d
, and
c
b
∗
,3
d
are the section-averaged, depth-averaged, and local
equilibrium suspended-load concentrations in the 1-D, 2-D, and 3-D subdomains;
h
3
d
and
U
3
d
are the flow depth and depth-averaged velocity calculated in the 3-D
subdomain; and
α
c
∗
is the adaptation coefficient introduced in Section 2.5.
At the interface between 1-D and 2-D subdomains, assuming
C
=
K
1
d
∗
,1
d
U
1
d
/(
m
and
C
U
2
d
/(
m
and substituing these relations
[
gR
1
d
ω
)
]
=
K
2
d
[
gh
2
d
ω
)
]
s
s
∗
,2
d
into Eq. (8.16) yields
0
U
2
d
h
2
d
(
U
2
d
/
m
dy
h
2
d
)
K
1
d
K
2
d
=
(8.17)
U
1
d
/
m
Q
1
d
(
R
1
d
)
Eq. (8.17) implies that if the same formula is used to determine the suspended-
load transport capacities in 1-D and 2-D models, its coefficients need to be adjusted
to satisfy condition (8.16). At the interface between 2-D and 3-D subdomains, the
situation is even more complicated, because the 3-D model involves the equilibrium
