Geoscience Reference
In-Depth Information
max
1
ψ
1
−
ψ
−
≤
θ
≤
D
r
,
1 with
C
r
+
ψ
C
r
−
(
1
−
ψ)
D
r
max
0, 1
C
r
D
r
−
≤
ψ
≤
1
(5.150)
where
C
r
is the Courant number
U
t
/
x
; and
D
r
is a scale factor of non-equilibrium
sediment transport, defined as
D
r
β
tk
is set to 1 here.
Condition (5.150) is sufficient but not necessary for the numerical stability of
Eq. (5.138). If
L
t
=
U
t
/
L
t
. Note that
x
,
D
r
C
r
, then
ψ
and
θ
should be given values close to 1.
5.3.3.4 Advantages of the coupled sediment calculation
procedure
Stabilities of explicit and implicit schemes for bed-material gradation
The decoupled and coupled sediment calculation procedures are compared by analyz-
ing the stabilities of the explicit and implicit schemes for the bed-material gradation
in Eq. (5.35). For convenience, Eqs. (5.140) and (5.141) are replaced by
Q
n
+
1
t
1
=[
θ
p
p
n
+
1
p
bk
,
i
+
1
]
Q
∗
n
+
1
tk
,
i
1
+
(
1
−
θ
p
)
(5.151)
∗
k
,
i
+
bk
,
i
+
+
1
θ
p
is the temporal weighting factor for bed-material gradation:
=
where
1 for the
implicit scheme (coupled calculation procedure), and 0 for the explicit scheme
(decoupled calculation procedure).
Inserting Eqs. (5.142), (5.143), and (5.151) into Eq. (5.135) yields the equation for
the bed-material gradation in the mixing layer:
A
n
+
1
m
,
i
p
∗
bk
,
i
+
1
A
m
,
i
+
1
−
(
−
A
b
,
i
+
1
)
+
1
p
n
+
1
bk
,
i
1
=
+
A
n
+
1
m
,
i
f
1
e
k
)θ
p
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
+
1
+
1
Q
∗
n
+
1
tk
,
i
A
m
,
i
+
1
−
(
p
bk
,
i
+
1
[
f
2
−
f
1
e
k
)(
1
−
θ
)
]
p
+
1
+
A
n
+
1
m
,
i
f
1
e
k
)θ
p
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
+
1
+
1
f
1
e
0
k
+
f
0
k
+
(5.152)
A
n
+
1
m
,
i
p
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)θ
+
1
+
1
To simplify the analysis, it is assumed that
A
n
+
1
m
A
m
. For deposition, usually
≈
A
n
+
1
m
,
i
A
m
,
i
+
1
1
, then
p
∗
bk
,
i
+
1
p
bk
,
i
+
1
, and the bed-material gradation
A
b
,
i
+
1
+
≥
=
+
error,
δ
, is governed by
n
A
n
+
1
Q
∗
n
+
1
tk
,
i
−
A
b
,
i
+
1
−
(
f
2
−
f
1
e
k
)(
1
−
θ
)
p
m
,
i
+
1
+
1
n
+
1
δ
=
δ
(5.153)
A
n
+
1
m
,
i
p
Q
∗
n
+
1
tk
,
i
+
(
f
2
−
f
1
e
k
)θ
+
1
+
1
