Geoscience Reference
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and bed material sorting is relatively low. Therefore, Wu (2004) suggested another
iteration procedure, which is described below.
Substituting Eqs. (6.70) and (6.72) into Eq. (6.67) yields
n
m
,
P
C
n
+
1
+
1
n
m
,
P
q
n
+
1
+
1
αω
sk
t
δ
+
t
δ
/
L
k
,
P
bk
,
P
z
bk
,
P
=
1
m
,
P
n
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
p
m
)δ
(
1
−
+
αω
+
/
L
sk
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
1
m
,
P
n
+
n
m
,
P
p
bk
,
P
+
(δ
n
m
,
P
p
∗
bk
,
P
]
[
αω
sk
+
/
L
][
δ
−
δ
)
−
n
1
m
,
P
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
p
m
)δ
(
1
−
+
αω
+
/
L
sk
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
p
∗
bk
,
P
[
αω
sk
+
/
L
]
+
L
z
b
,
P
(6.73)
n
1
m
,
P
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
p
m
)δ
(
1
−
+
αω
+
/
sk
Summing Eq. (6.73) over all size classes and using Eq. (6.68) yields the following
equation for the total change in bed elevation:
⎧
⎨
⎩
n
m
,
P
C
n
+
1
+
1
n
m
,
P
q
n
+
1
+
1
N
αω
t
δ
+
t
δ
/
L
sk
k
,
P
bk
,
P
z
b
,
P
=
1
m
,
P
n
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
(
1
−
p
m
)δ
+
αω
sk
+
bk
,
P
/
L
k
=
1
⎫
⎬
⎭
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
bk
,
P
m
,
P
p
bk
,
P
+
(δ
n
n
1
m
,
P
+
n
m
,
P
p
∗
bk
,
P
]
N
[
αω
+
/
L
][
δ
−
δ
)
sk
−
n
1
m
,
P
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
(
1
−
p
m
)δ
+
αω
sk
+
bk
,
P
/
L
k
=
1
⎨
⎩
⎬
⎭
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
p
∗
bk
,
P
N
[
αω
sk
+
bk
,
P
/
]
L
1
−
(6.74)
n
1
m
,
P
+
tC
∗
n
+
1
k
,
P
tq
∗
n
+
1
(
1
−
p
m
)δ
+
αω
sk
+
bk
,
P
/
L
k
=
1
Thus, the semi-coupled flow and sediment calculations using the suggested iteration
procedure are executed as follows:
(1) Calculate the flow field;
(2) Determine
z
b
,
P
using Eq. (6.74) with estimated
C
n
+
1
and
q
n
+
1
bk
;
k
(3) Compute
z
bk
,
P
using Eq. (6.73);
(4) Calculate
p
n
+
1
bk
using Eq. (6.70);
(5) Determine
C
n
+
1
∗
and
q
n
+
1
b
k
using Eq. (6.72);
k
∗
(6) Calculate
C
n
+
1
k
using Eq. (6.65);
(7) Calculate
q
n
+
1
bk
using Eq. (6.66);
(8) Use the calculated
C
n
+
1
k
and
q
n
+
1
bk
as new estimates and repeat steps (2)-(7) until
the convergent solution is obtained;
(9) Update the bed topography using Eq. (6.69) and the bed-material gradations in
the subsurface layers;
(10) Return to step (1) for the next time step until a specified time is reached.
Eq. (6.74) constitutes a tighter correlation among the unknown sediment variables.
