Geoscience Reference
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can be treated as “flow loop” and “sediment loop,” respectively, and then coupled
using this iteration procedure.
Fully coupled solution procedure
Fully coupled models can be found in Lyn and Goodwin (1987), Holly and Rahuel
(1990), Correia et al . (1992), and Yeh et al . (1995). In particular, Holly and
Rahuel (1990) proposed a fully coupled procedure for calculating non-uniform sed-
iment transport, which is herein extended to solve Eqs. (5.133), (5.135), (5.137),
(5.139) (5.141), and (5.180)-(5.182). To reduce the effort of matrix inversion
required in the fully coupled solution procedure, these equations are re-organized
as follows.
Inserting Eqs. (5.135) and (5.141) into Eq. (5.139) yields
tp bk , i + 1 Q n + 1
θ
tk , i
+
1
A bk , i + 1 =
A b , i + 1
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
p m )
θ
+ (
1
+
m , i
+
1
+
1
tA n + 1
m , i
θ
+
1
Q n + 1
tk , i
+
+
1
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
p m )
θ
1 + (
1
+
+
m , i
+
1
tQ n + 1
tk , i
A n + 1
m , i
A m , i + 1 p bk , i + 1 +
p bk , i + 1 (
A m , i + 1 ) ]
θ
1 [
+
+
1
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
p m )
θ
1 + (
1
+
+
m , i
+
1
tL n + 1
t , i
1 A n + 1
Q tk , i + 1
Q t k , i + 1 )/
L t , i + 1
(
θ)
(
1
+
m , i
+
1
+
(5.183)
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
θ
1 + (
1
p m )
+
+
m , i
+
1
and then summing Eq. (5.183) over all size classes and using Eq. (5.133) yields
N
r k , i + 1 Q n + 1
tk , i
A b , i + 1 =
1 +
s i + 1
(5.184)
+
k
=
1
where
tA n + 1
m , i
θ
1
λ
+
1
r k , i + 1 =
, and
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
p m )
θ
+ (
1
+
m , i
+
1
+
1
tQ n + 1
tk , i
A m , i + 1 p bk , i + 1 +
p bk , i + 1 (
A n + 1
m , i
A m , i + 1 ) ]
N
θ
1 [
1
λ
+
1
+
s i + 1
=−
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
θ
1 + (
1
p m )
+
+
m , i
+
1
k
=
1
tL n + 1
t , i
1 A n + 1
Q tk , i + 1
Q t k , i + 1
L t , i + 1
N
(
1
θ)
1
λ
+
m , i
+
1
+
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
θ
1 + (
1
p m )
+
+
m , i
+
1
k
=
1
tp bk , i + 1 Q n + 1
N
θ
tk , i
+
1
with
λ =
1
.
tQ n + 1
tk , i
L n + 1
t , i
1 A n + 1
p m )
θ
+ (
1
+
m , i
+
1
k
=
1
+
1
 
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