Geoscience Reference
In-Depth Information
Figure 5.7
Finite difference grid in 1-D channel model.
where
1.
In Eq. (5.42), the friction slope is represented by the arithmetic mean between
cross-sections
i
and
i
x
i
+
1
/
2
represents the length of the reach between cross-sections
i
and
i
+
+
1. It can also be represented by the harmonic mean
2
K
i
+
1
Q
i
+
1
|
K
i
Q
i
|
S
f
,
i
+
1
/
2
=
Q
i
+
1
|
+
(5.43)
Q
i
|
the geometric mean
Q
i
+
1
1
/
2
|
Q
i
+
1
|
Q
i
|
Q
i
|
S
f
,
i
+
1
/
2
=
(5.44)
K
i
+
1
K
i
or the conveyance mean
Q
i
+
1
2
+
Q
i
S
f
,
i
+
1
/
2
=
(5.45)
K
i
+
1
+
K
i
If the channel cross-section is suddenly expanded or contracted, a local head loss
should be considered and Eq. (5.42) is replaced by
Q
i
+
1
β
i
Q
i
2
gA
i
+
=
β
i
+
1
Q
i
+
1
2
gA
i
+
1
+
x
i
+
1
/
2
2
|
Q
i
+
1
|
Q
i
|
Q
i
|
z
s
,
i
+
z
s
,
i
+
1
+
K
i
+
1
K
i
β
i
+
1
Q
i
+
1
2
gA
i
+
1
−
β
i
Q
i
2
gA
i
+
λ
(5.46)
i
+
1
/
2
where
2
is the coefficient of local head loss due to channel expansion or
contraction in the reach between cross-sections
i
and
i
λ
i
+
1
/
+
1.
5.2.1.2 Solution of discretized steady flow equations
The solution procedure for Eq. (5.42) differs in cases of subcritical and supercritical
flows. For subcritical flow, a flow discharge is usually specified at the inlet and a water
