Geoscience Reference
In-Depth Information
Fig. 5.2
Illustration of the
different terms in
Equation (5.23), i.e.
the integrated shallow
water momentum
equation under steady
conditions.
V
1
2
(
2
g
)
S
f
δ
x
/
EGL
FS=HGL
V
2
2
(
2
g
)
/
h
1
h
2
δ
S
0
x
δ
x
two-dimensional, they assume the form (see, for example, Stoker, 1957)
∂
A
c
∂
∂
∂
t
+
(
VA
c
)
−
Q
l
=
0
(5.24)
x
and
∂
g
∂
S
0
V
∂
V
∂
V
h
Q
l
V
A
c
t
+
x
+
x
+
S
f
−
+
=
0
(5.25)
∂
∂
where
A
c
is the wetted cross-sectional area and
Q
l
the lateral inflow per unit length of
channel.
In hydrology the average velocity is often of less importance than the rate of flow,
=
VA
c
. For a wide channel of surface width
B
s
=
∂
A
c
/∂
Q
(
h
), (5.13) and (5.22) assume
the form
∂
A
c
∂
t
+
∂
Q
∂
x
−
Q
l
=
0
(5.26)
and
A
c
∂
x
+
gA
c
−
Q
2
B
s
∂
Q
∂
2
A
c
Q
∂
Q
h
gA
c
(
S
f
−
t
+
x
+
S
0
)
=
0
∂
∂
5.3
FRICTION SLOPE
In the determination of the friction slope in the shallow water equations, it is commonly
assumed that the resistance to flow, resulting from the last two terms of Equation (5.15),
acts in the same way in unsteady nonuniform flow as it does in steady uniform flow.
Thus, from inspection of (5.21) or (5.22) for such conditions one has
S
f
=
S
0
(5.27)
The definition of
S
f
for two-dimensional flow, as given in Equation (5.20), can be
written for uniform flow, when
∂
()
/∂
x
=
0, as
h
−
1
∂τ
zx
∂
S
f
=
dz
(5.28)
ρ
gh
z
0
