Geoscience Reference
In-Depth Information
This result was obtained for a wide channel of rectangular cross section; for other cross-
sectional geometries, one can formulate the rate of flow in terms of decomposed variables
in a way similar to Equation (5.116), namely
Q
0
+
Q
p
=
C
r
S
0
(
A
c0
+
A
cp
)
a
+
1
(
P
w0
+
P
wp
)
−
a
Q
0
1
a
+
1
1
−
a
A
cp
A
c0
P
wp
P
w0
=
+
+
Again, because
A
cp
<<
A
c0
and
P
wp
<<
P
w0
, one obtains the analog of (5.116)
Q
p
=
(
a
+
1)
V
0
A
cp
−
aQ
0
P
wp
P
w0
(5.121)
With the equation of continuity one can, as before, define a celerity, (
dQ
p
/
dA
cp
) which
now assumes the form
aQ
0
P
w0
dP
w
dA
c
c
k0
=
(
a
+
1)
V
0
−
(5.122)
Note that this result shows that also here (
dQ
p
/
dA
cp
)
=
(
dQ
0
/
dA
c0
).
Example 5.8. Semi-infinite channel with known upstream inflow
The general solution of Equation (5.118) (and of (5.117)) is especially simple in the absence
of lateral inflow, when
i
=
0, namely
q
p
=
q
p
(
x
−
c
k0
t
)
(5.123)
Equation (5.123) shows that, in a linear kinematic channel, an upstream disturbance is
merely translated downstream. Unlike a disturbance in a linear dynamic channel and in a
linear diffusion channel, it does not undergo any deformation as it propagates downstream.
Thus, in contrast to Equations (5.72) and (5.95), the unit response that is the outflow, at a
time
t
and at a distance
x
downstream from a point
x
=
0, where the flow disturbance at
t
=
0 is a unit impulse
δ
(
x
,
t
), is now simply
u
(
t
)
=
δ
(
x
−
c
k0
t
)
(5.124)
This describes a translation of the input without distortion, as it moves down along the
channel. But it should be remembered that this represents a disturbance over and above the
steady uniform flow
q
0
.
5.4.4
The lumped kinematic approach for free surface flow: a third approximation
Besides the approximations that led to the analysis of the kinematic wave in Section
5.4.3, the lumped formulation has the additional feature that the spatial dependency
of the continuity equation (5.13) (or (5.24)) is eliminated. This is accomplished by
integrating out the
x
-variable, so that
q
or
Q
becomes located on the boundaries of the
flow domain, in the form of inflows, and outflows, and
t
becomes the rate of change
of the water depth averaged over the entire flow domain, that is the stored water. As
already explained in Chapter 1, this produces the (lumped) storage equation
∂
h
/∂
dS
dt
Q
i
−
Q
e
=
(5.125)
