Geoscience Reference
In-Depth Information
1
a
=1
q
L
+
a
=2
0.5
0
0
0.5
1
1.5
t
+
=(
t
/
t
s
)
Fig. 6.11 Comparison between the rising hydrographs obtained with the lumped kinematic approach
(heavy lines) and the kinematic approach (thin lines). The same scaling is used in both cases.
The heavy curve for
a
=
1 represents the solution proposed by Horton (1938), and the heavy
curve for
a
=
2 is essentially the same as the solution used by Izzard (1946) to develop his
dimensionless hydrograph.
The rising hydrograph
For the case when
i
is applied starting at
t
0 on an initially dry plane, Equation (6.41) can
be rewritten in terms of the scaled variables defined behind (6.22); in fact, mathematically
this scaling appears to be the most obvious option. The resulting differential equation is
=
q
1
/
(
a
+
1)
L
+
(
a
+
1)
d
(
a
+
2)
1
−
q
L
+
=
(6.42)
dt
+
with the condition that
q
L
+
=
0 for
t
+
=
0. Equation (6.42) can be integrated in closed
form for values of (
a
+
1) equal to 1, 2, 3, 4, 3
/
2 and 4
/
3, but only 2 and 3 appear to
be relevant for surface runoff. As indicated in Chapter 5, the value (
a
+
1)
=
2 has been
derived from several data sets of overland flow on grass covered surfaces (see Wooding,
1965), and (
a
+
1)
=
3 is the theoretical value for laminar flow.
For
a
=
1 the solution of (6.42) is
q
L
=
tanh
2
(1.5
t
+
)
(6.43)
Similarly, for
a
=
2 the solution can be shown to be
t
+
=
0
.
125 ln[(1
+
y
+
y
2
) (1
−
y
)
−
2
]
+
(
√
3
/
4) tan
−
1
[(2
y
+
1)
/
√
3]
−
(
√
3
/
4) tan
−
1
[1
/
√
3]
(6.44)
in which
y
=
q
1
/
3
L
. Both (6.43) and (6.44) are shown in Figure 6.11, where they can be
compared with the results obtainable with Equation (6.22). Prior to the development of the
kinematic wave approach, these two solutions have been widely used in practical design.
Horton (1938) proposed Equation (6.43), with the justification that (
a
+
1)
=
2 repre-
sents a flow, which is 75% turbulent. The equation was subsequently used as the basis for the
+
