Geoscience Reference
In-Depth Information
1
0.8
q
L
+
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
+
Fig. 6.9
Comparison between scaled receding hydrograph obtained with the kinematic wave approach (6.28)
(solid line for laminar flow with
a
=
2 and dashed line for turbulent flow with
a
=
2
/
3) and
experimental data obtained by Izzard (1944) on a plane covered with turf. The solid line represents
t
+
=
(1
/
3)
q
−
2
/
3
L
(1
−
q
L
+
) and the dashed line
t
+
=
(3
/
5)
q
−
2
/
5
L
(1
−
q
L
+
). The experimental data
points were obtained for the same experimental combinations as those of Figure 6.6. (After Morgali,
1970.)
+
+
(6.23) becomes
i
)
h
a
+
1
1)
K
r
h
a
t
x
=
(
a
+
+
(
K
r
/
(6.24)
As before, it is convenient to recast this in dimensionless form, by scaling the water
depth with the equilibrium depth (i.e. the initial depth prior to the recession phase) at
x
=
L
. Thus Equation (6.24) assumes the form
x
L
=
h
a
+
1
+
1)
h
a
+
(
a
+
t
+
+
(6.25)
where
t
+
is defined behind (6.22); the dimensionless water depth is
h
+
=
(
h
/
h
s
L
)in
K
r
)
1
/
(
a
+
1)
, in accordance with
(6.19). Equation (6.25) is illustrated in Figure 6.8, and shows successive water surface
profiles for increasing values of the time
t
after the cessation of the lateral inflow
i
.
Upon substitution of
h
by means of Equation (6.8), at the outflow point
x
which the equilibrium depth at the outlet is
h
s
L
=
(
iL
/
=
L
, (6.24)
becomes
q
a
/
(
a
+
1)
L
1)
K
1
/
(
a
+
1)
r
L
=
(
a
+
t
+
q
L
/
i
(6.26)
This allows the calculation of the recession hydrograph
q
L
=
q
L
(
t
), or rather in this case
implicitly as
t
=
t
(
q
L
),
=
(
a
−
1
(
iL
iq
a
/
(
a
+
1)
L
1)
K
1
/
(
a
+
1)
r
t
+
−
q
L
)
(6.27)
which is the main result of this analysis. Henderson and Wooding (1964) found that
(6.21) and (6.27) gave a good description of the experimental data for a grass-covered
surface published by Hicks (1944) and that the best fit for his three cases was obtained
with
a
=
0
.
8
,
0
.
8 and1.0, respectively.
