Geoscience Reference
In-Depth Information
namely 0
L
. The time required to reach this steady state equilibrium is obtained
by combining (6.16) with (6.19) at
x
≤
x
≤
=
L
,or
K
r
i
a
)
1
/
(
a
+
1
)
t
s
=
(
L
/
(6.20)
When the duration of lateral inflow exceeds the time to equilibrium, the outflow hydro-
graph at
x
L
can be readily obtained. Prior to the time to equilibrium, it is obtained
from (6.8) combined with (6.16); once equilibrium is established it can be obtained from
(6.18). Thus the rising hydrograph at
x
=
=
L
is given by
K
r
i
a
+
1
t
a
+
1
for
t
≤
t
s
q
L
=
(6.21)
iL
for
t
≥
t
s
Equation (6.21) is the main result of this section. The performance of (6.20) has been
compared with experimental data on turbulent sheet flow from the literature by McCuen
and Sp
ies
s (1995); they concluded that its use should be restricted by the criterion
(
nL
/
√
S
0
)
30 m.
To generalize the result shown in Equation (6.21), it is useful to express it in terms
of dimensionless variables. The simplest way to proceed here is to take the equilibrium
discharge rate (
iL
) from the plane and the time to equilibrium
t
s
as scaling variables.
This reduces Equation (6.21) to
<
(
t
+
)
a
+
1
for
t
+
≤
1
q
L
+
=
(6.22)
1
for
t
+
≥
1
where now
q
L
+
=
(
q
L
/
q
sL
) and
t
+
=
(
t
/
t
s
)
,
in which
q
s
L
=
(
iL
) is the equilibrium out-
flow rate at
x
L
; this rising hydrograph is illustrated in Figure 6.5. Figures 6.6 and
6.7 show a comparison between the kinematic wave rising hydrograph and experimental
data of Izzard (1944, 1946) scaled in the same manner. It can be seen that some of the
hydrographs in Figure 6.6 initially start out as laminar flow, and change to turbulent
flow later on around
t
+
=
=
9 some dynamic effects, which are
neglected in the kinematic formulation, appear to enter into play.
0
.
4; also, around
t
+
=
0
.
Decay phase: recession hydrograph after rain stops
As soon as
i
0. Hence, to the
observer moving at a celerity given by (6.11) it now appears that
h
remains constant. In
other words, (6.11) is the velocity of a point of the water surface with the given value of
h
. Thus on the
h
=
0, according to Equation (6.10) one has (
dh
/
dt
)
=
x
plane the characteristics describe straight lines parallel to the surface
of the plane where
h
−
0. One such characteristic is shown in Figure 6.8, as going from
A
1
to A
2
,A
3
, etc., for successive values of time
t
after the lateral inflow has ceased.
Because
h
remains constant, Equation (6.11) can be integrated immediately to yield
=
1)
K
r
h
a
t
=
+
+
x
(
a
x
0
(6.23)
where
x
0
is the starting value of
x
, i.e. its initial value at the time
t
=
0, when the rain
stops and the recession starts.
In case the duration of the rain is longer than the time to equilibrium
t
s
, i.e.
D
t
s
,
initially the water surface has an equilibrium profile as given by Equation (6.19), so that
>
