Geoscience Reference
In-Depth Information
Again, with the same scaled variables as in (6.22), (6.27) can be expressed in a more
universal way, as follows
t
+
=
(
a
1)(
q
L
+
)
a
/
(
a
+
1)
−
1
(1
+
−
q
L
+
)
(6.28)
The recession hydrograph described by Equation (6.28) is illustrated graphically in
Figure 6.5 for the case of fully turbulent flow with
a
=
2
/
3. Figure 6.9 shows a com-
parison between (6.28), both for laminar flow with
a
=
2 and for turbulent flow with
a
3, and experimental data obtained by Izzard (1946) on a turf surface for the same
experimental set-up combinations shown in Figure 6.6; it can be seen that while the
rising hydrographs exhibited turbulent flow, the recessions were somewhat closer to the
laminar curve, except initially.
=
2
/
Runoff sequence for a short rainfall burst
In the case that the rainfall duration
D
is shorter than the time to equilibrium, i.e. for
D
<
t
s
,
the water surface profile at the end of the rain, (i.e. the initial profile at the beginning of the
decay phase) is typically represented by one of the profiles 0ABC shown in Figure 6.4. Let
in what follows the reference
t
=
0 indicate the beginning of the rainfall. If
h
=
h
0
(
=
iD
)
(cf. Equation (6.16)) denotes the largest depth achieved during the buildup phase, once the
rainfall stops, the point A moves downstream at a constant velocity [(
a
+
1)
K
r
h
0
] and it
will reach
x
=
L
at a time (see Equation (6.24))
D
+
t
p
=
D
+
L
−
(
K
r
/
i
)
h
a
+
0
(
a
+
1)
K
r
h
0
(6.29)
Thus as long as
D
≤
t
<
D
+
t
p
, the water depth and the outflow rate at
x
=
L
remain
constant at, respectively,
h
=
h
0
and
K
r
h
a
+
1
0
q
L
=
(6.30)
After that, for
t
≥
t
p
+
D
, the outflow rate is given by (6.27), but with the addition of a time
shift
D
to account for the duration of the lateral inflow.
To summarize, the hydrograph sequence for the case, when the rain stops before full
equilibrium is reached, is as follows in terms of scaled variables. As the lateral inflow starts,
the outflow rate at
x
=
L
is given by the first of Equations (6.22). At the moment
t
=
D
,
that is
t
+
(
=
t
/
t
s
)
=
D
+
, when the lateral inflow ceases, the outflow rate is
(
D
+
)
a
+
1
q
L
+
=
(6.31)
where, as before,
q
L
+
=
(
q
L
/
q
s
L
)
,
in which
q
s
L
=
iL
, and
D
+
=
(
D/t
s
). The rate of flow
at
x
=
L
will remain constant at the value given by (6.31) for a duration (cf. (6.29))
t
p
+
=
1
−
h
a
+
1
0
+
(
a
+
1)
h
0
+
−
1
(6.32)
where
t
p
+
=
(
t
p
/
t
s
), and
h
0
+
=
(
h
0
/
h
s
L
). Because
h
0
=
(
iD
), this duration of constant flow
can also be expressed more conveniently in terms of the relative rainfall duration
D
+
,as
follows
t
p
+
=
1
−
D
a
+
1
+
(
a
+
1)
D
a
+
−
1
(6.33)
From then on, i.e. for
t
+
≥
(
D
+
+
t
p
+
) after the onset of the rain, the rate of flow is given by
Equation (6.28). Since the time reference
t
=
0 is taken at the onset of the rain, here (6.28)
