Geoscience Reference
In-Depth Information
distance,
x
,
where the flow will change from laminar to turbulent. Assume that this occurs at a
critical Reynolds number,
Vh
/ν
=
500.
6.3
(a) Derive the water surface profile,
h
=
h
(
x
), once steady state has been established, on a plane,
L
=
30 m long, with a slope,
S
0
=
0
.
0015, resulting from a rainfall rate,
P
=
37 mm h
−
1
. Assume
that the kinematic wave assumption is valid. (b) What is the depth,
h
,
of the flow, at
x
=
L
=
30 m,
in this analysis?
6.4
In the previous problem, use is made of the kinematic wave assumption. If this simplification were
not valid, which terms would still be negligible in the shallow water equations (6.1) and (6.2)?
Thus, write these equations in their simplest form, which would allow the solution of the same
steady-state problem, but without the kinematic wave assumption. In doing so, express
S
f
in terms
of the dependent variables,
V
and
h
, for a wide channel.
6.5
Assume that free-surface flow over a plane uniform surface resulting from a steady rain intensity,
P
,
can be described by the kinematic-wave method, so that (6.1) is the governing equation, with
i
=
P
. Assume that the flow is fully turbulent, and that its dynamics can be described by the GM
equation, (5.41). With what celerity does a point on the free surface, with a given depth,
h
=
D
g
,
move downstream, after the rain ceases? Assume that the roughness coefficient
n
is not affected
by the impact of the raindrops. Show how you obtain this answer.
6.6
Show that the area, ABC, in Figure 6.5 represents, indeed, the volume stored on the plane under
equilibrium flow conditions, and that it is equal to the area DEF. (Hint: perform the integration
qdt
=
tdq
between the appropriate limits.)
6.7
A concrete pavement is
L
=
40 m long, and it has a slope of
S
0
=
0
.
0015. A uniform rainfall starts
at
t
=
0, and continues for a long time at a steady rate of
P
=
25mm h
−
1
. (a) First, determine
whether the maximum flow, at
x
=
L
,
is laminar or turbulent. (b) Compute the rising hydrograph
(in mm h
−
1
to make it comparable to the rainfall rate), at the lower end of the pavement, after this
rainfall starts. Use the kinematic wave method, with and without the effect of the rain. (c) Plot the
two results on one figure as
q
=
q
(
t
) in mm h
−
1
, and
t
in hours. Note: under turbulent conditions,
the effect of the rainfall on the flow is usually neglected. Under laminar conditions, (5.33) can be
used to incorporate this effect.
6.8
Same as the previous problem for a plane soil surface with a short grassy vegetation and with
L
=
45 m,
S
0
=
0
.
02, and
P
=
85 mm h
−
1
.
6.9
A concrete pavement is
L
005. A steady rainfall lasts
for a long time at a rate of
P
=
30 mm h
−
1
. (a) First, determine whether the maximum flow, at
x
=
L
,
is laminar or turbulent. (b) Compute the recession hydrograph (in cm h
−
1
) at the lower end
of the pavement after this rainfall stops. Use the kinematic wave method, and neglect the effect of
the raindrop impact and the temporary increase at
t
=
0 (when the rain stops). Plot the hydrograph
in mm h
−
1
to match the units of
P
. (c) On the basis of the result obtained in (b), what is the flow
rate, in mm h
−
1
,at
x
=
L
, 15 min after the rain stops?
=
35 m long, and it has a slope of
S
0
=
0
.
6.10
Consider the same situation as described in the previous problem. Initially, after the rain stops,
the runoff, at
x
=
L
,
is likely to increase a little on account of the decreased resistance. Calculate
