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of
h
does the mean velocity
V
occur? (Note: in hydrologic practice, this depth is often taken as
z
=
0
.
4
h
.
)
5.10
Consider a very wide channel in which the velocity distribution is given by the power-type equation
(5.37); assume
m
=
1
/
6. At what fraction of the depth
h
does the mean velocity
V
occur? (Note:
in hydrologic practice, this depth is often taken as
z
=
0
.
4
h
.)
5.11
In hydrologic practice, it is common to dete
rm
ine
th
e mean velocity,
V
,
as the average of mea-
surements at 0
.
2
h
and at 0
.
8
h
; thus,
V
=
(
u
0
.
2
+
u
0
.
8
)
/
2. Calculate what the error would be if
the velocity profile were logarithmic as given by Equation (5.34); assume that
d
0
=
0.
5.12
Demonstrate that Equation (5.51) is a solution of (5.49).
5.13
Consider the diffusion approximation describing flow in a wide open channel, as given by (5.87)
with
in which the parameters are defined in (5.43). Show for a linearized
channel that the coefficient of the second term on the left, namely (
bQ
/α
r
)(
d
α
r
/
α
r
=
(
C
r
h
a
+
1
)
−
1
/
b
,
dA
c
), is equivalent
with the Kleitz-Seddon equation for the celerity, i.e.
c
k0
=
(
dQ
0
/
dA
c0
). Hint: linearization is
accomplished by putting
A
c
=
A
c0
+
A
cp
and
Q
=
Q
0
+
Q
p
, and by assuming
A
cp
and
Q
p
to be relatively small perturbations.
5.14
The diffusion equation for open channel flow can be written in the form of (5.92) with (5.93) and
(5.94). Consider a flood routing problem, in which the equation becomes (in m and s):
∂
2
Q
∂
Q
∂
17
∂
Q
1365
∂
+
2
.
=
x
2
(a) Give a rough estimate of the mean velocity of the flow,
V
(not the flood wave), from this
equation. (b) Show how you derive the equation, which you use to estimate
V
. (c) What is the
significance of the magnitude of the coefficients (namely 2.17 and 1365 in this case) in regard to
the shape of the flood wave? In other words, what would be the effect of each coefficient, if it
were larger or smaller, on the evolution of the shape of the flood wave? Discuss the effect of each
coefficient separately (one sentence each).
t
∂
x
5.15
(a) Derive Equation (5.114) from the Kleitz-Seddon equation (5.112). (b) Implement (5.114) to
calculate the celerity of a kinematic wave in a channel with a triangular cross section. Make use
of the GM equation, (5.41), in which
S
f
=
S
0
.
5.16
Multiple choice. Indicate which of the following statements are correct. The derivation of the
shallow-water equations, (5.13) and (5.22), requires the following.
(a)
The lateral inflow does not depend on
x
; in other words, it must be uniform along the
direction of flow.
(b)
The pressure distribution is hydrostatic
al
ong
z
, the direction normal to the bottom.
az
m
, where
a
and
m
are constants.
(c)
The velocity profile obeys a power law,
u
=
(d)
The slope of the channel bottom must be small, in the direction of flow, to allow substitution
of
−
sin
θ
by
S
0
.
(e)
The roughness of the channel must be constant in the direction of flow.
