Geoscience Reference
In-Depth Information
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PROBLEMS
5.1
Derive the
x
-component of the Reynolds equations, namely (5.14), from the corresponding com-
ponent of the Navier-Stokes equations (1.12).
5.2
Derive Equation (5.35) from (5.34); show the intermediate steps.
5.3
Calculate the Boussinesq correction factor,
β
c
, for the logarithmic velocity profile in a wide open
channel. Thus, integrate (5.19), in which the velocities are given by (5.34) and (5.36). What is its
value for (
h
/
z
0
)
=
100?
5.4
Calculate the Boussinesq correction factor,
β
c
, for the power-type velocity profile in a wide, open
channel. Thus, integrate (5.19), in which the velocities are given by (5.37) and (5.38). What is its
value for
m
=
1
/
6?
5.5
Write down the expression for the hydraulic radius,
R
h
using Equation (5.40) for a channel (a)
with a triangular cross section (
B
b
=
,
=
0)
, and (b) with a rectangular cross section (
B
s
B
b
).
5.6
What would be the values of the powers
a
and
b
in Equation (5.43), if the velocity profile in a very
wide, open channel were given by (5.37) with the classical value,
m
=
1
/
7.
5.7
Consider flow in a channel with a trapezoidal cross section and side banks having a slope, 1 vertical
to 2 horizontal, and a bottom width,
B
b
=
5 m; the channel roughness coefficient is
n
=
0
.
015
,
and the bed slope is
S
0
=
0
.
001. Given the water depth, at the center as
h
=
2 m, under uniform,
steady-state conditions, calculate the velocity,
V
; the rate of flow,
Q
=
VA
c
; and the Reynolds
number, Re
=
VR
h
/ν
.
60 m
3
s
−
1
,
5.8
For a channel with the same characteristics as in Problem 5.7, with a flow rate of
Q
=
calculate the depth of flow,
h
, at the center of the channel. Use trial and error.
5.9
Consider a very wide open channel in which the velocity profile is given by the logarithmic
equation (5.34); assume that the displacement height is negligible or
d
0
=
0. At what fraction
