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Problem 10.C1
February 1998 saw the failure of the earthen dam impounding California Jim’s Pond in southern
Rhode Island. The resulting flood raised temporary havoc in the nearby village of Peace Dale.
The pond is 17 acres in area and 15 ft deep and was full from heavy rains. The breach in the dam
was 22 ft wide and 15 ft deep. Estimate the time required to drain the pond to a depth of 2 ft.
Solution 10.C1
Unfortunately, Table 10.2, item b, does not really apply, because the breach is so deep:,
H = y 0.5Y = 0. Nevertheless, it’s all we have, and ponds don’t rupture every day, so let’s use
it! A control volume around the pond yields
Problem 10.C2
A circular, unfinished concrete drainpipe is laid on a slope of 0.0025 and is planned to carry
from 50 to 300 ft3/s of run-off water. Design constraints are that (1) the water depth should be no
more than three-fourths of the diameter; and (2) the flow should always be subcritical. What is
the appropriate pipe diameter to satisfy these requirements? If no commercial pipe is exactly this
calculated size, should you buy the next smallest or the next largest pipe?
Solution 10.C2
For unfinished concrete n 0.014. From the geometry of Fig. 10.6, 3/4- full corresponds to an angle
= 120. This level should be able to carry the maximum 300 ft3/s flow:
Problem 10.C3
Extend Prob. 10.72, whose solution was V2 ≈ 4.33 m/s. (a) Use gradually-varied theory to estimate
the water depth 10 m down at section (3) for the 5 unfinished concrete slope shown in Fig. P10.72.
(b) Repeat your calculations for an upward (adverse) slope of 5 . (c) When you find that part (b) is
impossible with gradually varied theory, explain why and repeat for an adverse slope of 1.
Problem 10.72*
Water approaches the wide sluice gate of Figure P10.72 at V1 = 0.2 m/s and y1 = 1 m.
Accounting for upstream kinetic energy, estimate, at outlet section 2, (a) depth; (b) velocity; and
(c) Froude number.
Solution 10.C3
For unfinished concrete take n 0.014. Note that y(0) = 0.0462 m. For the given flow rate
q = 0.2 m3/sm, first calculate reference depths:
Problem 10.C4
It is desired to meter an asphalt rectangular channel of width 1.5 m which is designed for
uniform flow at a depth of 70 cm and a slope of 0.0036. The vertical sides of the channel are
1.2 m high. Consider using a thin-plate rectangular weir, either full or partial width (Table 10.2a,
b) for this purpose. Sturm [7, p. 51] recommends, for accurate correlation, that such a weir have
Y 9 cm and H/Y 2.0. Determine the feasibility of installing such a weir which will be accurate
and yet not cause the water to overflow the sides of the channel.
Solution 10.C4
For asphalt take n = 0.016. We have only one partial-width formula, and that is from
Table 10.2b. We are slightly outside the limits of applicability, but we will use it anyway:
Problem 10.C5
Figure C10.5 shows a hydraulic model of a compound weir, that is, one which combines two
different shapes. (a) Other than measurement, for which it might be poor, what could be the
engineering reason for such a weir? (b) For the prototype river, assume that both sections have
sides at a 70 angle to the vertical, with the bottom section having a base width of 2 m and the
upper section having a base width of 4.5 m, including the cut-out portion. The heights of lower
and upper horizontal sections are 1 m and 2 m, respectively. Use engineering estimates and make
a plot of upstream water depth versus Petaluma River flow rate in the range 0 to 4 m3/s. (c) For
what river flow rate will the water overflow the top of the dam?
Solution 10.C5
We have no formulas in the text for a compound weir shape, but we can still use the concept of
weir flow and estimate the discharge for various water depths.
Problem 10.C6
Figure C10.6 shows a horizontal flow of water through a sluice gate, a hydraulic jump, and over
a 6-ft-high sharp-crested weir. Channel, gate, jump, and weir are all 8 ft wide unfinished
concrete. Determine (a) the flow rate, (b) the normal depth, (c) y2, (d) y3, and (e) y4.
Solution 10.C6
[Note to instructor: This problem combines the three parts of Problems P10.76, 10.90, and
10.114.]
Problem 10.C7
Consider the V-shaped channel in Fig, C10.7, with an arbitrary angle
. Make a continuity and
momentum analysis of a small disturbance
y << y, as in Fig. 10.4. Show that the wave
propagation speed in this channel is independent of
and does not equal the wide-channel result
co = (gy)1/2.
Solution 10.C7
NOTE: This derivation is heavy algebra, but the result is rewarding. A much easier derivation is
Problem 10.W1
Free-surface problems are driven by gravity. Why do so many of the formulas in this chapter
contain the square root of the acceleration of gravity?
Solution 10.W1
Problem 10.W2
Explain why the flow under a sluice gate, Fig. 10.10, either is or is not analogous to
compressible gas flow through a converging–diverging nozzle, Fig. 9.12.
Solution 10.W2
Problem 10.W3
In uniform open-channel flow, what is the balance of forces? Can you use such a force balance to
derive the Chézy equation (10.13)?
Solution 10.W3
Problem 10.W4
A shallow-water wave propagates at the speed c0 ≈(gy)1/2. What makes it propagate? That is,
what is the balance of forces in such wave motion? In which direction does such a wave
propagate?
Solution 10.W4
Problem 10.W5
Why is the Manning friction correlation, Eq. (10.16), used almost universally by hydraulics
engineers, instead of the Moody friction factor?
Problem 10.W6
During horizontal channel flow over a bump, is the specific energy constant? Explain.
Solution 10.W5
Problem 10.W7
Cite some similarities, and perhaps some dissimilarities, between a hydraulic jump and a gas
dynamic normal shock wave.
Solution 10.W7
Problem 10.W8
Give three examples of rapidly varied flow. For each case, cite reasons why it does not satisfy
one or more of the five basic assumptions of gradually varied flow theory.
Solution 10.W8
Problem 10.W9
Is a free overfall, Fig. 10.15e, similar to a weir? Could it be calibrated versus flow rate in the
same manner as a weir? Explain.
Solution 10.W9
Problem 10.W10
Cite some similarities, and perhaps some dissimilarities, between a weir and a Bernoulli
Problem 10.W11
Is a bump, Fig. 10.9a, similar to a weir? If not, when does a bump become large enough, or sharp
enough, to be a weir?
Solution 10.W11
The solution to this word problem is not provided.
Problem 10.W12
After doing some reading and/or thinking, explain the design and operation of a long-throated
flume.
Solution 10.W12
Problem 10.W13
Describe the design and operation of a critical-depth flume. What are its advantages compared
to the venturi flume of Prob. P10.113?
Problem 10.113
Figure P10.113 shows a channel contraction section often called a venturi flume [23, p. 167],
because measurements of y1 and y2 can be used to meter the flow rate. Show that if losses are
neglected and the flow is one-dimensional and subcritical, the flow rate is given by
( ) ( )
1/2
12
2 2 2 2
2 2 1 1
2 ( )
11
g y y
Q
b y b y
−
=
−
Apply this to the special case b1 = 3 m, b2 = 2 m, and y1 = 1.9 m. Find the flow rate (a) if
y2 = 1.5 m. (b) Also find the depth y2 for which the flow becomes critical in the throat.
Solution 10.W13
Problem 10.1
The formula for shallow-water wave propagation speed, Eq. (10.9) or (10.10), is independent of
the physical properties of the liquid, i.e., density, viscosity, or surface tension. Does this mean
that waves propagate at the same speed in water, mercury, gasoline, and glycerin? Explain.
Solution 10.1
The shallow-water wave formula,
o
c (gy),=
is valid for any fluid except for viscosity and
Problem 10.2
Water at 20C flows in a 30-cm-wide rectangular channel at a depth of 10 cm and a flow rate of
80,000 cm3/s. Estimate (a) the Froude number; and (b) the Reynolds number.
Solution 10.2
For water, take
= 998 kg/m3 and
Problem 10.3
Narragansett Bay is approximately 21 (statute) mi long and has an average depth of 42 ft. Tidal
charts for the area indicate a time delay of 30 min between high tide at the mouth of the bay
(Newport, Rhode Island) and its head (Providence, Rhode Island). Is this delay correlated with
the propagation of a shallow-water tidal-crest wave through the bay? Explain.
Solution 10.3
If it is a simple shallow-water wave phenomenon, the time delay would be
Problem 10.4
The water flow in Fig. P10.4 has a free surface in three places. Does it qualify as an open-
channel flow? Explain. What does the dashed line represent?
Solution 10.4
No, this is not an open-channel flow. The open tubes are merely piezometer or pressure-
Problem 10.5
Water flows down a rectangular channel that is 4 ft wide and 2 ft deep. The flow rate is
20,000 gal/min. Estimate the Froude number of the flow.
Solution 10.5
Stay in BG units. Convert 20,000 gal/min to 44.56 ft3/s. The velocity and wave speed are:
Problem 10.6
Pebbles dropped successively at the same point, into a water-channel flow of depth 42 cm,
create two circular ripples, as in Fig. P10.6. From this information, estimate (a) the Froude
number; and (b) the stream velocity.
Problem 10.7
Pebbles dropped successively at the same point, into a water-channel flow of depth 65 cm,
create two circular ripples, as in Fig. P10.7. From this information, estimate (a) the Froude
number; and (b) the stream velocity.
Solution 10.6
The center of each circle moves at stream velocity V. For the small circle,
sAns.
Solution 10.7
If the pebble-drop-site is at distance X ahead of the small-circle center,
Problem 10.8
An earthquake near the Kenai Peninsula, Alaska, creates a single “tidal” wave (called a
‘tsunami’) which propagates south across the Pacific Ocean. If the average ocean depth is 4 km
and seawater density is 1025 kg/m3, estimate the time of arrival of this tsunami in Hilo, Hawaii.
Solution 10.8
Everyone get out your Atlases, how far is it from Kenai to Hilo? Well, it’s about 2800 statute
Problem 10.9
Equation (10.10) is for a single disturbance wave. For periodic small-amplitude surface waves of
wavelength
and period T, inviscid theory [8 to 10] predicts a wave propagation speed
2
0
2
tanh
2
gy
c
=
where y is the water depth and surface tension is neglected. (a) Determine if this expression is
affected by the Reynolds number, Froude number, or Weber number. Derive the limiting values
of this expression for (b)
y
and (c)
.y
(d) Also for what ratio y/
is the wave speed
within 1 percent of limit (c)?
Solution 10.9
(a) Obviously there is no effect in this theory for Reynolds number or Weber number, because
Problem 10.10
If surface tension is included in the analysis of Prob. 10.9, the resulting wave speed is
[8 to 10]:
2
0
22
tanh
2
g Y y
c
=+
(a) Determine if this expression is affected by the Reynolds number, Froude number, or Weber
number. Derive the limiting values of this expression for (b)
y
and (c)
.y
(d) Finally
determine the wavelength
crit for a minimum value of c0, assuming that
.y
Problem 10.9
Equation (10.10) is for a single disturbance wave. For periodic small-amplitude surface waves of
wavelength
and period T, inviscid theory [8 to 10] predicts a wave propagation speed
2
0
2
tanh
2
gy
c
=
where y is the water depth and surface tension is neglected. (a) Determine if this expression is
affected by the Reynolds number, Froude number, or Weber number. Derive the limiting values
of this expression for (b)
y
and (c)
.y
(d) Also for what ratio y/
is the wave speed
within 1 percent of limit (c)?
Solution 10.10
(a) Obviously there is no effect in this theory for Reynolds number, because viscosity is not
Problem 10.11
A rectangular channel is 2 m wide and contains water 3 m deep. If the slope is 0.85 and the
lining is corrugated metal, estimate the discharge for uniform flow.
Solution 10.11
For corrugated metal, take Manning’s n 0.022. Get the hydraulic radius:
Problem 10.12
a) For laminar draining of a wide thin sheet of water on pavement sloped at angle
, as in
Fig. P4.36, show that the flow rate is given by
3sin
3
gbh
Q
=
where b is the sheet width and h its depth. (b) By (somewhat laborious) comparison with
Eq. (10.13), show that this expression is compatible with a friction factor f = 24/Re, where
Re = Vavh/
.
