Fluid Mechanics

Frank M. White

Chapter 10

Open-Channel Flow - all with Video Answers

Educators

Chapter Questions

Problem 1

The formula for shallow-water wave propagation speed, Eq. (10.9) or $(10.10),$ is independent of the physical properties of the liquid, like density, viscosity, or surface tension. Does this mean that waves propagate at the same speed in water, mercury, gasoline, and glycerin? Explain.

Abid Hussain

Abid Hussain

Numerade Educator

Problem 2

Water at $20^{\circ} \mathrm{C}$ flows in a 30 -cm-wide rectangular channel at a depth of $10 \mathrm{cm}$ and a flow rate of $80,000 \mathrm{cm}^{3} / \mathrm{s}$. Estimate
(a) the Froude number and
$(b)$ the Reynolds number.

Ma Ednelyn Lim

Numerade Educator

Problem 3

Narragansett Bay is approximately 21 (statute) $\mathrm{mi}$ 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.

Abid Hussain

Numerade Educator

Problem 4

The water flow in Fig. $P 10.4$ has a free surface in three places. Does it qualify as an open-channel flow? Explain. What does the dashed line represent?

Abid Hussain

Numerade Educator

Problem 5

Water flows down a rectangular channel that is $4 \mathrm{ft}$ wide and $2 \mathrm{ft}$ deep. The flow rate is 20,000 gal/min. Estimate the Froude number of the flow.

Narayan Hari

Numerade Educator

Problem 6

Pebbles dropped successively at the same point, into a water channel flow of depth $42 \mathrm{cm}$, create two circular ripples, as in Fig. P10.6. From this information estimate
$(a)$ the Froude number and $(b)$ the stream velocity.

Abid Hussain

Numerade Educator

Problem 7

Pebbles dropped successively at the same point, into a water channel flow of depth $65 \mathrm{cm}$, create two circular ripples, as in Fig. $\mathrm{P} 10.7 .$ From this information estimate
(a) the Froude number and ( $b$ ) the stream velocity.

Abid Hussain

Numerade Educator

Problem 8

An earthquake near the Kenai Peninsula, Alaska, creates a single "tidal" wave (called a tsunami ) that propagates southward across the Pacific Ocean. If the average ocean depth is $4 \mathrm{km}$ and seawater density is $1025 \mathrm{kg} / \mathrm{m}^{3},$ estimate the time of arrival of this tsunami in Hilo, Hawaii.

Abid Hussain

Numerade Educator

Problem 9

Equation (10.10) is for a single disturbance wave. For periodic small-amplitude surface waves of wavelength $\lambda$
and period $T,$ inviscid theory $[8 \text { to } 10]$ predicts a wave propagation speed
\[
c_{0}^{2}=\frac{g \lambda}{2 \pi} \tanh \frac{2 \pi y}{\lambda}
\]
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 $(b) y \ll \lambda$ and $(c) y>r$ $\lambda .(d)$ For what ratio $y / \lambda$ is the wave speed within 1 percent of limit $(c) ?$

Abid Hussain

Numerade Educator

Problem 10

If surface tension $Y$ is included in the analysis of Prob. P10.9, the resulting wave speed is [8 to $10]$
\[
c_{0}^{2}=\left(\frac{g \lambda}{2 \pi}+\frac{2 \pi \mathrm{Y}}{\rho \lambda}\right) \tanh \frac{2 \pi y}{\lambda}
\]
(a) Determine if this expression is affected by the Reynolds number, Froude number, or Weber number. Derive the limiting values of this expression for $(b) y \ll \lambda$ and $(c) y \gg$
$\lambda .(d)$ Finally, determine the wavelength $\lambda_{\text {crit }}$ for a minimum value of $c_{0},$ assuming that $y \geqslant \lambda$.

Abid Hussain

Numerade Educator

Problem 11

A rectangular channel is $2 \mathrm{m}$ wide and contains water $3 \mathrm{m}$ deep. If the slope is $0.85^{\circ}$ and the lining is corrugated metal, estimate the discharge for uniform flow.

Rory Naguib

Numerade Educator

Problem 12

(a) For laminar draining of a wide, thin sheet of water on pavement sloped at angle $\theta$, as in Fig. $\mathrm{P} 4.36$, show that the flow rate is given by
\[
Q=\frac{\rho g b h^{3} \sin \theta}{3 \mu}
\]
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 $f=24 / \mathrm{Re}$
\[
\text { where } \operatorname{Re}=V_{\mathrm{av}} h / \nu
\].

Penny Riley

Numerade Educator

Problem 13

A large pond drains down an asphalt rectangular channel that is $2 \mathrm{ft}$ wide. The channel slope is 0.8 degrees. If the flow is uniform, at a depth of 21 in, estimate the time to drain 1 acre-foot of water.

Prabhat Tyagi

Numerade Educator

Problem 14

The Chézy formula (10.18) is independent of fluid density and viscosity. Does this mean that water, mercury, alcohol, and SAE 30 oil will all flow down a given open channel at the same rate? Explain.

Abid Hussain

Numerade Educator

Problem 15

The painted-steel channel of Fig. $P 10.15$ is designed, without the barrier, for a flow rate of $6 \mathrm{m}^{3} / \mathrm{s}$ at a normal depth of 1 $\mathrm{m}$. Determine $(a)$ the design slope of the channel and
(b) the reduction in total flow rate if the proposed painted-steel central barrier is installed.

Penny Riley

Numerade Educator

Problem 16

Water flows in a brickwork rectangular channel $2 \mathrm{m}$ wide, on a slope of $5 \mathrm{m} / \mathrm{km}$
(a) Find the flow rate when the normal depth is $50 \mathrm{cm}$
(b) If the normal depth remains $50 \mathrm{cm},$ find the channel width which will triple the flow rate. Comment on this result.

Victor Salazar

Numerade Educator

Problem 17

The trapezoidal channel of Fig. $P 10.17$ is made of brickwork and slopes at $1: 500 .$ Determine the flow rate if the normal depth is $80 \mathrm{cm}$.

James Kiss

Numerade Educator

Problem 18

A V-shaped painted steel channel, similar to Fig. E10.6 has an included angle of $90^{\circ} .$ If the slope, in uniform flow, is $3 \mathrm{m}$ per $\mathrm{km}$, estimate $(a)$ the flow rate, in $\mathrm{m}^{3} / \mathrm{s}$ and
$(b)$ the average wall shear stress. Take $y=2 \mathrm{m}$.

Chai Santi

Numerade Educator

Problem 19

Modify Prob. $\mathrm{P} 10.18$, the $90^{\circ} \mathrm{V}$ channel, to let the surface be clean earth, which erodes if the average velocity exceeds 6 ft/s. Find the maximum depth that avoids erosion. The slope is still 3 m per km.

Narayan Hari

Numerade Educator

Problem 20

An unfinished concrete sewer pipe, of diameter $4 \mathrm{ft}$, is flowing half-full at 39,500 U.S. gallons per minute. If this is the normal depth, what is the pipe slope, in degrees?

Ajay Singhal

Numerade Educator

Problem 21

An engineer makes careful measurements with a weir (see Sec. 10.7 ) that monitors a rectangular unfinished concrete channel laid on a slope of $1^{\circ} .$ She finds, perhaps with surprise, that when the water depth doubles from 2 ft 2 inches to $4 \mathrm{ft}$ 4 inches, the normal flow rate more than doubles, from 200 to $500 \mathrm{ft}^{3} / \mathrm{s}$. ( $a$ ) Is this plausible?
(b) If so, estimate the channel width.

Prabhat Tyagi

Numerade Educator

Problem 22

For more than a century, woodsmen harvested trees in Skowhegan, ME, elevation $171 \mathrm{ft}$, and floated the logs down the Kennebec River to Bath, ME, elevation $62 \mathrm{ft}$, a distance of 72 miles. The river has an average depth of $14 \mathrm{ft}$ and an average width of $400 \mathrm{ft}$. Assuming uniform flow and a stony bottom, estimate the travel time required for this trip.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 23

It is desired to excavate a clean-earth channel as a trapezoidal cross section with $\theta=60^{\circ}$ (see Fig. 10.7 ). The expected flow rate is $500 \mathrm{ft}^{3} / \mathrm{s}$, and the slope is $8 \mathrm{ft}$ per mile. The uniform flow depth is planned, for efficient performance, such that the flow cross section is half a hexagon. What is the appropriate bottom width of the channel?

James Kiss

Numerade Educator

Problem 24

A rectangular channel, laid out on a $0.5^{\circ}$ slope, delivers a flow rate of 5000 gal/min in uniform flow when the depth is $1 \mathrm{ft}$ and the width is $3 \mathrm{ft}$
(a) Estimate the value of Manning's factor $n$
(b) What water depth will triple the flow rate?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 25

The equilateral-triangle channel in Fig. $\mathrm{P} 10.25$ has constant slope $S_{\circ}$ and constant Manning factor $n .$ If $y=a / 2$ find an analytic expression for the flow rate $Q$.

Victor Salazar

Numerade Educator

Problem 26

In the spirit of Fig. $10.6 b,$ analyze a rectangular channel in uniform flow with constant area $A=b y,$ constant slope, but varying width $b$ and depth $y .$ Plot the resulting flow rate $Q$ normalized by its maximum value $Q_{\max },$ in the range $0.2<b / y<4.0,$ and comment on whether it is crucial for discharge efficiency to have the channel flow at a depth exactly equal to half the channel width.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

A circular corrugated-metal water channel has a slope of 1: 800 and a diameter of $6 \mathrm{ft}$. (a) Estimate the normal discharge, in gal/min, when the water depth is 4 ft. ( $b$ ) For this condition, calculate the average wall shear stress.

Abid Hussain

Numerade Educator

Problem 28

A new, finished-concrete trapezoidal channel, similar to Fig. $10.7,$ has $b=8 \mathrm{ft}, y_{n}=5 \mathrm{ft},$ and $\theta=50^{\circ} .$ For this depth, the discharge is $500 \mathrm{ft}^{3} / \mathrm{s}$. ( $a$ ) What is the slope of the channel?
(b) As years pass, the channel corrodes and
$n$ doubles. What will be the new normal depth for the same flow rate?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Suppose that the trapezoidal channel of Fig. P10.17 contains sand and silt that we wish not to erode. According to an empirical correlation by A. Shields in 1936 , the average wall shear stress $\tau_{\mathrm{cni}}$ required to erode sand particles of diameter $d_{p}$ is approximated by $$\frac{\tau_{\mathrm{crit}}}{\left(\rho_{s}-\rho\right) g d_{p}} \approx 0.5$$ where $\rho_{s}=2400 \mathrm{kg} / \mathrm{m}^{3}$ is the density of sand. If the slope of the channel in Fig. $\mathrm{P} 10.17$ is 1: 900 and $n \approx 0.014$, determine the maximum water depth to keep from eroding particles of $1-\mathrm{mm}$ diameter.

James Kiss

Numerade Educator

Problem 30

A clay tile V-shaped channel, with an included angle of $90^{\circ},$ is $1 \mathrm{km}$ long and is laid out on a 1: 400 slope. When running at a depth of $2 \mathrm{m}$, the upstream end is suddenly closed while the lower end continues to drain. Assuming quasi-steady normal discharge, find the time for the channel depth to drop to $20 \mathrm{cm}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 31

An unfinished-concrete 6-ft-diameter sewer pipe flows half full. What is the appropriate slope to deliver 50,000 gal/min of water in uniform flow?

Victor Salazar

Numerade Educator

Problem 32

Does half a V-shaped channel perform as well as a full V-shaped channel? The answer to Prob. 10.18 is $Q=$ $12.4 \mathrm{m}^{3} / \mathrm{s} .$ (Do not reveal this to your friends still working on $\mathrm{P} 10.18 .$ For the painted-steel half-V in Fig. $\mathrm{P} 10.32,$ at the same slope of $3: 1000,$ find the flow area that gives the same $Q$ and compare with $\mathrm{P} 10.18$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

Five sewer pipes, each a 2 -m-diameter clay tile pipe running half full on a slope of $0.25^{\circ},$ empty into a single asphalt pipe, also laid out at $0.25^{\circ}$. If the large pipe is also to run half full, what should be its diameter?

Victor Salazar

Numerade Educator

Problem 34

A brick rectangular channel with $S_{0}=0.002$ is designed to carry $230 \mathrm{ft}^{3} / \mathrm{s}$ of water in uniform flow. There is an argument over whether the channel width should be 4 or $8 \mathrm{ft}$ Which design needs fewer bricks? By what percentage?

Narayan Hari

Numerade Educator

Problem 35

In flood stage a natural channel often consists of a deep main channel plus two floodplains, as in Fig. P10.35. The floodplains are often shallow and rough. If the channel has the same slope everywhere, how would you analyze this situation for the discharge? Suppose that $y_{1}=20 \mathrm{ft}, y_{2}=$ $5 \mathrm{ft}, b_{1}=40 \mathrm{ft}, b_{2}=100 \mathrm{ft}, n_{1}=0.020,$ and $n_{2}=0.040$
with a slope of $0.0002 .$ Estimate the discharge in $\mathrm{ft}^{3} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 36

The Blackstone River in northern Rhode Island normally flows at about $25 \mathrm{m}^{3} / \mathrm{s}$ and resembles Fig. $\mathrm{P} 10.35$ with a clean-earth center channel, $b_{1} \approx 20 \mathrm{m}$ and $y_{1}=3 \mathrm{m}$. The bed slope is about $2 \mathrm{ft} / \mathrm{mi}$. The sides are heavy brush with $b_{2} \approx 150 \mathrm{m} .$ During Hurricane Carol in $1954,$ a record flow rate of $1000 \mathrm{m}^{3} / \mathrm{s}$ was estimated. Use this information to estimate the maximum flood depth $y_{2}$ during this event.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A triangular channel (see Fig. $\mathrm{E} 10.6$ ) is to be constructed of corrugated metal and will carry $8 \mathrm{m}^{3} / \mathrm{s}$ on a slope of 0.005 The supply of sheet metal is limited, so the engineers want to minimize the channel surface. What are
$(a)$ the best included angle $\theta$ for the channel, $(b)$ the normal depth for part $(a),$ and $(c)$ the wetted perimeter for part $(b) ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

For the half-Vee channel in Fig. $\mathrm{P} 10.32$, let the interior angle of the Vee be $\theta$. For a given value of area, slope, and
$n,$ find the value of $\theta$ for which the flow rate is a maximum. To avoid cumbersome algebra, simply plot $Q$ versus $\theta$ for constant $A$.

Victor Salazar

Numerade Educator

Problem 39

A trapezoidal channel has $n=0.022$ and $S_{0}=0.0003$ and is made in the shape of a half-hexagon for maximum efficiency. What should the length of the side of the hexagon be if the channel is to carry $225 \mathrm{ft}^{3} / \mathrm{s}$ of water? What is the discharge of a semicircular channel of the same crosssectional area and the same $S_{0}$ and $n ?$

Supratim Pal

Numerade Educator

Problem 40

Using the geometry of Fig. $10.6 a$, prove that the most efficient circular open channel (maximum hydraulic radius for a given flow area) is a semicircle.

James Kiss

Numerade Educator

Problem 41

Determine the most efficient value of $\theta$ for the $V$ -shaped channel of Fig. $P 10.41$.

Thomas Thompson

Thomas Thompson

Numerade Educator

Problem 42

It is desired to deliver 30,000 gal/min of water in a brickwork channel laid on a slope of $1: 100 .$ Which would require fewer bricks, in uniform flow: $(a)$ a $\mathrm{V}$ channel with $\theta=45^{\circ},$ as in Fig. $\mathrm{P} 10.41,$ or $(b)$ an efficient rectangular channel with $b=2 y ?$

Narayan Hari

Numerade Educator

Problem 43

Determine the most efficient dimensions for a clay tile rectangular channel to carry 110,000 gal/min on a slope of 0.002.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

What are the most efficient dimensions for a half-hexagon cast iron channel to carry 15,000 gal/min on a slope of $0.16^{\circ} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Calculus tells us that the most efficient wall angle for a V-shaped channel (Fig. $P 10.41)$ is $\theta=45^{\circ} .$ It yields the highest normal flow rate for a given area. But is this a sharp or a flat maximum? For a flow area of $1 \mathrm{m}^{2}$ and an unfin-
ished-concrete channel with a slope of $0.004,$ plot the normal flow rate $Q,$ in $\mathrm{m}^{3} / \mathrm{s},$ versus angle for the range $30^{\circ} \leq$ $\theta \leq 60^{\circ}$ and comment.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

It is suggested that a channel that reduces erosion has a parabolic shape, as in Fig. P10.46. Formulas for area and perimeter of the parabolic cross section are as follows $[7, p .36]$
\[
\begin{array}{c}
A=\frac{2}{3} b h_{0} ; P=\frac{b}{2}\left[\sqrt{1+\alpha^{2}}+\frac{1}{\alpha} \ln (\alpha+\sqrt{1+\alpha^{2}})\right] \\
\text { where } \alpha=\frac{4 h_{0}}{b}
\end{array}
\]
For uniform flow conditions, determine the most efficient ratio $h_{0} / b$ for this channel (minimum perimeter for a given constant area).

James Kiss

Numerade Educator

Problem 47

Calculus tells us that the most efficient water depth for a rectangular channel (such as Fig. E10.1) is $y / b=1 / 2 .$ It yields the highest normal flow rate for a given area. But is this a sharp or a flat maximum? For a flow area of $1 \mathrm{m}^{2}$ and a clay tile channel with a slope of $0.006,$ plot the normal flow rate $Q,$ in $\mathrm{m}^{3} / \mathrm{s},$ versus $y / b$ for the range $0.3 \leq$ $y / b \leq 0.7$ and comment.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

A wide, clean-earth river has a flow rate $q=150 \mathrm{ft}^{3} /(\mathrm{s} \cdot \mathrm{ft})$ What is the critical depth? If the actual depth is $12 \mathrm{ft}$, what is the Froude number of the river? Compute the critical slope by
(a) Manning's formula and
$(b)$ the Moody chart.

Narayan Hari

Numerade Educator

Problem 49

Find the critical depth of the brick channel in Prob. P10.34 for both the 4 - and 8 -ft widths. Are the normal flows subcritical or supercritical?

Prabhat Tyagi

Numerade Educator

Problem 50

A pencil point piercing the surface of a rectangular channel flow creates a wedgelike $25^{\circ}$ half-angle wave, as in Fig. $\mathrm{P} 10.50 .$ If the channel surface is painted steel and the depth is $35 \mathrm{cm},$ determine ( $a$ ) the Froude number, (b) the critical depth, and ( $c$ ) the critical slope for uniform flow.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

An unfinished concrete duct, of diameter $1.5 \mathrm{m}$, is flowing half-full at $8.0 \mathrm{m}^{3} / \mathrm{s}$. ( $a$ ) Is this a critical flow? If not, what is $(b)$ the critical flow rate, $(c)$ the critical slope, and $(d)$ the Froude number? ( $e$ ) If the flow is uniform, what is the slope of the duct?

Ronald Prasad

Numerade Educator

Problem 52

Water flows full in an asphalt half-hexagon channel of bottom width $W$. The flow rate is $12 \mathrm{m}^{3} / \mathrm{s}$. Estimate $W$ if the Froude number is exactly 0.60.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 53

For the river flow of Prob. $\mathrm{P} 10.48$, find the depth $y_{2}$ that has the same specific energy as the given depth $y_{1}=12 \mathrm{ft}$ These are called conjugate depths. What is $\mathrm{Fr}_{2}$ ?

Victor Salazar

Numerade Educator

Problem 54

A clay tile $V$ -shaped channel has an included angle of $70^{\circ}$ and carries $8.5 \mathrm{m}^{3} / \mathrm{s}$. Compute $(a)$ the critical depth,
(b) the critical velocity, and (c) the critical slope for uniform flow.

James Kiss

Numerade Educator

Problem 55

A trapezoidal channel resembles Fig. 10.7 with $b=1 \mathrm{m}$ and $\theta=50^{\circ} .$ The water depth is $2 \mathrm{m},$ and the flow rate is $32 \mathrm{m}^{3} / \mathrm{s}$. If you stick your fingernail in the surface, as in Fig. $\mathrm{P} 10.50,$ what half-angle wave might appear?

Mayukh Banik

Numerade Educator

Problem 56

A 4 -ft-diameter finished-concrete sewer pipe is half full of water. (a) In the spirit of Fig. $10.4 a$, estimate the speed of propagation of a small-amplitude wave propagating along the channel.
(b) If the water is flowing at 14,000 gal/min, calculate the Froude number.

Averell Hause

Carnegie Mellon University

Problem 57

Consider the V-shaped channel of arbitrary angle in Fig. $P 10.41 .$ If the depth is $y,(a)$ find an analytic expression for the propagation speed $c_{0}$ of a small-disturbance wave along this channel. [Hint: Eliminate flow rate from the analyses in Sec. 10.4 .1 If $\theta=45^{\circ}$ and the depth is $1 \mathrm{m}$ determine $(b)$ the propagation speed and $(c)$ the flow rate if the channel is running at a Froude number of $1 / 3$.

Narayan Hari

Numerade Educator

Problem 58

For a half-hexagon channel running full, find an analytic expression for the propagation speed of a small-disturbance wave travelling along this channel. Denote the bottom width as $b$ and use Fig. 10.7 as a guide.

Ameer Said

Numerade Educator

Problem 59

Uniform water flow in a wide brick channel of slope $0.02^{\circ}$ moves over a 10 -cm bump as in Fig. P10.59. A slight depression in the water surface results. If the minimum water depth over the bump is $50 \mathrm{cm},$ compute ( $a$ ) the velocity over the bump and ( $b$ ) the flow rate per meter of width.

Narayan Hari

Numerade Educator

Problem 60

Water, flowing in a rectangular channel $2 \mathrm{m}$ wide, encounters a bottom bump $10 \mathrm{cm}$ high. The approach depth is $60 \mathrm{cm},$ and the flow rate $4.8 \mathrm{m}^{3} / \mathrm{s}$. Determine $(a)$ the water depth,
(b) velocity, and
(c) Froude number above the bump. Hint: The change in water depth is rather slight, only about $8$ $\mathrm{cm}$.

Vidhi Bhatt

Numerade Educator

Problem 61

Modify Prob. P10.59 as follows: Again assuming uniform subcritical approach flow $\left(V_{1}, y_{1}\right),$ find $(a)$ the flow rate and
(b) $y_{2}$ for which the flow at the crest of the bump is exactly critical $\left(\mathrm{Fr}_{2}=1.0\right)$.

Vidhi Bhatt

Numerade Educator

Problem 62

Consider the flow in a wide channel over a bump, as in Fig. $P 10.62 .$ One can estimate the water depth change or transition with frictionless flow. Use continuity and the Bernoulli equation to show that
\[
\frac{d y}{d x}=-\frac{d h / d x}{1-V^{2} /(g y)}
\]
Is the drawdown of the water surface realistic in Fig. $P 10.62 ?$ Explain under what conditions the surface might rise above its upstream position $y_{0}$.

Mahnoor Amin

Numerade Educator

Problem 63

In Fig. $P 10.62$ let $V_{0}=1 \mathrm{m} / \mathrm{s}$ and $y_{0}=1 \mathrm{m}$. If the maximum bump height is $15 \mathrm{cm}$, estimate ( $a$ ) the Froude number over the top of the bump and ( $b$ ) the maximum depression in the water surface.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 64

For the rectangular channel in Prob. P10.60, the Froude number over the bump is about $1.37,$ which is 17 percent less than the approach value. For the same entrance conditions, find the bump height $\Delta h$ that causes the bump Froude number to be 1.00.

James Kiss

Numerade Educator

Problem 65

Program and solve the differential equation of "frictionless flow over a bump," from Prob. P10.62, for entrance conditions $V_{0}=1 \mathrm{m} / \mathrm{s}$ and $y_{0}=1 \mathrm{m} .$ Let the bump have the convenient shape $h=0.5 h_{\max }[1-\cos (2 \pi x / L)],$ which
simulates Fig. P10.62. Let $L=3 \mathrm{m},$ and generate a numerical solution for $y(x)$ in the bump region $0<x<L$ If you have time for only one case, use $h_{\max }=15 \mathrm{cm}$ (Prob. P10.63), for which the maximum Froude number is 0.425 If more time is available, it is instructive to examine a complete family of surface profiles for $h_{\max } \approx 1 \mathrm{cm}$ up to $35 \mathrm{cm}$ (which is the solution of Prob. P10.64).

Narayan Hari

Numerade Educator

Problem 66

In Fig. $\mathrm{P} 10.62,$ let $V_{\mathrm{o}}=5.5 \mathrm{m} / \mathrm{s}$ and $y_{\mathrm{o}}=90 \mathrm{cm} .(a)$ Will the
water rise or fall over the bump? ( $b$ ) For a bump height of $30 \mathrm{cm},$ determine the Froude number over the bump.
$(c)$ Find the bump height that will cause critical flow over the bump.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 67

7 Modify Prob. P10.63 so that the 15-cm change in bottom level is a depression, not a bump. Estimate (a) the Froude number above the depression and ( $b$ ) the maximum change in water depth.

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 68

Modify Prob. P10.65 to have a supercritical approach condition $V_{0}=6 \mathrm{m} / \mathrm{s}$ and $y_{0}=1 \mathrm{m} .$ If you have time for only one case, use $\left.h_{\max }=35 \mathrm{cm} \text { (Prob. } \mathrm{P} 10.66\right),$ for which the maximum Froude number is 1.47 . If more time is available, it is instructive to examine a complete family of surface profiles for $1 \mathrm{cm}<h_{\max }<52 \mathrm{cm}$ (which is the solution to Prob. $\mathrm{P} 10.67$ ).

Hast Aggarwal

Numerade Educator

Problem 69

Given is the flow of a channel of large width $b$ under a sluice gate, as in Fig. P10.69. Assuming frictionless steady flow with negligible upstream kinetic energy, derive a formula for the dimensionless flow ratio $Q^{2} /\left(y_{1}^{3} b^{2} g\right)$ as a function of the ratio $y_{2} / y_{1}$. Show by differentiation that the maximum flow rate occurs at $y_{2}=2 y_{1} / 3$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A periodic and spectacular water release, in China's Henan province, flows through a giant sluice gate. Assume that the gate is $23 \mathrm{m}$ wide, and its opening is $8 \mathrm{m}$ high. The water depth far upstream is $32 \mathrm{m}$. Assuming free discharge, estimate the volume flow rate through the gate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

In Fig. $\mathrm{P} 10.69$ let $y_{1}=95 \mathrm{cm}$ and $y_{2}=50 \mathrm{cm} .$ Estimate the flow rate per unit width if the upstream kinetic energy is
(a) neglected and ( $b$ ) included.

Averell Hause

Carnegie Mellon University

Problem 72

Water approaches the wide sluice gate of Fig. $\mathrm{P} 10.72$ at $V_{1}=0.2 \mathrm{m} / \mathrm{s}$ and $y_{1}=1 \mathrm{m} .$ Accounting for upstream kinetic energy, estimate at the outlet, section $2,$ the
$(a)$ depth, $(b)$ velocity, and
$(c)$ Froude number.

Narayan Hari

Numerade Educator

Problem 73

In Fig. $\mathrm{P} 10.69$, let $y_{1}=6 \mathrm{ft}$ and the gate width $b=8 \mathrm{ft}$ Find the gate opening $H$ that would allow a free-discharge flow of 30,000 gal/min under the gate.

Hunza Gilgit

Numerade Educator

Problem 74

With respect to Fig. $\mathrm{P} 10.69,$ show that, for frictionless flow, the upstream velocity may be related to the water levels by
\[
V_{1}=\sqrt{\frac{2 g\left(y_{1}-y_{2}\right)}{K^{2}-1}}
\]
where $K=y_{1} / y_{2}$.

James Kiss

Numerade Educator

Problem 75

A tank of water $1 \mathrm{m}$ deep, $3 \mathrm{m}$ long, and $4 \mathrm{m}$ wide into the paper has a closed sluice gate on the right side, as in Fig. $\mathrm{P} 10.75$ At $t=0$ the gate is opened to a gap of $10 \mathrm{cm} .$ Assuming quasisteady sluice gate theory, estimate the time required for the water level to drop to $50 \mathrm{cm}$. Assume free outflow.

Hunza Gilgit

Numerade Educator

Problem 76

6 Figure $\mathrm{P} 10.76$ shows a horizontal flow of water through a sluice gate, a hydraulic jump, and over a 6 -ft sharp-crested weir. Channel, gate, jump, and weir are all 8 ft wide unfinished concrete. Determine (a) the flow rate in $\mathrm{ft}^{3} / \mathrm{s}$ and
(b) the normal depth.

James Kiss

Numerade Educator

Problem 77

Equation (10.41) for sluice gate discharge is for free outflow. If the outflow is drowned, as in Fig. $10.10 c$, there is dissipation, and $C_{d}$ drops sharply, as shown in Fig. $\mathrm{P} 10.77,$ taken from Ref. $2 .$ Use this data to restudy Prob. $10.73,$ with $H=9$ in. Plot the estimated flow rate, in gal/min, versus $y_{2}$ in the range $0.5 \mathrm{ft}<y_{2}<5 \mathrm{ft}$.

Ameer Said

Numerade Educator

Problem 78

In Fig. $P 10.69,$ free discharge, a gate opening of $0.72 \mathrm{ft}$ will allow a flow rate of 30,000 gal/min. Recall $y_{1}=6 \mathrm{ft}$ and the gate width $b=8 \mathrm{ft}$. Suppose that the gate is drowned (Fig. $P 10.77$ ), with $y_{2}=4 \mathrm{ft}$. What gate opening would then be required?

Hunza Gilgit

Numerade Educator

Problem 79

Show that the Froude number downstream of a hydraulic jump will be given by
\[
\mathrm{Fr}_{2}=8^{1 / 2} \mathrm{Fr}_{1} /\left[\left(1+8 \mathrm{Fr}_{1}^{2}\right)^{1 / 2}-1\right]^{3 / 2}
\]
Does the formula remain correct if we reverse subscripts 1 and $2 ?$ Why?

Dominador Tan

Numerade Educator

Problem 80

Water flowing in a wide channel $25 \mathrm{cm}$ deep suddenly jumps to a depth of $1 \mathrm{m}$. Estimate ( $a$ ) the downstream Froude number; $(b)$ the flow rate per unit width; $(c)$ the critical depth; and ( $d$ ) the percentage of dissipation.

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 81

Water flows in a wide channel at $q=25 \mathrm{ft}^{3} /(\mathrm{s} \cdot \mathrm{ft}), y_{1}=1 \mathrm{ft}$ and then undergoes a hydraulic jump. Compute $y_{2}, V_{2}, \mathrm{Fr}_{2}$ $h_{f}$ the percentage of dissipation, and the horsepower dissipated per unit width. What is the critical depth?

Victor Salazar

Numerade Educator

Problem 82

Downstream of a wide hydraulic jump the flow is $4 \mathrm{ft}$ deep and has a Froude number of $0.5 .$ Estimate
$(a) y_{1}$
$(b) V_{1}$
$(c) \mathrm{Fr}_{1},(d)$ the percentage of dissipation, and $(e) y_{c}$.

Vidhi Bhatt

Numerade Educator

Problem 83

A wide-channel flow undergoes a hydraulic jump from 40 to $140 \mathrm{cm} .$ Estimate
$(a) V_{1},(b) V_{2},(c)$ the critical depth, in $\mathrm{cm},$ and $(d)$ the percentage of dissipation.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 84

Consider the flow under the sluice gate of Fig. P10.84. If $y_{1}=10 \mathrm{ft}$ and all losses are neglected except the dissipation in the jump, calculate $y_{2}$ and $y_{3}$ and the percentage of dissipation, and sketch the flow to scale with the EGL included. The channel is horizontal and wide.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 85

The analogy between a hydraulic jump and a normal shock equates Mach number and Froude number, air density and water depth, air pressure and the square of the water depth. Test this analogy for $\mathrm{Ma}_{1}=\mathrm{Fr}_{1}=4.0$ and comment on the results.

Chai Santi

Numerade Educator

Problem 86

A bore is a hydraulic jump that propagates upstream into a still or slower-moving fluid, as in Fig. $\mathrm{P} 10.86,$ on the Sée-Sélune channel, near Mont Saint Michel in northwest France. The bore is moving at about $10 \mathrm{ft} / \mathrm{s}$ and is about one foot high. Estimate $(a)$ the depth of the water in this area and $(b)$ the velocity induced by the wave.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 87

A tidal bore may occur when the ocean tide enters an estuary against an oncoming river discharge, such as on the Severn River in England. Suppose that the tidal bore is $10 \mathrm{ft}$ deep and propagates at $13 \mathrm{mi} / \mathrm{h}$ upstream into a river that is 7 ft deep. Estimate the river current in $\mathrm{kn}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 88

Consider supercritical flow, $\mathrm{Fr}_{1}>1,$ down a shallow flat water channel toward a wedge of included angle $2 \theta$, as in Fig. $\mathrm{P} 10.88 .$ By the compressible flow analogy, hydraulic jumps should form, similar to the shock waves in Fig. $\mathrm{P} 9.132 a .$ Using an approach similar to Fig. 9.20 develop and explain the equations that could be used to find the wave angle $\beta$ and $\mathrm{Fr}_{2}$.

Victor Salazar

Numerade Educator

Problem 89

Water $30 \mathrm{cm}$ deep is in uniform flow down a $1^{\circ}$ unfinished concrete slope when a hydraulic jump occurs, as in Fig. $P 10.89 .$ If the channel is very wide, estimate the water depth $y_{2}$ downstream of the jump.

Victor Salazar

Numerade Educator

Problem 90

For the gate/jump/weir system sketched in Fig. P10.76, the flow rate was determined to be $379 \mathrm{ft}^{3} / \mathrm{s}$. Determine ( $a$ ) the water depths $y_{2}$ and $y_{3},$ and $(b)$ the Froude numbers $\mathrm{Fr}_{2}$ and $\mathrm{Fr}_{3}$ before and after the hydraulic jump.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 91

Follow up Prob. P10.88 numerically with flow down a shallow, flat water channel $1 \mathrm{cm}$ deep at an average velocity of $0.94 \mathrm{m} / \mathrm{s}$. The wedge half-angle $\theta$ is $20^{\circ}$. Calculate
$(a) \beta ;(b) \mathrm{Fr}_{2} ;$ and $(c) y_{2}$.

Anand Jangid

Numerade Educator

Problem 92

A familiar sight is the circular hydraulic jump formed by a faucet jet falling onto a flat sink surface, as in Fig. P10.92. Because of the shallow depths, this jump is strongly dependent on bottom friction, viscosity, and surface tension [35] It is also unstable and can form remarkable noncircular shapes, as shown in the website <http://web.mit.edu/jeffa/ www/jump.htm>.
For this problem, assume that two-dimensional jump theory is valid. If the water depth outside the jump is $4 \mathrm{mm}$ the radius at which the jump appears is $R=3 \mathrm{cm},$ and the faucet flow rate is $100 \mathrm{cm}^{3} / \mathrm{s}$, find the conditions just upstream of the jump.

Narayan Hari

Numerade Educator

Problem 93

Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic jump, as in Fig. $\mathrm{P} 10.93 .$ If $y_{1}=1 \mathrm{m}$ and $y_{3}=40 \mathrm{cm},$ estimate $(a) V_{1}$
(b) $V_{3},(c) y_{4},$ and $(d)$ the bump height $h$.

Victor Salazar

Numerade Educator

Problem 94

In Fig. 10.11 , the upstream flow is only $2.65 \mathrm{cm}$ deep. The channel is $50 \mathrm{cm}$ wide, and the flow rate is $0.0359 \mathrm{m}^{3} / \mathrm{s}$
Determine $(a)$ the upstream Froude number, $(b)$ the downstream velocity, $(c)$ the downstream depth, and $(d)$ the percent dissipation.

Narayan Hari

Numerade Educator

Problem 95

A 10 -cm-high bump in a wide horizontal water channel creates a hydraulic jump just upstream and the flow pattern in Fig. P10.95. Neglecting losses except in the jump, for the case $y_{3}=30 \mathrm{cm},$ estimate
$(a) V_{4},(b) y_{4},(c) V_{1},$ and
$(d) y_{1}$.

Victor Salazar

Numerade Educator

Problem 96

For the circular hydraulic jump in Fig. $\mathrm{P} 10.92$, the water depths before and after the jump are $2 \mathrm{mm}$ and $4 \mathrm{mm}$ respectively. Assume that two-dimensional jump theory is valid. If the faucet flow rate is $150 \mathrm{cm}^{3} / \mathrm{s}$, estimate the radius $R$ at which the jump will appear.

Narayan Hari

Numerade Educator

Problem 97

A brickwork rectangular channel $4 \mathrm{m}$ wide is flowing at $8.0 \mathrm{m}^{3} / \mathrm{s}$ on a slope of $0.1^{\circ} .$ Is this a mild, critical, or steep slope? What type of gradually varied solution curve are we on if the local water depth is
$(a) 1 \mathrm{m},(b) 1.5 \mathrm{m},$ and
$(c) 2 m ?$

James Kiss

Numerade Educator

Problem 98

A gravelly earth wide channel is flowing at $10 \mathrm{m}^{3} / \mathrm{s}$ per meter of width on a slope of $0.75^{\circ} .$ Is this a mild, critical, or steep slope? What type of gradually varied solution curve are we on if the local water depth is
$(a) 1 \mathrm{m},(b) 2 \mathrm{m},$ or $(c) 3 \mathrm{m} ?$

Prabhat Tyagi

Numerade Educator

Problem 99

A clay tile V-shaped channel of included angle $60^{\circ}$ is flowing at $1.98 \mathrm{m}^{3} / \mathrm{s}$ on a slope of $0.33^{\circ} .$ Is this a mild, critical, or steep slope? What type of gradually varied solution curve are we on if the local water depth is $(a) 1 \mathrm{m}$
$(b) 2 m,$ or $(c) 3 m ?$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 100

If bottom friction is included in the sluice gate flow of Prob. P10.84, the depths $\left(y_{1}, y_{2}, y_{3}\right)$ will vary with $x$ Sketch the type and shape of gradually varied solution curve in each region $(1,2,3),$ and show the regions of rapidly varied flow.

Satpal Satpal

Numerade Educator

Problem 101

Consider the gradual change from the profile beginning at point $a$ in Fig. $P 10.101$ on a mild slope $S_{01}$ to a mild but steeper slope $S_{02}$ downstream. Sketch and label the curve $y(x)$ expected.

Carson Merrill

Numerade Educator

Problem 102

The wide-channel flow in Fig. $\mathrm{P} 10.102$ changes from a steep slope to one even steeper. Beginning at points $a$ and
$b,$ sketch and label the water surface profiles expected for gradually varied flow.

Abid Hussain

Numerade Educator

Problem 103

A gravelly rectangular channel, $7 \mathrm{m}$ wide and $2 \mathrm{m}$ deep, is flowing at $75 \mathrm{m}^{3} / \mathrm{s}$ on a slope of $0.013 .(a)$ Is this on a mild, critical, or steep curve?
(b) Approximately how many meters downstream will the gradually varied solution reach the normal depth?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 104

The rectangular-channel flow in Fig. P10.104 expands to a cross section 50 percent wider. Beginning at points $a$ and $b,$ sketch and label the water surface profiles expected for gradually varied flow.

Narayan Hari

Numerade Educator

Problem 105

In Prob. $\mathrm{P} 10.84$ the frictionless solution is $y_{2}=0.82 \mathrm{ft}$ which we denote as $x=0$ just downstream of the gate. If the channel is horizontal with $n=0.018$ and there is no hydraulic jump, compute from gradually varied theory the downstream distance where $y=2.0 \mathrm{ft}$.

Victor Salazar

Numerade Educator

Problem 106

A rectangular channel with $n=0.018$ and a constant slope of 0.0025 increases its width linearly from $b$ to $2 b$ over a distance $L$, as in Fig. P10.106.
$(a)$ Determine the variation $y(x)$ along the channel if $b=4 \mathrm{m}, L=250 \mathrm{m}$ the initial depth is $y(0)=1.05 \mathrm{m},$ and the flow rate is $7 \mathrm{m}^{3} / \mathrm{s}$. ( $b$ ) Then, if your computer program is running well, determine the initial depth $y(0)$ for which the exit flow will be exactly critical.

Dominador Tan

Numerade Educator

Problem 107

A clean-earth wide-channel flow is climbing an adverse slope with $S_{0}=-0.002 .$ If the flow rate is $q=4.5 \mathrm{m}^{3} /(\mathrm{s} \cdot \mathrm{m})$
use gradually varied theory to compute the distance for the depth to drop from 3.0 to $2.0 \mathrm{m}$.

Narayan Hari

Numerade Educator

Problem 108

Water flows at $1.5 \mathrm{m}^{3} / \mathrm{s}$ along a straight, riveted-steel $90^{\circ}$ V-shaped channel (see Fig. P10.41, $\theta=45^{\circ}$ ). At section $1,$ the water depth is $1.0 \mathrm{m}$. (a) As we proceed downstream, will the water depth rise or fall? Explain.
(b) Depending upon your answer to part ( $a$ ), calculate, in one numerical swoop, from gradually varied theory, the distance downstream for which the depth rises (or falls) $0.1 \mathrm{m}$.

Narayan Hari

Numerade Educator

Problem 109

Figure $\mathrm{P} 10.109$ illustrates a free overfall or $d$ropdown flow pattern, where a channel flow accelerates down a slope and falls freely over an abrupt edge. As shown, the flow reaches critical just before the overfall. Between $y_{c}$ and the edge the flow is rapidly varied and does not satisfy gradually varied theory. Suppose that the flow rate is $q=1.3 \mathrm{m}^{3} /(\mathrm{s} \cdot \mathrm{m})$ and the surface is unfinished concrete. Use Eq. (10.51) to estimate the water depth $300 \mathrm{m}$ upstream as shown.

James Kiss

Numerade Educator

Problem 110

We assumed frictionless flow in solving the bump case, Prob. P10.65, for which $V_{2}=1.21 \mathrm{m} / \mathrm{s}$ and $y_{2}=0.826 \mathrm{m}$ over the crest when $h_{\max }=15 \mathrm{cm}, V_{1}=1 \mathrm{m} / \mathrm{s},$ and
$y_{1}=1 \mathrm{m} .$ However, if the bump is long and rough, friction may be important. Repeat Prob. P10.65 for the same bump shape, $h=0.5 h_{\max }[1-\cos (2 \pi x / L)],$ to compute
conditions $(a)$ at the crest and $(b)$ at the end of the bump, $x=L .$ Let $h_{\max }=15 \mathrm{cm}$ and $L=100 \mathrm{m},$ and assume a
clean-earth surface.

Averell Hause

Carnegie Mellon University

Problem 111

The Rolling Dam on the Blackstone River has a weedy bottom and an average flow rate of $900 \mathrm{ft}^{3} / \mathrm{s}$. Assume the river upstream is $150 \mathrm{ft}$ wide and slopes at $10 \mathrm{ft}$ per statute mile. The water depth just upstream of the dam is $7.7 \mathrm{ft}$ Calculate the water depth one mile upstream ( $a$ ) for the given initial depth, $7.7 \mathrm{ft} ;$ and
(b) if flashboards on the dam raise this depth to $10.7 \mathrm{ft}$.

Narayan Hari

Numerade Educator

Problem 112

The clean-earth channel in Fig. $\mathrm{P} 10.112$ is $6 \mathrm{m}$ wide and slopes at $0.3^{\circ} .$ Water flows at $30 \mathrm{m}^{3} / \mathrm{s}$ in the channel and enters a reservoir so that the channel depth is $3 \mathrm{m}$ just before the entry. Assuming gradually varied flow, how far is the distance $L$ to a point in the channel where $y=2 \mathrm{m} ?$ What type of curve is the water surface?

Victor Salazar

Numerade Educator

Problem 113

Figure $\mathrm{P} 10.113$ shows a channel contraction section often called a venturi flume $[23, \mathrm{p} .167]$ because measurements of $y_{1}$ and $y_{2}$ 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
\[
Q=\left[\frac{2 g\left(y_{1}-y_{2}\right)}{1 /\left(b_{2}^{2} y_{2}^{2}\right)-1 /\left(b_{1}^{2} y_{1}^{2}\right)}\right]^{1 / 2}
\]
Apply this to the special case $b_{1}=3 \mathrm{m}, b_{2}=2 \mathrm{m},$ and $y_{1}=1.9 \mathrm{m} .(a)$ Find the flow rate if $y_{2}=1.5 \mathrm{m} .(b)$ Also find the depth $y_{2}$ for which the flow becomes critical in the throat.

James Kiss

Numerade Educator

Problem 114

For the gate/jump/weir system sketched in Fig. $\mathrm{P} 10.76$ the flow rate was determined to be $379 \mathrm{ft}^{3} / \mathrm{s}$. Determine the water depth $y_{4}$ just upstream of the weir.

Victor Salazar

Numerade Educator

Problem 115

Gradually varied theory, Eq. (10.49), neglects the effect of width changes, dbldx, assuming that they are small. But they are not small for a short, sharp contraction such as the venturi flume in Fig. P10.113. Show that, for a rectangular section with $b=b(x),$ Eq. (10.49) should be modified as follows:
\[
\frac{d y}{d x} \approx \frac{S_{0}-S+\left[V^{2} /(g b)\right](d b / d x)}{1-\mathrm{Fr}^{2}}
\]
Investigate a criterion for reducing this relation to Eq. (10.49).

James Kiss

Numerade Educator

Problem 116

A Cipolletti weir, popular in irrigation systems, is trapezoidal, with sides sloped at 1: 4 horizontal to vertical, as in Fig. P10.1 $16 .$ The following are flow-rate values, from the U.S. Dept. of Agriculture, for a few different system parameters:
Use this data to correlate a Cipolletti weir formula with a reasonably constant weir coefficient.

Victor Salazar

Numerade Educator

Problem 117

A popular flow-measurement device in agriculture is the Parshall flume $[33],$ Fig. $P 10.117,$ named after its inventor, Ralph L. Parshall, who developed it in 1922 for the
U.S. Bureau of Reclamation. The subcritical approach flow is driven, by a steep constriction, to go critical $\left(y=y_{c}\right)$ and then supercritical. It gives a constant reading $H$ for a wide range of tailwaters. Derive a formula for estimating $Q$ from measurement of $H$ and knowledge of constriction width $b$. Neglect the entrance velocity head.

Mahnoor Amin

Numerade Educator

Problem 118

Using a Bernoulli-type analysis similar to Fig. $10.16 a$ show that the theoretical discharge of the V-shaped weir in Fig. $\mathrm{P} 10.118$ is given by
\[
Q=0.7542 g^{1 / 2} \tan \alpha H^{5 / 2}
\]

Chai Santi

Numerade Educator

Problem 119

Data by A. T. Lenz for water at $20^{\circ} \mathrm{C}$ (reported in Ref. 23 ) show a significant increase of discharge coefficient of V-notch weirs (Fig. $P 10.118$ ) at low heads. For $\alpha=20^{\circ}$ some measured values are as follows:
$$\begin{array}{l|c|c|c|c|c}
H, \mathrm{ft} & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\
\hline C_{d} & 0.499 & 0.470 & 0.461 & 0.456 & 0.452
\end{array}$$
Determine if these data can be correlated with the
Reynolds and Weber numbers vis-ã-vis Eq. (10.61) . If not, suggest another correlation.

Satpal Satpal

Numerade Educator

Problem 120

The rectangular channel in Fig. $\mathrm{P} 10.120$ contains a V-notch weir as shown. The intent is to meter flow rates between 2.0 and $6.0 \mathrm{m}^{3} / \mathrm{s}$ with an upstream hook gage set to measure water depths between 2.0 and $2.75 \mathrm{m}$ What are the most appropriate values for the notch height $Y$ and the notch half-angle $\alpha ?$

Victor Salazar

Numerade Educator

Problem 121

Water flow in a rectangular channel is to be metered by a thin-plate weir with side contractions, as in Table $10.2 b$ with $L=6 \mathrm{ft}$ and $Y=1 \mathrm{ft} .$ It is desired to measure flow rates between 1500 and 3000 gal/min with only a 6 -in change in upstream water depth. What is the most appropriate length for the weir width $b ?$

Prabhakar Kumar

Prabhakar Kumar

Numerade Educator

Problem 122

In $1952 \mathrm{E}$
S. Crump developed the triangular weir shape shown in Fig. $\mathrm{P} 10.122[23, \text { Chap. } 4] .$ The front slope is 1: 2 to avoid sediment deposition, and the rear slope is 1: 5 to maintain a stable tailwater flow. The beauty of the design is that it has a unique discharge correlation up to near-drowning conditions, $H_{2} / H_{1} \leq 0.75$
\[
\begin{array}{c}
Q=C_{d} b g^{1 / 2}\left(H_{1}+\frac{V_{1}^{2}}{2 g}-k_{h}\right)^{3 / 2} \\
\text { where } C_{d} \approx 0.63 \quad \text { and } \quad k_{h} \approx 0.3 \mathrm{mm}
\end{array}
\]
The term $k_{h}$ is a low-head loss factor. Suppose that the weir is $3 \mathrm{m}$ wide and has a crest height $Y=50 \mathrm{cm} .$ If the water depth upstream is $65 \mathrm{cm}$, estimate the flow rate in $\operatorname{gal/min}$.

Ajay Singhal

Numerade Educator

Problem 123

Water in a 20 -ft-wide rectangular channel, flowing at $120 \mathrm{ft}^{3} / \mathrm{s}$ and a depth of $10 \mathrm{ft},$ is to be metered by a rectangular weir with side contractions, as in Table $10.2 b$ Suggest some appropriate design values of $b, Y$, and $H$ to match the table conditions for this weir.

Victor Salazar

Numerade Educator

Problem 124

Water flows at $600 \mathrm{ft}^{3} / \mathrm{s}$ in a rectangular channel $22 \mathrm{ft}$ wide with $n \approx 0.024$ and a slope of $0.1^{\circ} .$ A dam increases the depth to $15 \mathrm{ft}$, as in Fig. P10.124. Using gradually varied theory, estimate the distance $L$ upstream at which the water depth will be $10 \mathrm{ft}$. What type of solution curve are we on? What should be the water depth asymptotically far upstream?

Victor Salazar

Numerade Educator

Problem 125

The Tupperware dam on the Blackstone River is $12 \mathrm{ft}$ high, $100 \mathrm{ft}$ wide, and sharp-edged. It creates a backwater similar to Fig. $\mathrm{P} 10.124 .$ Assume that the river is a weedy-earth rectangular channel $100 \mathrm{ft}$ wide with a flow rate of $800 \mathrm{ft}^{3} / \mathrm{s}$. Estimate the water depth 2 mi upstream of the dam if $S_{0}=0.001$.

Lucas Finney

Numerade Educator

Problem 126

Suppose that the rectangular channel of Fig. $\mathrm{P} 10.120$ is made of riveted steel and carries a flow of $8 \mathrm{m}^{3} / \mathrm{s}$ on a slope of $0.15^{\circ} .$ If the V-notch weir has $\alpha=30^{\circ}$ and $Y=$ $50 \mathrm{cm},$ estimate, from gradually varied theory, the water depth $100 \mathrm{m}$ upstream.

Victor Salazar

Numerade Educator

Problem 127

A clean-earth river is $50 \mathrm{ft}$ wide and averages $600 \mathrm{ft}^{3} / \mathrm{s}$. It contains a dam that increases the water depth to $8 \mathrm{ft}$, to provide head for a hydropower plant. The bed slope is 0.0025
(a) What is the normal depth of this river?
(b) Engineers propose putting flashboards on the dam to raise the water level to $10 \mathrm{ft}$. Residents a half mile upstream are worried about flooding above their present water depth of about $2.2 \mathrm{ft}$. Using Eq. (10.52) in one big half-mile step, estimate the new water depth upstream.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 128

A rectangular channel $4 \mathrm{m}$ wide is blocked by a broad-crested weir $2 \mathrm{m}$ high, as in Fig. $\mathrm{Pl} 0.128$. The channel is horizontal for $200 \mathrm{m}$ upstream and then slopes at $0.7^{\circ}$ as shown. The flow rate is $12 \mathrm{m}^{3} / \mathrm{s}$, and $n=0.03$. Compute the water depth $y$ at 300 m upstream from gradually varied theory.

James Kiss

Numerade Educator