Figure P10.76 shows a horizontal fl ow of water through a sluice gate, a hydraulic jump

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

Water at 208C fl ows in a 30-cm-wide rectangular channel at a depth of 10 cm and a fl ow rate of 80,000 cm3 /s. Estimate (a) the Froude number and (b) the Reynolds number.

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

The water fl ow in Fig. P10.4 has a free surface in three places. Does it qualify as an open-channel fl ow? Explain. What does the dashed line represent?

Water fl ows down a rectangular channel that is 4 ft wide and 2 ft deep. The fl ow rate is 20,000 gal/min. Estimate the Froude number of the fl ow

Pebbles dropped successively at the same point, into a water channel fl ow 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. V 4 m 9 m 6 m

Pebbles dropped successively at the same point, into a water channel fl ow 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. P10.7 V 9 m 4 m 3 m

An earthquake near the Kenai Peninsula, Alaska, creates a single tidal wave (called a tsunami) that propagates southward across the Pacifi c 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.

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 c2 0 5 g 2 tanh 2y 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 ) For what ratio y/ is the wave speed within 1 percent of limit (c)?

If surface tension Y is included in the analysis of Prob. P10.9, the resulting wave speed is [8 to 10] c 2 0 5 a g 2 1 2Y b tanh 2y (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 @ .

A rectangular channel is 2 m wide and contains water 3 m deep. If the slope is 0.858 and the lining is corrugated metal, estimate the discharge for uniform fl ow.

(a) For laminar draining of a wide, thin sheet of water on pavement sloped at angle , as in Fig. P4.36, show that the fl ow rate is given by Q 5 gbh3 sin 3 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 5 24/Re, where Re 5 Vavh/

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

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

The painted-steel channel of Fig. P10.15 is designed, without the barrier, for a fl ow rate of 6 m3 /s at a normal depth of 1 m. Determine (a) the design slope of the channel and (b) the reduction in total fl ow rate if the proposed paintedsteel central barrier is installed.

Water fl ows in a brickwork rectangular channel 2 m wide, on a slope of 5 m/km. (a) Find the fl ow rate when the normal depth is 50 cm. (b) If the normal depth remains 50 cm, fi nd the channel width which will triple the fl ow rate. Comment on this result

7 The trapezoidal channel of Fig. P10.17 is made of brickwork and slopes at 1:500. Determine the fl ow rate if the normal depth is 80 cm. 30 2 m 30

A V-shaped painted steel channel, similar to Fig. E10.6, has an included angle of 908. If the slope, in uniform fl ow, is 3 m per km, estimate (a) the fl ow rate, in m3 /s and (b) the average wall shear stress. Take y 5 2 m.

Modify Prob. P10.18, the 908 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

An unfi nished concrete sewer pipe, of diameter 4 ft, is fl owing half-full at 39,500 U.S. gallons per minute. If this is the normal depth, what is the pipe slope, in degrees?

An engineer makes careful measurements with a weir (see Sec. 10.7) that monitors a rectangular unfi nished concrete channel laid on a slope of 18. She fi nds, perhaps with surprise, that when the water depth doubles from 2 ft 2 inches to 4 ft 4 inches, the normal fl ow rate more than doubles, from 200 to 500 ft3 /s. (a) Is this plausible? (b) If so, estimate the channel width.

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

It is desired to excavate a clean-earth channel as a trapezoidal cross section with 5 608 (see Fig. 10.7). The expected fl ow rate is 500 ft3 /s, and the slope is 8 ft per mile. The uniform fl ow depth is planned, for effi cient performance, such that the fl ow cross section is half a hexagon. What is the appropriate bottom width of the channel?

A rectangular channel, laid out on a 0.58 slope, delivers a fl ow rate of 5000 gal/min in uniform fl ow when the depth is 1 ft and the width is 3 ft. (a) Estimate the value of Mannings factor n. (b) What water depth will triple the fl ow rate?

The equilateral-triangle channel in Fig. P10.25 has constant slope So and constant Manning factor n. If y 5 a/2, fi nd an analytic expression for the fl ow rate Q. P10.25 a a y a

In the spirit of Fig. 10.6b, analyze a rectangular channel in uniform fl ow with constant area A 5 by, constant slope, but varying width b and depth y. Plot the resulting fl ow rate Q, normalized by its maximum value Qmax, in the range 0.2 , b/y , 4.0, and comment on whether it is crucial for discharge effi ciency to have the channel fl ow at a depth exactly equal to half the channel width

A circular corrugated-metal water channel has a slope of 1:800 and a diameter of 6 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

A new, fi nished-concrete trapezoidal channel, similar to Fig. 10.7, has b 5 8 ft, yn 5 5 ft, and 5 508. For this depth, the discharge is 500 ft3 /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 fl ow rate?

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 crit required to erode sand particles of diameter dp is approximated by crit (s2)g dp < 0.5 where s < 2400 kg/m3 is the density of sand. If the slope of the channel in Fig. P10.17 is 1:900 and n < 0.014, determine the maximum water depth to keep from eroding particles of 1-mm diameter

A clay tile V-shaped channel, with an included angle of 908, is 1 km long and is laid out on a 1:400 slope. When running at a depth of 2 m, the upstream end is suddenly closed while the lower end continues to drain. Assuming quasi-steady normal discharge, fi nd the time for the channel depth to drop to 20 cm

An unfi nished-concrete 6-ft-diameter sewer pipe fl ows half full. What is the appropriate slope to deliver 50,000 gal/min of water in uniform fl ow?

Does half a V-shaped channel perform as well as a full V-shaped channel? The answer to Prob. 10.18 is Q 5 12.4 m3 /s. (Do not reveal this to your friends still working on P10.18.) For the painted-steel half-V in Fig. P10.32, at the same slope of 3:1000, fi nd the fl ow area that gives the same Q and compare with P10.18. P10.32 yn 45

Five sewer pipes, each a 2-m-diameter clay tile pipe running half full on a slope of 0.258, empty into a single asphalt pipe, also laid out at 0.258. If the large pipe is also to run half full, what should be its diameter?

A brick rectangular channel with S0 5 0.002 is designed to carry 230 ft3 /s of water in uniform fl ow. There is an argument over whether the channel width should be 4 or 8 ft. Which design needs fewer bricks? By what percentage?

In fl ood stage a natural channel often consists of a deep main channel plus two fl oodplains, as in Fig. P10.35. The fl oodplains are often shallow and rough. If the channel has the same slope everywhere, how would you analyze this situation for the discharge? Suppose that y1 5 20 ft, y2 5 5 ft, b1 5 40 ft, b2 5 100 ft, n1 5 0.020, and n2 5 0.040, with a slope of 0.0002. Estimate the discharge in ft3 /s. y2 n2 b2 y 1 n1 b1 y 1 b2 y2 n2

The Blackstone River in northern Rhode Island normally fl ows at about 25 m3 /s and resembles Fig. P10.35 with a clean-earth center channel, b1 < 20 m and y1 < 3 m. The bed slope is about 2 ft/mi. The sides are heavy brush with b2 < 150 m. During Hurricane Carol in 1954, a record fl ow rate of 1000 m3 /s was estimated. Use this information to estimate the maximum fl ood depth y2 during this event

A triangular channel (see Fig. E10.6) is to be constructed of corrugated metal and will carry 8 m3 /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 for the channel, (b) the normal depth for part (a), and (c) the wetted perimeter for part (b)?

For the half-Vee channel in Fig. P10.32, let the interior angle of the Vee be . For a given value of area, slope, and n, fi nd the value of for which the fl ow rate is a maximum. To avoid cumbersome algebra, simply plot Q versus for constant A

For the half-Vee channel in Fig. P10.32, let the interior angle of the Vee be . For a given value of area, slope, and n, fi nd the value of for which the fl ow rate is a maximum. To avoid cumbersome algebra, simply plot Q versus for constant A

Using the geometry of Fig. 10.6a, prove that the most effi - cient circular open channel (maximum hydraulic radius for a given fl ow area) is a semicircle.

Determine the most effi cient value of for the V-shaped channel of Fig. P10.41. P10.41 y

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 fl ow: (a) a V channel with 5 458, as in Fig. P10.41, or (b) an effi cient rectangular channel with b 5 2y?

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 fl ow: (a) a V channel with 5 458, as in Fig. P10.41, or (b) an effi cient rectangular channel with b 5 2y?

What are the most effi cient dimensions for a half-hexagon cast iron channel to carry 15,000 gal/min on a slope of 0.168?

Calculus tells us that the most effi cient wall angle for a V-shaped channel (Fig. P10.41) is 5 458. It yields the highest normal fl ow rate for a given area. But is this a sharp or a fl at maximum? For a fl ow area of 1 m2 and an unfi nished-concrete channel with a slope of 0.004, plot the normal fl ow rate Q, in m3 /s, versus angle for the range 308 # # 608 and comment

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]: A 5 2 3 bh0; P 5 b 2 c21 1 2 1 1 ln( 1 21 1 2 )d where 5 4 h0 b For uniform fl ow conditions, determine the most effi cient ratio h0/b for this channel (minimum perimeter for a given constant area). z Parabola z = b b 2 b 2 h0 h(z)

Calculus tells us that the most effi cient water depth for a rectangular channel (such as Fig. E10.1) is y/b 5 1/2. It yields the highest normal fl ow rate for a given area. But is this a sharp or a fl at maximum? For a fl ow area of 1 m2 and a clay tile channel with a slope of 0.006, plot the normal fl ow rate Q, in m3 /s, versus y/b for the range 0.3 # y/b # 0.7 and comment.

A wide, clean-earth river has a fl ow rate q 5 150 ft3 /(s ? ft). What is the critical depth? If the actual depth is 12 ft, what is the Froude number of the river? Compute the critical slope by (a) Mannings formula and (b) the Moody chart.

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

A pencil point piercing the surface of a rectangular channel fl ow creates a wedgelike 258 half-angle wave, as in Fig. P10.50. If the channel surface is painted steel and the depth is 35 cm, determine (a) the Froude number, (b) the critical depth, and (c) the critical slope for uniform fl ow.

An unfi nished concrete duct, of diameter 1.5 m, is fl owing half-full at 8.0 m3 /s. (a) Is this a critical fl ow? If not, what is (b) the critical fl ow rate, (c) the critical slope, and (d ) the Froude number? (e) If the fl ow is uniform, what is the slope of the duct?

Water fl ows full in an asphalt half-hexagon channel of bottom width W. The fl ow rate is 12 m3 /s. Estimate W if the Froude number is exactly 0.60

For the river fl ow of Prob. P10.48, fi nd the depth y2 that has the same specifi c energy as the given depth y1 5 12 ft. These are called conjugate depths. What is Fr2?

A clay tile V-shaped channel has an included angle of 708 and carries 8.5 m3 /s. Compute (a) the critical depth, (b) the critical velocity, and (c) the critical slope for uniform fl ow

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

A trapezoidal channel resembles Fig. 10.7 with b 5 1 m and 5 508. The water depth is 2 m, and the fl ow rate is 32 m3 /s. If you stick your fi ngernail in the surface, as in Fig. P10.50, what half-angle wave might appear?

Modify Prob. P10.59 as follows: Again assuming uniform subcritical approach fl ow (V1, y1), fi nd (a) the fl ow rate and (b) y2 for which the fl ow at the crest of the bump is exactly critical (Fr2 5 1.0)

Consider the fl ow in a wide channel over a bump, as in Fig. P10.62. One can estimate the water depth change or transition with frictionless fl ow. Use continuity and the Bernoulli equation to show that dy dx 5 2 dh/dx 1 2 V2 /(gy) Is the drawdown of the water surface realistic in Fig. P10.62? Explain under what conditions the surface might rise above its upstream position y0

Consider the fl ow in a wide channel over a bump, as in Fig. P10.62. One can estimate the water depth change or transition with frictionless fl ow. Use continuity and the Bernoulli equation to show that dy dx 5 2 dh/dx 1 2 V2 /(gy) Is the drawdown of the water surface realistic in Fig. P10.62? Explain under what conditions the surface might rise above its upstream position y0

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, fi nd the bump height Dh that causes the bump Froude number to be 1.00.

Program and solve the differential equation of frictionless fl ow over a bump, from Prob. P10.62, for entrance conditions V0 5 1 m/s and y0 5 1 m. Let the bump have the convenient shape h 5 0.5hmax[1 2 cos (2x/L)], which simulates Fig. P10.62. Let L 5 3 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 hmax 5 15 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 profi les for hmax < 1 cm up to 35 cm (which is the solution of Prob. P10.64).

In Fig. P10.62, let Vo 5 5.5 m/s and yo 5 90 cm. (a) Will the water rise or fall over the bump? (b) For a bump height of 30 cm, determine the Froude number over the bump. (c) Find the bump height that will cause critical fl ow over the bump

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.

Modify Prob. P10.65 to have a supercritical approach condition V0 5 6 m/s and y0 5 1 m. If you have time for only one case, use hmax 5 35 cm (Prob. P10.66), for which the maximum Froude number is 1.47. If more time is available, it is instructive to examine a complete family of surface profi les for 1 cm , hmax , 52 cm (which is the solution to Prob. P10.67).

Modify Prob. P10.65 to have a supercritical approach condition V0 5 6 m/s and y0 5 1 m. If you have time for only one case, use hmax 5 35 cm (Prob. P10.66), for which the maximum Froude number is 1.47. If more time is available, it is instructive to examine a complete family of surface profi les for 1 cm , hmax , 52 cm (which is the solution to Prob. P10.67).

A periodic and spectacular water release, in Chinas Henan province, fl ows through a giant sluice gate. Assume that the gate is 23 m wide, and its opening is 8 m high. The water depth far upstream is 32 m. Assuming free discharge, estimate the volume fl ow rate through the gate.

In Fig. P10.69 let y1 5 95 cm and y2 5 50 cm. Estimate the fl ow rate per unit width if the upstream kinetic energy is (a) neglected and (b) included.

Water approaches the wide sluice gate of Fig. P10.72 at V1 5 0.2 m/s and y1 5 1 m. Accounting for upstream kinetic energy, estimate at the outlet, section 2, the (a) depth, (b) velocity, and (c) Froude number. (1) (2) (3)

3 In Fig. P10.69, let y1 5 6 ft and the gate width b 5 8 ft. Find the gate opening H that would allow a free-discharge fl ow of 30,000 gal/min under the gate.

With respect to Fig. P10.69, show that, for frictionless fl ow, the upstream velocity may be related to the water levels by V1 5 B 2g(y1 2 y2) K2 2 1

A tank of water 1 m deep, 3 m long, and 4 m wide into the paper has a closed sluice gate on the right side, as in Fig. P10.75. At t 5 0 the gate is opened to a gap of 10 cm. Assuming quasisteady sluice gate theory, estimate the time required for the water level to drop to 50 cm. Assume free outfl ow. Gate closed Gate raised to a gap of 10 cm 1 m 3 m P10.75

Figure P10.76 shows a horizontal fl ow 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 unfi nished concrete. Determine (a) the fl ow rate in ft3 /s and (b) the normal depth.

Equation (10.41) for sluice gate discharge is for free outfl ow. If the outfl ow is drowned, as in Fig. 10.10c, there is dissipation, and Cd drops sharply, as shown in Fig. P10.77, taken from Ref. 2. Use this data to restudy Prob. 10.73, with H 5 9 in. Plot the estimated fl ow rate, in gal/min, versus y2 in the range 0.5 ft , y2 , 5 ft. 0.6 0.5 0.4 0.3 0.2 0.1 Cd y1 H y2 H = 0 2 4 6 8 10 12 14 16 Drowned tailwater Fig. 10.10c Free outflow

In Fig. P10.69, free discharge, a gate opening of 0.72 ft will allow a fl ow rate of 30,000 gal/min. Recall y1 5 6 ft and the gate width b 5 8 ft. Suppose that the gate is drowned (Fig. P10.77), with y2 5 4 ft. What gate opening would then be required?

Show that the Froude number downstream of a hydraulic jump will be given by Fr2 5 81/2 Fr1/ 3(1 1 8 Fr 2 1 ) 1/2 2 14 3/2

Water fl owing in a wide channel 25 cm deep suddenly jumps to a depth of 1 m. Estimate (a) the downstream Froude number; (b) the fl ow rate per unit width; (c) the critical depth; and (d ) the percentage of dissipation.

Water fl ows in a wide channel at q 5 25 ft3 /(s ? ft), y1 5 1 ft, and then undergoes a hydraulic jump. Compute y2, V2, Fr2, hf, the percentage of dissipation, and the horsepower dissipated per unit width. What is the critical depth?

Downstream of a wide hydraulic jump the fl ow is 4 ft deep and has a Froude number of 0.5. Estimate (a) y1, (b) V1, (c) Fr1, (d ) the percentage of dissipation, and (e) yc

Downstream of a wide hydraulic jump the fl ow is 4 ft deep and has a Froude number of 0.5. Estimate (a) y1, (b) V1, (c) Fr1, (d ) the percentage of dissipation, and (e) yc

Consider the fl ow under the sluice gate of Fig. P10.84. If y1 5 10 ft and all losses are neglected except the dissipation in the jump, calculate y2 and y3 and the percentage of dissipation, and sketch the fl ow to scale with the EGL included. The channel is horizontal and wide. P10.84 V1 = 2 ft/s y2 Jump y1 y3

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 Ma1 5 Fr1 5 4.0 and comment on the results.

A bore is a hydraulic jump that propagates upstream into a still or slower-moving fl uid, as in Fig. P10.86, on the Se-Slune channel, near Mont Saint Michel in northwest France. The bore is moving at about 10 ft/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. P10.86 Tidal bore on the Se-Slune river channel in northwest France. (Courtesy of Prof. Hubert Chanson, University of Queensland.)

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 ft deep and propagates at 13 mi/h upstream into a river that is 7 ft deep. Estimate the river current in kn

Consider supercritical fl ow, Fr1 . 1, down a shallow fl at water channel toward a wedge of included angle 2, as in Fig. P10.88. By the compressible fl ow analogy, hydraulic jumps should form, similar to the shock waves in Fig. P9.132a. Using an approach similar to Fig. 9.20, develop and explain the equations that could be used to fi nd the wave angle and Fr2. P10.88 Jumps Fr2 Fr1 > 1

Water 30 cm deep is in uniform fl ow down a 18 unfi nished concrete slope when a hydraulic jump occurs, as in Fig. P10.89. If the channel is very wide, estimate the water depth y2 downstream of the jump. y1 = 30 cm Jump y2 ? Unfinished concrete, 1 slope

For the gate/jump/weir system sketched in Fig. P10.76, the fl ow rate was determined to be 379 ft3 /s. Determine (a) the water depths y2 and y3, and (b) the Froude numbers Fr2 and Fr3 before and after the hydraulic jump

Follow up Prob. P10.88 numerically with fl ow down a shallow, fl at water channel 1 cm deep at an average velocity of 0.94 m/s. The wedge half-angle is 208. Calculate (a) ; (b) Fr2; and (c) y2.

A familiar sight is the circular hydraulic jump formed by a faucet jet falling onto a fl at 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

Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic jump, as in Fig. P10.93. If y1 5 1 m and y3 5 40 cm, estimate (a) V1, (b) V3, (c) y4, and (d ) the bump height h. Jump 1 2 3 4

Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic jump, as in Fig. P10.93. If y1 5 1 m and y3 5 40 cm, estimate (a) V1, (b) V3, (c) y4, and (d ) the bump height h. Jump 1 2 3 4

A 10-cm-high bump in a wide horizontal water channel creates a hydraulic jump just upstream and the fl ow pattern in Fig. P10.95. Neglecting losses except in the jump, for the case y3 5 30 cm, estimate (a) V4, (b) y4, (c) V1, and (d ) y1

For the circular hydraulic jump in Fig. P10.92, the water depths before and after the jump are 2 mm and 4 mm, respectively. Assume that two-dimensional jump theory is valid. If the faucet fl ow rate is 150 cm3 /s, estimate the radius R at which the jump will appear.

A brickwork rectangular channel 4 m wide is fl owing at 8.0 m3 /s on a slope of 0.18. 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 m, (b) 1.5 m, and (c) 2 m?

A gravelly earth wide channel is fl owing at 10 m3 /s per meter of width on a slope of 0.758. 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 m, (b) 2 m, or (c) 3 m?

A clay tile V-shaped channel of included angle 608 is fl owing at 1.98 m3 /s on a slope of 0.338. 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 m, (b) 2 m, or (c) 3 m?

If bottom friction is included in the sluice gate fl ow of Prob. P10.84, the depths (y1, y2, y3) 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 fl ow

Consider the gradual change from the profi le beginning at point a in Fig. P10.101 on a mild slope S01 to a mild but steeper slope S02 downstream. Sketch and label the curve y(x) expected. P10.101 a ? yn2 y c Mild Mild but steeper

The wide-channel fl ow in Fig. P10.102 changes from a steep slope to one even steeper. Beginning at points a and b, sketch and label the water surface profi les expected for gradually varied fl ow. a b Steep Steeper yc yn1 yn2

A gravelly rectangular channel, 7 m wide and 2 m deep, is fl owing at 75 m3 /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?

The rectangular-channel fl ow in Fig. P10.104 expands to a cross section 50 percent wider. Beginning at points a and b, sketch and label the water surface profi les expected for gradually varied fl ow. a b Steep 50% increase in channel width yc1 yn1 yc2 yn2

In Prob. P10.84 the frictionless solution is y2 5 0.82 ft, which we denote as x 5 0 just downstream of the gate. If the channel is horizontal with n 5 0.018 and there is no hydraulic jump, compute from gradually varied theory the downstream distance where y 5 2.0 ft.

A rectangular channel with n 5 0.018 and a constant slope of 0.0025 increases its width linearly from b to 2b over a distance L, as in Fig. P10.106. (a) Determine the variation y(x) along the channel if b 5 4 m, L 5 250 m, the initial depth is y(0) 5 1.05 m, and the fl ow rate

A clean-earth wide-channel fl ow is climbing an adverse slope with S0 5 20.002. If the fl ow rate is q 5 4.5 m3 /(s ? m), use gradually varied theory to compute the distance for the depth to drop from 3.0 to 2.0 m

A clean-earth wide-channel fl ow is climbing an adverse slope with S0 5 20.002. If the fl ow rate is q 5 4.5 m3 /(s ? m), use gradually varied theory to compute the distance for the depth to drop from 3.0 to 2.0 m

Figure P10.109 illustrates a free overfall or dropdown fl ow pattern, where a channel fl ow accelerates down a slope and falls freely over an abrupt edge. As shown, the fl ow reaches critical just before the overfall. Between yc and the edge the fl ow is rapidly varied and does not satisfy gradually varied theory. Suppose that the fl ow rate is q 5 1.3 m3 /(s ? m) and the surface is unfi nished concrete. Use Eq. (10.51) to estimate the water depth 300 m upstream as shown. 300 m S0 = 0.06 y? y

We assumed frictionless fl ow in solving the bump case, Prob. P10.65, for which V2 5 1.21 m/s and y2 5 0.826 m over the crest when hmax 5 15 cm, V1 5 1 m/s, and y1 5 1 m. However, if the bump is long and rough, friction may be important. Repeat Prob. P10.65 for the same bump shape, h 5 0.5hmax[1 2 cos (2x/L)], to compute conditions (a) at the crest and (b) at the end of the bump, x 5 L. Let hmax 5 15 cm and L 5 100 m, and assume a clean-earth surface

The Rolling Dam on the Blackstone River has a weedy bottom and an average fl ow rate of 900 ft3 /s. Assume the river upstream is 150 ft wide and slopes at 10 ft per statute mile. The water depth just upstream of the dam is 7.7 ft. Calculate the water depth one mile upstream (a) for the given initial depth, 7.7 ft; and (b) if fl ashboards on the dam raise this depth to 10.7 ft.

The Rolling Dam on the Blackstone River has a weedy bottom and an average fl ow rate of 900 ft3 /s. Assume the river upstream is 150 ft wide and slopes at 10 ft per statute mile. The water depth just upstream of the dam is 7.7 ft. Calculate the water depth one mile upstream (a) for the given initial depth, 7.7 ft; and (b) if fl ashboards on the dam raise this depth to 10.7 ft.

The Rolling Dam on the Blackstone River has a weedy bottom and an average fl ow rate of 900 ft3 /s. Assume the river upstream is 150 ft wide and slopes at 10 ft per statute mile. The water depth just upstream of the dam is 7.7 ft. Calculate the water depth one mile upstream (a) for the given initial depth, 7.7 ft; and (b) if fl ashboards on the dam raise this depth to 10.7 ft.

For the gate/jump/weir system sketched in Fig. P10.76, the fl ow rate was determined to be 379 ft3 /s. Determine the water depth y4 just upstream of the weir

Gradually varied theory, Eq. (10.49), neglects the effect of width changes, db/dx, assuming that they are small. But they are not small for a short, sharp contraction such as the venturi fl ume in Fig. P10.113. Show that, for a rectangular section with b 5 b(x), Eq. (10.49) should be modifi ed as follows: dy dx < S0 2 S 1 3V 2 /(gb)4(db/dx) 1 2 Fr2 Investigate a criterion for reducing this relation to Eq. (10.49)

A Cipolletti weir, popular in irrigation systems, is trapezoidal, with sides sloped at 1:4 horizontal to vertical, as in Fig. P10.116. The following are fl ow-rate values, from the U.S. Dept. of Agriculture, for a few different system parameters: H 4 1 1 4 b P10.116 H, ft 0.8 1.0 1.35 1.5 b, ft 1.5 2.0 2.5 3.5 Q, gal/min 1620 3030 5920 9740 Source: U.S. Dept of Agriculture. Use this data to correlate a Cipolletti weir formula with a reasonably constant weir coeffi cient.

A popular fl ow-measurement device in agriculture is the Parshall fl ume [33], Fig. P10.117, named after its inventor, Ralph L. Parshall, who developed it in 1922 for the U.S. Bureau of Reclamation. The subcritical approach fl ow is driven, by a steep constriction, to go critical (y 5 yc) 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. Q b H Plan view Elevation view yc

Using a Bernoulli-type analysis similar to Fig. 10.16a, show that the theoretical discharge of the V-shaped weir in Fig. P10.118 is given by Q 5 0.7542g1/2 tan H5/2 H

Data by A. T. Lenz for water at 208C (reported in Ref. 23) show a signifi cant increase of discharge coeffi cient of V-notch weirs (Fig. P10.118) at low heads. For 5 208, some measured values are as follows: H, ft 0.2 0.4 0.6 0.8 1.0 Cd 0.499 0.470 0.461 0.456 0.452 Determine if these data can be correlated with the Reynolds and Weber numbers vis--vis Eq. (10.61). If not, suggest another correlation.

The rectangular channel in Fig. P10.120 contains a V-notch weir as shown. The intent is to meter fl ow rates between 2.0 and 6.0 m3 /s with an upstream hook gage P10.120 Flow 2 m Y

Water fl ow in a rectangular channel is to be metered by a thin-plate weir with side contractions, as in Table 10.2b, with L 5 6 ft and Y 5 1 ft. It is desired to measure fl ow 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?

In 1952 E. S. Crump developed the triangular weir shape shown in Fig. P10.122 [23, Chap. 4]. The front slope is 1:2 to avoid sediment deposition, and the rear slope is 1:5 to maintain a stable tailwater fl ow. The beauty of the design is that it has a unique discharge correlation up to near-drowning conditions, H2/H1 # 0.75: Q 5 Cd bg1/2aH1 1 V2 1 2g 2 khb 3/2 where Cd < 0.63 and kh < 0.3 mm The term kh is a low-head loss factor. Suppose that the weir is 3 m wide and has a crest height Y 5 50 cm. If the water depth upstream is 65 cm, estimate the fl ow rate in gal/min. Flow 1:2 slope Hydraulic jump 1:5 slope H2

Water in a 20-ft-wide rectangular channel, fl owing at 120 ft3 /s and a depth of 10 ft, is to be metered by a rectangular weir with side contractions, as in Table 10.2b. Suggest some appropriate design values of b, Y, and H to match the table conditions for this weir.

Water fl ows at 600 ft3 /s in a rectangular channel 22 ft wide with n < 0.024 and a slope of 0.18. A dam increases the depth to 15 ft, as in Fig. P10.124. Using gradually varied theory, estimate the distance L upstream at which the water depth will be 10 ft. What type of solution curve are we on? What should be the water depth asymptotically far upstream? P10.124 Backwater curve 10 ft 15 ft L = ?

Water fl ows at 600 ft3 /s in a rectangular channel 22 ft wide with n < 0.024 and a slope of 0.18. A dam increases the depth to 15 ft, as in Fig. P10.124. Using gradually varied theory, estimate the distance L upstream at which the water depth will be 10 ft. What type of solution curve are we on? What should be the water depth asymptotically far upstream? P10.124 Backwater curve 10 ft 15 ft L = ?

Suppose that the rectangular channel of Fig. P10.120 is made of riveted steel and carries a fl ow of 8 m3 /s on a slope of 0.158. If the V-notch weir has 5 308 and Y 5 50 cm, estimate, from gradually varied theory, the water depth 100 m upstream

A clean-earth river is 50 ft wide and averages 600 ft3 /s. It contains a dam that increases the water depth to 8 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 fl ashboards on the dam to raise the water level to 10 ft. Residents a half mile upstream are worried about fl ooding above their present water depth of about 2.2 ft. Using Eq. (10.52) in one big half-mile step, estimate the new water depth upstream.

8 A rectangular channel 4 m wide is blocked by a broad-crested weir 2 m high, as in Fig. P10.128. The channel is horizontal for 200 m upstream and then slopes at 0.78 as shown. The fl ow rate is 12 m3 /s, and n 5 0.03. Compute the water depth y at 300 m upstream from gradually varied theory. y? y(x) Slope 0.7 12 m3/s 200 m 100 m