Aeronautical Engineering Chapter 10 Homework Consider the flow in a wide channel over a bump

Problem 10.67

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.

Problem 10.63

In Fig. P10.62, let Vo = 1 m/s and yo = 1 m. If the maximum bump height is 15 cm, estimate

(a) the Froude number over the top of the bump; and (b) the maximum depression in the water

surface.

Problem 10.62

Consider the flow in a wide channel over a bump, as in Fig. P10.62. One can estimate the water–

depth change or transition with frictionless flow. Use continuity and the Bernoulli equation to

show that

2

/

1 / ( )

dy dh dx

dx V 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 yo.

Solution 10.67

The upstream Froude number is V1/(gy1)1/2 = 0.319, subcritical. Use Eq. (10.39):

Problem 10.68

Modify Prob. 10.65 to have a supercritical approach condition Vo = 6 m/s and yo = 1 m. If you have

time for only one case, use hmax = 35 cm (Prob. 10.66), for which the maximum Froude number

is 1.47. If more time is available, it is instructive to examine a complete family of surface profiles

for 1 cm  hmax  52 cm (which is the solution to Prob. 10.67).

Problem 10.65

Program and solve the differential equation of “frictionless flow over a bump,” from Prob. 10.62,

for entrance conditions Vo = 1 m/s and yo = 1 m. Let the bump have the convenient shape h =

0.5hmax[1 − cos(2



x/L)], which simulates Fig. P10.62. Let L = 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 = 15 cm (Prob. 10.63), 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 hmax  1 cm up to

35 cm (which is the solution of Prob. 10.64).

Problem 10.66*

In Fig. P10.62 let Vo = 5.5 m/s and yo = 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 flow over the bump.

Solution 10.68

This is quite similar to the subcritical display in Prob. 10.65. The new family of supercritical-

flow profiles is shown below:

Problem 10.69*

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

2 3 2

1

/( )Q y b g

as a function of the ratio y2/y1. Show by differentiation that

the maximum flow rate occurs at y2 = 2y1/3.

Solution 10.69

With upstream kinetic energy neglected, the energy equation becomes

Problem 10.70

A periodic and spectacular water release, in China’s Henan province, flows through a giant

sluice gate. Assume the gate is 23 m wide with an opening 8 m high, with the water depth far

upstream assumed equal to 32 m. Assuming free discharge, estimate the volume flow rate

through the gate.

Solution 10.70

Let us proceed, although the writer has no firm idea if the sluice gate formula,

Problem 10.71

In Fig. P10.69 let y1 = 95 cm and y2 = 50 cm. Estimate the flow rate per unit width if the

upstream kinetic energy is (a) neglected; and (b) included.

Problem 10.69*

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

2 3 2

1

/( )Q y b g

as a function of the ratio y2/y1. Show by differentiation that

the maximum flow rate occurs at y2 = 2y1/3.

Solution 10.71

The result of Prob. 10.69 gives an excellent answer to part (a):

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 10.72

(a) If we assume frictionless flow, the gap size is immaterial, and Eq. (10.40) applies:

Problem 10.73

In Fig. P10.69, let y1 = 6 ft and the gate width b = 8 ft. Find (a) the gate opening H that would

allow a free-discharge flow of 30,000 gal/min; and (b) the depth y2.

Problem 10.69*

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

2 3 2

1

/( )Q y b g

as a function of the ratio y2/y1. Show by differentiation that

the maximum flow rate occurs at y2 = 2y1/3.

Solution 10.73

(a) The gate-opening problem is handled by Eq. (10.41):

Problem 10.74

With respect to Fig. P10.69, show that, for frictionless flow, the upstream velocity may be related

to the water levels by

12

2 ( )

1

g y y

VK

−

=−

where K = y1/y2.

Problem 10.69*

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

2 3 2

1

/( )Q y b g

as a function of the ratio y2/y1. Show by differentiation that

the maximum flow rate occurs at y2 = 2y1/3.

Solution 10.74

We have already shown this beautifully in Prob. 10.71b:

Problem 10.75

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 = 0 the gate is opened to a gap of 10 cm. Assuming quasi-

steady sluice-gate theory, estimate the time required for the water level to drop to 50 cm. Assume

free outflow.

Solution 10.75

Use a control volume surrounding the tank with Eq. 10.41 for the gate flow:

Problem 10.76

Figure P10.76 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 ft3/s, and (b) the normal depth.

Solution 10.76

This problem is concerned only with the sluice gate. (Other questions will be asked later.)

(a) The flow rate follows from the sluice gate correlation, Eq. (10.41):

Problem 10.77

Equation (10.41) for sluice gate discharge is for free outflow. If the outflow 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 = 9 inches. Plot the estimated flow rate, in

gal/min, versus y2 in the range 0.5 ft < y2 < 5 ft.

Problem 10.73

In Fig. P10.69, let y1 = 6 ft and the gate width b = 8 ft. Find (a) the gate opening H that would

allow a free-discharge flow of 30,000 gal/min; and (b) the depth y2.

Problem 10.69*

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

2 3 2

1

/( )Q y b g

as a function of the ratio y2/y1. Show by differentiation that

the maximum flow rate occurs at y2 = 2y1/3.

Solution 10.77

We pick all our discharge coefficients from the single vertical line y1/H = 6 ft/0.75 ft = 8 in

0

5000

10000

15000

20000

25000

30000

35000

0 1 2 3 4 5

Q – gallons per minute

y2 – feet

Problem 10.78

In Fig. P10.69, free discharge, a gate opening of 0.72 ft will allow a flow rate of 30,000 gal/min.

Recall y1 = 6 ft and the gate width b = 8 ft. Suppose that the gate is drowned (Fig. P10.77), with

y2 = 4 ft. What gate opening would then be required?

Solution 10.78

We need to find Cd, which varies considerably in Fig.P10.77, such that

Problem 10.79

Show that the Froude number downstream of a hydraulic jump will be given by

1/2 2 1/2 3/2

2 1 1

Fr 8 Fr / [(1 8Fr ) 1] .= + −

Does the formula remain correct if we reverse subscripts 1 and 2? Why?

Solution 10.79

Take the ratio of Froude numbers, use continuity, and eliminate y2/y1:

Problem 10.80

Water flowing in a wide channel 25 cm deep suddenly jumps to a depth of 1 m. Estimate (a) the

downstream Froude number; (b) the flow rate per unit width; (c) the critical depth; and (d) the

percentage of dissipation.

Solution 10.80

(a) First find the upstream and then the downstream Froude number:

Problem 10.81

Water flows in a wide channel at q = 25 ft3/(s·ft) and y1 = 1 ft and undergoes a hydraulic jump.

Compute y2, V2, Fr2, hf, the percentage dissipation, and the horsepower dissipated per unit

width. What is the critical depth?

Solution 10.81

This is a series of straightforward calculations:

Problem 10.82

Downstream of a wide hydraulic jump the flow is 4 ft deep and has a Froude number of 0.5.

Estimate (a) y1; (b) V1; (c) Fr1; (d) the percent dissipation; and (e) yc.

Solution 10.82

As shown in Prob. 10.79, the hydraulic jump formulas are reversible. Thus, with Fr1 known,

Problem 10.83

A wide channel flow undergoes a hydraulic jump from 40 cm to 140 cm. Estimate (a) V1;

(b) V2; (c) the critical depth, in cm; and (d) the percent dissipation.

Solution 10.83

With the jump-height-ratio known, use Eq. 10.43:

Problem 10.84*

Consider the flow under the sluice gate of Fig. P10.84. If y1 = 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 flow to scale with the EGL included. The channel is horizontal and wide.

Solution 10.84

First get the conditions at “2” by assuming a frictionless acceleration:

Problem 10.85

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 Ma1 = Fr1 = 4.0 and comment on the results.

Solution 10.85

The normal shock wave results are right in Table B.2:

Problem 10.86

A bore is a hydraulic jump that propagates upstream into a still or slower-moving fluid, as in

Fig. P10.86, on the Sée-Sélune 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.

Solution 10.86

The bore moves at speed C =10 ft/s

Problem 10.87

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 ft deep and

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

Solution 10.87

Modify the analysis in 10.86 by superimposing a river velocity Vr onto the flow. Then, as shown,

Problem 10.88*

Consider supercritical flow, Fr1 > 1, down a shallow flat water channel toward a wedge of

included angle 2θ, as in Fig. P10.88. By the compressible flow analogy, hydraulic jumps should

form, similar to the shock waves in Fig. P9.132-A. Using an approach similar to Fig. 9.20,

develop and explain the equations that could be used to find the wave angle β and Fr2.