Aeronautical Engineering Chapter 10 Homework the wave causes a change in normal Froude number

Solution 10.88*

As with the oblique shock wave, the wave causes a change in normal Froude number. There is

no change in Froude number parallel to the wave.

Problem 10.89

Water 30 cm deep is in uniform flow down a 1 unfinished-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.

Solution 10.89

For unfinished concrete, take n  0.014. Compute the upstream velocity:

Problem 10.90

For the gate/jump/weir system sketched earlier in Fig. P10.76, the flow 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.

Solution 10.90

With the flow rate known, we can compute V1 and E1 and equate them to section 2 values:

Problem 10.91*

Follow up Prob. P10.88 numerically with flow down a shallow, flat water channel 1 cm deep at

an average velocity of 0.94 m/s. The wedge half-angle θ is 20º. Calculate (a) β; (b) Fr2 ; and

(c) y2.

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.

Solution 10.91*

The system of equations was developed in Prob. P10.88, with the hydraulic jump across the

oblique wave based upon the Froude numbers normal to the jump. The Froude numbers parallel

Problem 10.92

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

non-circular shapes, as shown in the website http://www-math.mit.edu/~bush/jump.htm. For

this problem, assume that two-dimensional jump theory is valid. If the water depth outside the

jump is 5 mm, the radius at which the jump appears is R = 3 cm, and the faucet flow rate is

100 cm3/s, find the conditions just upstream of the jump.

Solution 10.92

Compute the upstream Froude number, jump theory, and one-dimensional continuity:

Problem 10.93

Water in a horizontal channel accelerates smoothly over a bump and then undergoes a hydraulic

jump, as in Fig. P10.93. If y1 = 1 m and y3 = 40 cm, estimate (a) V1; (b) V3; (c) y4; and (d) the

bump height h.

Solution 10.93

Assume frictionless flow except in the jump. From point 1 to point 3:

Problem 10.94

In Fig. 10.11, the upstream flow is only 2.65 cm deep. The channel is 50 cm wide, and the flow

rate is 0.0359 m3/s. Determine (a) the upstream Froude number; (b) the downstream velocity;

(c) the downstream depth; and (d) the percent dissipation.

Solution 10.94

(a) We have enough information to find the upstream Froude number:

Problem 10.95

A 10-cm-high bump in a wide horizontal channel creates a hydraulic jump just upstream and the

flow pattern in Fig. P10.95. Neglect losses except in the jump. If y3 = 30 cm, estimate (a) V4;

(b) y4; (c) V1; and (d) y1.

Solution 10.95

Since section “2” is subcritical and “4” is supercritical, assume “3” is critical:

Problem 10.96

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

flow rate is 150 cm3/s, estimate the radius R at which the jump will appear.

Solution 10.96

The proper radius R must yield the upstream and downstream velocities that satisfy two-

dimensional jump theory:

Problem 10.97

A brickwork rectangular channel 4 m wide is flowing at 8.0 m3/s on a slope of 0.1. 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; (c) 2 m?

Solution 10.97

For brickwork, take n  0.015. Then, with q = Q/b = 8/4 = 2.0 m3/sm,

Problem 10.98

A gravelly-earth wide channel is flowing at 10.0 m3/s per meter on a slope of 0.75. 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; (c) 3 m?

Solution 10.98

For gravelly earth, take n  0.025. Then, with Rh = y itself,

nc

Problem 10.99

A clay tile V-shaped channel, of included angle 60, is flowing at 1.98 m3/s on a slope of 0.33.

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; (c) 3 m?

Solution 10.99

For clay tile, take n  0.014. For a 60 Vee-channel, from Example 10.5 of the text, A = y2cot60

and Rh = (y/2)cos60. For uniform flow,

Problem 10.100

If bottom friction is included in the sluice gate flow of Prob. 10.84, 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 varying flow.

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 10.100

The expected curves are all of the “H” (horizontal) type and are shown below:

Problem 10.101

Consider the gradual change from the profile beginning at point a in Fig. P10.101 on a mild slope So1

to a mild but steeper slope So2 downstream. Sketch and label the curve y(x) expected.

Solution 10.101

There are two possible profiles, depending upon whether or not the initial M-2 profile slips

below the new normal depth yn2. These are shown on the next page:

Problem 10.102*

The wide channel flow 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 profiles which are expected for

gradually varied flow.

Solution 10.102

The point-a curve will approach each normal depth in turn. Point-b curves, depending upon initial

Problem 10.103

A gravelly rectangular channel, 7 m wide and 2 m deep, is flowing 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?

Solution 10.103

For a gravelly surface, from Table 10.1, n = 0.025. (a) Find the critical and normal depths:

Problem 10.104

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 which are expected for

gradually varied flow.

Solution 10.104

Three types of dual curves are possible: S2/S2, S3/S2, and S3/S3, as shown:

Problem 10.105

In Prob. 10.84 the frictionless solution is y2 = 0.82 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 ft.

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 10.105

Given q = Vy = 20 ft3/s·ft, the critical depth is yc = (q2/32.2)1/3 = 2.32 ft, hence we are on an