3-8 Summary 81

3.3. A trapezoidal channel having a bottom width of 20 m and side slopes of2H:1V is carrying 60 m3/s. Assuming = 1.1, determine the critical depth.

3.4. For the channel section shown in Fig. 3-11, determine the critical depthfor a flow of 80 m3/s.

Fig. 3-11. Cross section for Prob. 3-4

3.5. For a discharge of 850 m3/s, compute the critical depth in a tunnel havinga standard horseshoe section (Fig. 3-12). The flow area, A, top water-surfacewidth, B, and hydraulic radius, R, at different flow depths, y, are listed in thefollowing table:

Fig. 3-12. Horseshoe section

3.6. If y1 and y2 are the alternate depths in a rectangular channel, prove that

yc =3

2(y1y2)2

y1 + y2

82 3 CRITICAL FLOW

y A R B(m) (m2) (m) (m)

0 0.0 0.00 0.02 23.4 1.34 16.64 58.3 2.68 18.26 95.7 3.70 19.28 134.8 4.50 19.810 174.6 5.15 20.012 214.4 5.65 19.614 252.5 5.99 18.316 287.0 6.13 16.018 315.4 6.01 12.020 331.7 5.08 0.0

3.7. Derive expressions for the critical depth in a prismatic channel havingthe following cross sections and assuming in each case that the slope of thechannel bottom is small:

i. Trapezoidalii. Triangulariii. Circular

3.8. A 50-m wide rectangular channel is carrying a flow of 250 m3/s at a flowdepth of 5 m. To produce critical flow in this channel, determine:

i. The height of the step in the channel bottom if the width remains constantii. The reduction in the channel width if the channel-bottom level remains

unchangediii. A combination of the width reduction and the bottom step.

3.9. The drainage canal shown in Fig. 3-13 has a flow of 96 m3/s. If the flowdepth at Section 1 is 4.22 m, what is the depth at Section 2? Assume thereare no losses in the transition.

Determine the flow depth at the downstream end if the canal ends in afree overfall. Assume that critical depth occurs at the overfall.

3.10. Write a general-purpose computer program to determine the criticaldepth in a channel for a specified discharge and for either of the followingchannel cross sections:

i. Circular;ii. Trapezoidal;iii. Triangular; andiv. Horseshoe.

3-8 Summary 83

Fig. 3-13. Drainage canal of Prob. 3-9

3.11. Show that the critical depth in a channel having a triangular, rectangu-lar, or parabolic cross section may be determined from the following explicitequation:

yc =

[Q2

g

(m+ 1)3

4k2

]1/(2m+3)in which the x and y coordinates of the sides of the half-section may be definedas x = kym. For a triangular cross section, k = s, m = 1; for a rectangularcross section, k = 12Bo, m = 0; and for a parabolic cross section, k = (1/a)

m,and m = 1/n with the equation for the parabola being y = axn.

3.12. A mountain creek has a parabolic cross section with a top water surfacewidth of 9 ft at a depth of 3 ft. Determine the critical depth for a flow of 50ft3/s.

3.13. An 8-ft diameter concrete-lined sewer is laid at a bottom slope of 1ft/mile. Compute the critical depth for a discharge of 100 cfs.

3.14. A trapezoidal irrigation channel is 10 ft wide at the bottom and hasside slopes of 1 V: 2H. For a flow of 100 ft3/s, determine the critical depth.

3.15. A 5-ft dia circular culvert carries a flow of 15 ft3/s. Determine thecritical depth.

3.16. For a horse-shoe tunnel shown in Fig. 1-17 (do = 30 ft), determine thecritical depth for a discharge of 300 cfs.

3.17. A 4-ft diameter culvert barral carries a flow of 10 ft3/sec and dischargesfree into a lake. What is the depth just upstream of the free fall?

84 3 CRITICAL FLOW

3.18. A high-level rectangular outlet at a dam is 10 ft wide, 15 ft high and500 ft long. The invert level at the entrance is at El. 122 and the botom slopeis 0.001. A sluice gate at the entrance controls the flow through the outlet. Ifthe coefficient of discharge, Cd, is 0.7 (Q = CdA

2gH), determine the thrust

on the gate for the reservoir level of El 182. The water level in the river intowhich the outlet discharges is at El. 131 for this flow.

3.19. In order to reduce the flow velocity at a section, a fisheries biologisttied a 6-in diameter tree log at the bottom of a stream. The flow velocityand the flow depth prior to the installation of the log were 2 ft/sec and 4 ftrespectively. Determine the change in the flow velocity and flow depth justdownstream of the log.

3.20. For the channel of Problem 3-8, what is the minimum channel widthwithout affecting the upstream water level.

3.21. The reservoir level upstream of an overflow spillway is at El. 400 ft. Thedownstream water level for the design flow of 80,000 cfs is at El. 220 ft. If thespillway width at the entrance to the stilling basin is 200 ft, determine theinvert level of the basin so that a hydraulic jump is formed in the basin atdesign flow. No bae piers, chute blocks, or end sill is to be provided.

3.22. Write a computer program to determine the critical depth in a channelwith circular cross section. Use the bisection and the Newton methods.

3.23. The flow velocity and flow depth in a 5-m wide rectangular channel are1.5 m/s and 4 m respectively. Design a converging transition so that the flowis critical in the transition. Assume the channel bottom to be horizontal, andlosses in the transition to be negligible.

References

Black, R. G., 1982, Discussion of Blalock and Sturm [1981], Jour. Hyd. Div.,Amer. Soc. Civ. Engrs., vol. 108, pp. 798-800.

Blalock, M. E. and Sturm, T. W., 1981, Minimum Specific Energy in Com-pound Channel, Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 107, pp.699-717 (see also closure, vol. 109, 1983, pp. 483-486).

Chapra, S. C. and Canale, R. P., 2006, Numerical Methods for Engineers, 5thed., McGraw Hill, New York, NY.

Chaudhry, M. H., and Bhallamudi, S. M., 1988, Computation of Critical Depthin Compound Channels, Jour., Hydraulic Research , Inter. Assoc. for Hy-draulic Research., vol. 26, no. 4, pp. 377-395.

Chow, V. T., 1959, Open-Channel Hydraulics , McGraw-Hill Book Company,New York, N. Y.