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Trapezoidal channel of Chow's Examples 6-2 and 10-1: normal depth, critical depth and the M1 backwater profile behind a dam

Tool	Open-Channel Flow
Source	Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill, New York. Example 6-2 (normal depth by Manning's equation), Example 10-1 and Table 10-4 (direct-step method, as reproduced in the Nashville Stormwater Management Manual, Vol. 2, Ch. 3, Table 3-9).
Level	Undergraduate

Problem

A trapezoidal channel with a 20-ft bottom, 2:1 side slopes, n = 0.025 and a bed slope of 0.0016 carries 400 ft³/s. Find the normal depth and velocity, the critical depth (energy coefficient α = 1.10), the flow regime, and the water-surface profile upstream of a dam that holds the water 5 ft deep, to the point where the depth is within 1 % of normal.

Inputs to Enter

Shape trapezoidal, b = 20 ft, z = 2, n = 0.025 (custom), S_0 = 0.0016, α = 1.10
Solve for the normal depth with Q = 400 ft³/s; hydraulic jump from y_1 = 1.0 ft (illustrative); profile from the control depth 5 ft over a 2,400-ft reach in 14 steps

Open the Open-Channel Flow tool (loaded with this example)

Answers

Normal depth y_n = 3.361 ft (book 3.36), A = 89.8 ft², R = 2.564 ft, V = 4.454 ft/s (book 4.45), Fr = 0.503: subcritical; τ_0 = 0.256 lb/ft², conveyance 10,000 ft³/s
Critical depth y_c = 2.212 ft (book 2.22 with α = 1.10; 2.15 with α = 1), V_c = 7.40 ft/s, E_min = 3.149 ft, critical slope 0.00704: the slope is mild
Specific energy at the normal depth 3.700 ft; alternate (supercritical) depth 1.542 ft
Jump from 1.0 ft: Fr_1 = 3.35 (oscillating), conjugate depth 3.891 ft by the momentum function (scipy 3.891376), energy loss 2.526 ft (38 %), 114.7 hp dissipated, length 22.2 ft (USBR L/y_2 = 5.69)
M1 profile: Chow's Table 10-4 reaches 3.40 ft at 2,375 ft upstream (tool with Chow's depth sequence 2,380 ft; exact integration 2,385 ft); the tool's 14 geometric steps give 3.399 ft at 2,400 ft

Notes

The tool's default inputs are this case; press Submit. Chow's Table 10-4 rows (A, E, x at every depth) are reproduced within 1 % in test_open_channel_flow.php, with White's Chapter 10 examples (10.2 to 10.13), USBR Engineering Monograph 25 and Water Measurement Manual values, Bengtson's course examples and the scipy script References/Open-Channel-Flow/channel_check.py.