Problem

Consider a starting (assumed) water surface elevation of 5717.6 ft at cross-section 1 for...

Consider a starting (assumed) water surface elevation of 5717.6 ft at cross-section 1 for example 5.4.5 and determine the computed water surface elevation at cross-section 4.

EXAMPLE 5.4.5

A plan view of the Red Fox River in California is shown in Figure 5.4.3, along with the location of four cross-sections. Perform the standard step calculations to determine the water surface elevation at cross-section 3 for a discharge of 6500 ft3/s. Figures 5.4.4a, b, and c are plots of cross-sections at 1, 2, and 3, respectively. Figures 5.4.5a, b, and c are the area and hydraulic radius curves for cross-sections 1, 2, and 3, respectively. Use expansion and contraction coefficients of 0.3 and 0.1, respectively. Maiming’s roughness factors are presented in Figure 5.4.4. The downstream starting water surface elevation at cross-section 1 is 5710.5 ft above mean sea level. This example was originally adapted by the U.S. Army Corps of Engineers from material developed by the U.S. Bureau of Reclamation (1957). Distance between cross-sections 1 and 2 is 500 ft, between cross-sections 2 and 3 is 400 ft, and between cross-sections 3 and 4 is 400 ft.

SOLUTION

The computations for this example are illustrated in Table 5.4.2.

Table 5.4.2 Standard Step Backwater Computation for Red Fox River

(1) Cross-section	(2) Water surface elevation Wk + 1	(3) Area A (ft2)	(4) Hydraulic radius R (ft)	(5) Manning roughness n	(6) Conveyance K	(7) Average conveyance	(8) Average friction slope Sf(10−3) (ft/ft)	(9) Friction loss hL (ft)	(10) (106)	(11) Energy correction factor α	(12) Velocity V (ft/sec)	(13) (ft)	(14) (ft)	(15) ho (ft)	(16) Wk (ft)
1	5710.5*	420	7.0	0.03	76,100					1.0	15.5	3.72
2	5714.7	470	7.6	0.03	90,100				3311.1
		260	2.5	0.05	14,200				42.0
		730			104,300	90,200	5.19	2.60	3353.5	1.58	8.90	1.95	+ 1.77	0.18	5715.07**
	5715.0
		500	7.85	0.03	97,800				3741.8
		300	2.7	0.05	17,300				57.5
		800			115,100	95,600	4.62	2.31	3799.3	1.59	8.13	1.63	+ 2.09	0.21	5715.1
3	5718.0	1145	5.85	0.03	184,100	149,600	1.89	0.76		1.0	5.68	0.50	+ 1.13	0.11	5717.1
	5717.1	970	5.6	0.03	151,500	133,300	2.38	0.95		1.0	6.70	0.70	+ .93	0.09	5717.1

*Known starting water surface elevation.

**Wk + 1 = 5710.7 + 1.77 + 2.60 + 0.18 = 5715.07 = 5715.1; for Δ(αV22g) < 0 (loss due to channel expansion); ho = Cc|Δ(αV2/2g)| for Δ(αV2/2g) > 0 (loss due to channel contraction); Wk + 1 = Wk + Δ(αV2/2g) + hL + ho.

Source: Hoggan (1997).

Figure 5.4.3 Map of the Red Fox River indicating cross-sections for water surface profile analysis (from U.S. Bureau of Reclamation (1957)).

Figure 5.4.4 Cross-sections of the Red Fox River (from U.S. Bureau of Reclamation (1957)).

Figure 5.4.5 Area elevation and hydraulic radius—elevation curves for cross-sections 1 to 4. (a) Cross-section 1; (b) Cross-section 2; (c) Cross-section 3; (d) Cross-section 4 (from U.S. Bureau of Reclamation (1957)).