The rectangular channel in Fig. 12–27a is made of unfinished concrete and has a slope of S0 = 0.035. When it reaches the check dam, the flow backs up as shown in Fig. 12–27b. At a specific location before the dam, the water has a depth of 1.25 m and the flow is Q = 0.75 m³/s. Classify the surface

Question

The rectangular channel in Fig. 12–27a is made of unfinished concrete and has a slope of [latex]S_0[/latex] = 0.035. When it reaches the check dam, the flow backs up as shown in Fig. 12–27b. At a specific location before the dam, the water has a depth of 1.25 m and the flow is Q = 0.75 m³/s. Classify the surface profile for the flow. Take n = 0.014.

Accepted Answer

Fluid Description. The flow is steady and nonuniform. The water is considered incompressible.
Analysis. To classify the surface profile, we must determine the critical depth [latex]y_c[/latex], the normal flow depth [latex]y_n[/latex], and the critical slope [latex]S_c[/latex]. Using Eq. 12–7, with q = Q/b = (0.75 m³/s)/(2 m) = 0.375 m²/s,

[latex]y_c = (\frac{q^2}{g})^{1/3} = [\frac{(0.375 m^2/s)^2}{9.81 m/s^2}]^{1/3} = 0.2429 m[/latex]

Since y = 1.25 m > [latex]y_c[/latex] = 0.2429 m, the flow is subcritical. The depth [latex]y_n[/latex] that will produce normal or uniform flow for Q = 0.75 m³/s is determined from the Manning equation. Since
P = (2[latex]y_n[/latex] + 2 m)
then Eq. 12–20 becomes

[latex]Q = \frac{kA^{5/3} S_0 ^{1/2}}{n P^{2/3}}; 0.75 m^3 /s = \frac{(1)[(2 m)y_n]^{5/3} (0.035)^{1/2}}{0.014(2y_n + 2 m)^{2/3}}[/latex]

[latex] \frac{y_n^{5/3}}{(2y_n + 2 m)^{2/3}} = 0.017678[/latex]

Solving by trial and error, or using a numerical
procedure, we get
[latex]y_n[/latex] = 0.1227 m
The critical slope is now determined from Eq. 12–22,

[latex]S_c = \frac{n^2gA_c}{k^2b_{top}R_{hc}^{4/3}} = \frac{(0.014)^2 (9.81 m/s^2)(2 m)(0.2429 m)}{(1)^2 (2 m)[\frac{2 m(0.2429 m)}{2(0.2429 m) + 2 m}]^{4/3}}[/latex]

= 0.004118

Since y = 1.25 m > [latex]y_c[/latex] > [latex]y_n[/latex] and [latex]S_0[/latex] = 0.035 > [latex]S_c[/latex], then according to Table 12–2 the water surface in Fig. 12–27 is an S1 profile.

Question 12.12: The rectangular channel in Fig. 12–27a is made of unfinished......

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