--- title: "Hump and constriction: critical depth, 1.8 m channel" subject: "Hydraulics" chapter: 7 key: "semester_4/hydraulics:hump-sidewall-constriction-critical-depth-1-8m-wide" source: https://www.bctengineer.com/q/semester_4/hydraulics/hump-sidewall-constriction-critical-depth-1-8m-wide site: BCT Engineering --- # Hump and constriction: critical depth, 1.8 m channel Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow - Asked in: 2070 Bhadra, 9 marks - Video: [Change in depths due to hump in a rectangular channel | Open Channel Flow](https://www.youtube.com/watch?v=zyxOadh2BzM) — The Study Sphere - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/hump-sidewall-constriction-critical-depth-1-8m-wide - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7 ## Question A flow of 2 m³/s is carried in a rectangular channel 1.8 m wide at a depth of 1.0 m. Will critical depth occur at a section where (a) a frictionless hump 15 cm high is installed across the bed? (b) a frictionless sidewall reduces the channel width to 1.3 m? (c) the hump and the sidewall constriction are installed together? ## Short answer Critical depth does not occur for the hump alone (available energy 0.913 m > E_c 0.752 m) nor for the sidewall contraction alone (1.063 m > 0.934 m). With both together, E_c + Δz = 1.084 m > approach energy 1.063 m, so flow chokes: critical depth 0.623 m occurs and upstream depth rises to 1.024 m. ## Answer **Given:** $Q = 2\ \text{m}^3/\text{s}$, $B = 1.8\ \text{m}$, $y_1 = 1.0\ \text{m}$. The approach flow is subcritical. **Approach specific energy** $$ \begin{aligned} q_1 &= \frac{2}{1.8} = 1.111\ \text{m}^2/\text{s},\quad V_1 = 1.111\ \text{m/s} \\ E_1 &= 1.0 + \frac{1.111^2}{2 \times 9.81} = 1.063\ \text{m} \end{aligned} $$ Rule: critical depth occurs at the section only if the energy available there is not more than the minimum energy needed, $E_1 - \Delta z \le E_c = 1.5\,y_c$ (with $y_c = (q^2/g)^{1/3}$). #### (a) Hump of 0.15 m $$ y_c = \left(\frac{1.111^2}{9.81}\right)^{1/3} = 0.501\ \text{m},\quad E_c = 0.752\ \text{m} $$ Available energy over the hump $= 1.063 - 0.15 = 0.913\ \text{m} > 0.752\ \text{m}$. **No critical depth.** The flow stays subcritical over the hump with $y \approx 0.82\ \text{m}$. #### (b) Side-wall contraction to 1.3 m $$ q_2 = \frac{2}{1.3} = 1.538\ \text{m}^2/\text{s},\quad y_c = \left(\frac{1.538^2}{9.81}\right)^{1/3} = 0.623\ \text{m},\quad E_c = 0.934\ \text{m} $$ Available energy $= 1.063\ \text{m} > 0.934\ \text{m}$. **No critical depth.** The depth in the throat is about 0.92 m (subcritical). #### (c) Hump and contraction together $$ E_c + \Delta z = 0.934 + 0.15 = 1.084\ \text{m} > E_1 = 1.063\ \text{m} $$ The approach flow does not have enough energy. **Critical depth does occur** at the throat ($y = y_c = 0.623\ \text{m}$) and the flow chokes. The upstream depth rises until $E_1' = 1.084\ \text{m}$: $$ y_1' + \frac{1.111^2}{2 g\, y_1'^2} = 1.084 \Rightarrow y_1' = 1.024\ \text{m} $$ **Answer:** (a) No, (b) No, (c) Yes. In (c) the upstream depth is raised from 1.00 m to about 1.02 m. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the rule used to decide if critical depth occurs at the section? A. Critical depth occurs if the approach Froude number exceeds 0.5 B. Critical depth occurs if E1 - dz is less than or equal to Ec = 1.5 yc C. Critical depth occurs whenever the width is reduced D. Critical depth occurs if the hump is higher than yc Answer: B. If the available energy is not more than the minimum energy needed, the section must pass critical flow. (Key point: Rule for critical depth) 2. For the 0.15 m hump, why is there no critical depth? A. The hump is too low to change the depth B. Available energy 0.913 m exceeds Ec = 0.752 m C. Available energy 0.913 m is below Ec = 0.752 m D. Width is unchanged so nothing can happen Answer: B. E1 - 0.15 = 0.913 m is greater than Ec = 0.752 m, so flow remains subcritical over the hump. (Key point: Hump alone) 3. In part (b), the throat q rises to 1.538 m^2/s. What is the consequence for the minimum energy? A. Ec falls to 0.752 m B. Ec rises to 0.934 m, but E1 = 1.063 m is still larger, so no critical depth C. Ec rises to 1.2 m, so choking occurs D. Ec is unchanged because Q is unchanged Answer: B. Larger q gives a larger yc (0.623 m) and Ec = 0.934 m, still below the available 1.063 m. (Key point: Side-wall contraction alone) 4. Why does critical depth occur when hump and contraction act together? A. Because yc becomes equal to y1 B. Because the hump raises q C. Because the required Ec + dz = 1.084 m exceeds E1 = 1.063 m D. Because the Froude number upstream becomes 1 Answer: C. Energy needed at the throat (0.934 + 0.15) is more than the approach flow has, so critical flow forms. (Key point: Combined effect) 5. In case (c) what happens upstream? A. The depth rises until the specific energy is 1.084 m (about 1.02 m) B. The depth falls to yc C. A hydraulic jump forms at the throat only D. Nothing; the discharge reduces to 1.8 m^3/s Answer: A. The flow cannot pass with E1 = 1.063 m, so upstream energy increases to 1.084 m by backing up. 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