---
title: "Hump check using given specific energy curve"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:hump-adequate-specific-energy-curve-given"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hump-adequate-specific-energy-curve-given
site: BCT Engineering
---

# Hump check using given specific energy curve

Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow

- Asked in: 2080 Chaitra, 2+6 marks
- Video: [Specific Energy and Specific Energy Curve - Flow through Open Channels - Applied Hydraulics](https://www.youtube.com/watch?v=oiahLm1_se4) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hump-adequate-specific-energy-curve-given
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A rectangular channel of width 2 m is designed for irrigating land downstream. The channel is nearly horizontal, except that there is a smooth, short hump of 0.5 m in the channel bottom. The energy loss due to the hump is negligible. The point A in figure represents section A just upstream of hump. From the provided specific energy diagram below for a constant discharge of 6 m³/s, i) Determine whether the flow has adequate specific energy to sustain 6 m³/s through the hump with explanation. ii) Based on your result of i), what will be the flow profile just upstream and downstream of the hump to sustain the required discharge 6 m³/s? Show final results on the curve. [Figure: specific energy diagram of y (m) against E (m); point A on the subcritical limb at y = 1.63 m and point A' on the supercritical limb at y = 0.62 m, both at E = 1.8 m]

## Short answer

(i) No, available specific energy over the 0.5 m hump is 1.30 m, less than the required minimum 1.46 m. (ii) Upstream depth rises to 1.82 m; depth over the hump is critical at 0.97 m; downstream depth becomes supercritical at 0.58 m.

## Answer

#### Given data
$b=2$ m, $Q=6$ m³/s, so $q=3$ m²/s. Hump height $\Delta z=0.5$ m, no energy loss.

Point A (upstream, subcritical): $y_A=1.63$ m, $E_A=1.63+\dfrac{3^2}{19.62\times1.63^2}=1.8026\approx1.80$ m. Its alternate depth is A' ($y=0.62$ m, supercritical), with the same $E=1.80$ m.

#### Critical state on the curve
$$
y_c=\sqrt[3]{\frac{q^2}{g}}=\sqrt[3]{\frac{9}{9.81}}=0.972\ \text{m},\qquad E_{min}=\frac32y_c=1.458\ \text{m}
$$

#### (i) Is the specific energy adequate?
Energy available over the hump:
$$
E_{hump}=E_A-\Delta z=1.80-0.50=1.30\ \text{m}
$$
Since $E_{hump}=1.30<E_{min}=1.458$ m, **the flow does not have adequate specific energy** to pass 6 m³/s over the 0.5 m hump. (The maximum hump height the present upstream state allows is $E_A-E_{min}=1.8026-1.4575=0.345$ m, which is less than 0.5 m.)

#### (ii) What happens (profile upstream and downstream)
The hump acts as a control. Upstream, the water surface rises (backwater) until the upstream specific energy is just enough for critical flow over the hump:
$$
E_1'=E_{min}+\Delta z=1.4575+0.50=1.9575\ \text{m}
$$
Upstream depth (subcritical root of $y+\dfrac{9}{19.62y^2}=1.9575$): $y_1'=1.819$ m.

- **On the hump:** critical flow, $y=y_c=0.972$ m.
- **Downstream of the hump:** the flow passes through critical depth and becomes supercritical. After the hump the bed returns to its original level and the specific energy is again $1.9575$ m, so the supercritical depth is $y_3=0.576$ m. This may be followed by a hydraulic jump further downstream if the tailwater is deep enough.

```text
  y (m)
  1.82 |            * A1 (upstream after rise, E=1.96)
  1.63 |        * A (original upstream, E=1.80)
       |
  0.97 |    * critical on hump (E=1.46)
  0.62 |        * A' (E=1.80)
  0.58 |            * downstream supercritical (E=1.96)
       +----+----+----+----+--> E (m)
          1.3  1.46 1.80 1.96
```
Final results on the curve: A moves up to 1.82 m (E = 1.96 m); the hump point is at the nose of the curve (0.97 m, 1.46 m); the downstream supercritical point is at 0.58 m (E = 1.96 m).

**Answer: (i) No, since $E=1.30$ m $<E_{min}=1.46$ m; (ii) the upstream depth rises to 1.82 m, the depth over the hump is critical (0.97 m), and the flow downstream is supercritical at 0.58 m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is adequacy of specific energy over the hump judged?
   A. Compare E_A - Δz with Emin for q = 3 m^2/s
   B. Compare the hump height with the depth
   C. Compare y_A with 1 m
   D. Compare A and A' depths
   Answer: A. Energy over the hump is 1.80 - 0.50 = 1.30 m, which must be at least Emin = 1.458 m. (Key point: Adequacy test)

2. Since 1.30 m < Emin = 1.458 m, the flow
   A. is adequate and critical
   B. is adequate and subcritical
   C. becomes supercritical upstream
   D. does not have adequate specific energy to pass 6 m^3/s over the 0.5 m hump
   Answer: D. The maximum allowable hump is about 0.345 m, which is less than 0.5 m. (Key point: Conclusion for (i))

3. To sustain 6 m^3/s, what happens upstream of the hump?
   A. The depth falls to the alternate depth 0.62 m
   B. The depth is unchanged
   C. Water backs up until E1' = Emin + Δz = 1.9575 m (depth 1.82 m)
   D. The discharge decreases
   Answer: C. The hump acts as a control, so the upstream surface rises until energy is Emin + Δz. (Key point: Upstream response)

4. On and just downstream of the hump the flow is
   A. subcritical both on and after
   B. critical on the hump (0.972 m), then supercritical (0.576 m) downstream
   C. supercritical on the hump, critical downstream
   D. critical on and after the hump
   Answer: B. Flow passes through critical depth at the hump crest and continues on the supercritical limb with E = 1.9575 m. (Key point: Flow on and after the hump)

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