---
title: "Flow over a 0.2 m step rise and drop"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:step-rise-and-drop-downstream-depth"
source: https://www.bctengineer.com/q/semester_4/hydraulics/step-rise-and-drop-downstream-depth
site: BCT Engineering
---

# Flow over a 0.2 m step rise and drop

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

- Asked in: 2076 Baisakh, 4+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/step-rise-and-drop-downstream-depth
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

The flow depth and the flow velocity upstream of a 0.2-m sudden step rise in the bottom of a 5-m wide rectangular channel are 5 m and 4 m/s respectively. Assuming there are no losses in the transition, determine: i) The flow depth downstream of the step and the change in the water level; ii) The flow depth and the water level downstream of the step if the channel bottom has a 0.2-m drop instead of the rise, as in (i).

## Short answer

Given E1 = 5.8155 m, q = 20 m²/s, yc = 3.442 m. (i) 0.2 m rise: E2 = 5.6155 m, y2 = 4.69 m; water level drops 0.11 m. (ii) 0.2 m drop: E2 = 6.0155 m, y2 = 5.29 m; water level rises 0.09 m.

## Answer

#### Given data
Rectangular channel $b=5$ m, upstream $y_1=5$ m, $V_1=4$ m/s, no losses.
$$
q=V_1y_1=4\times5=20\ \text{m}^2/\text{s},\quad Q=100\ \text{m}^3/\text{s}
$$
$$
E_1=y_1+\frac{V_1^2}{2g}=5+\frac{16}{19.62}=5.8155\ \text{m}
$$
$$
Fr_1=\frac{4}{\sqrt{9.81\times5}}=0.571\ (\text{subcritical}),\quad y_c=\sqrt[3]{\frac{20^2}{9.81}}=3.442\ \text{m},\quad E_{min}=5.163\ \text{m}
$$

#### (i) Step rise of 0.2 m
Energy equation (datum at the upstream bed): $E_1=\Delta z+E_2$
$$
E_2=5.8155-0.2=5.6155\ \text{m}>E_{min}
$$
so flow is possible, and since the approach flow is subcritical, the flow stays subcritical:
$$
y_2+\frac{20^2}{19.62\,y_2^2}=y_2+\frac{20.387}{y_2^2}=5.6155\;\Rightarrow\;y_2=4.688\ \text{m}
$$
Water surface elevation above the upstream bed $=0.2+4.688=4.888$ m.
Change in water level $=4.888-5.000=-0.112$ m, i.e. the **water surface falls by 0.112 m**.

#### (ii) Step drop of 0.2 m
$$
E_2=E_1+0.2=6.0155\ \text{m}
$$
Subcritical root: $y_2+\dfrac{20.387}{y_2^2}=6.0155\Rightarrow y_2=5.286\ \text{m}$

Water surface elevation above the upstream bed $=5.286-0.2=5.086$ m; change in water level $=5.086-5=+0.086$ m, i.e. the **water surface rises by 0.086 m**.

```text
 (i) rise:   y1=5 ____        (ii) drop:  y1=5 ____
                  \___ y2=4.69               ____/ y2=5.29
         step 0.2 |___|                    |___| drop 0.2
```
**Answer: (i) $y_2=4.69$ m, water level falls by 0.11 m; (ii) $y_2=5.29$ m, water level rises by 0.09 m.**

(Subcritical flow: the surface drops over a rise and goes up over a drop, the opposite of the bed movement.)

## Check yourself

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

1. For a step rise of Δz with no loss, the specific energy downstream is
   A. E2 = E1 + Δz
   B. E2 = E1 - Δz
   C. E2 = E1
   D. E2 = E1 x Δz
   Answer: B. Total energy is conserved; with the bed raised by Δz, E1 = Δz + E2. (Key point: Energy equation at a step)

2. Why is the subcritical root of y + q^2/(2g y^2) = E2 used downstream?
   A. The approach flow is subcritical (Fr1 = 0.57), so it stays on the same limb unless choked
   B. Supercritical roots are mathematically invalid
   C. The step always makes flow subcritical
   D. Because E2 < Emin
   Answer: A. E2 = 5.62 m exceeds Emin = 5.16 m, so no choking and the flow stays on the subcritical limb. (Key point: Choice of root)

3. For subcritical flow over a 0.2 m step rise, the water surface
   A. rises by 0.2 m
   B. stays at the same level
   C. falls, by 0.112 m here
   D. rises by more than 0.2 m
   Answer: C. Depth falls from 5 to 4.688 m, so surface = 0.2 + 4.688 = 4.888 m, a fall of 0.112 m. (Key point: Water surface response to a rise)

4. For a 0.2 m step drop in subcritical flow, the water surface
   A. falls
   B. rises slightly (by 0.086 m here)
   C. stays unchanged
   D. becomes critical
   Answer: B. E2 = E1 + 0.2 gives y2 = 5.286 m, so surface = 5.086 m, a rise of 0.086 m; the surface moves opposite to the bed. (Key point: Step drop behaviour)

## Related in Hydraulics

- [Contraction and bed rise, 3.5 m channel (1 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1m-depth) — Hydraulics, chapter 7, asked 2×
- [Direct step, 16 m wide channel, 40 m³/s](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Agvf-profile-40m3s-16m-wide-direct-step) — Hydraulics, chapter 8
- [Trapezoidal to rectangular transition with bed lowering](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Atrapezoid-to-rectangular-transition-bed-lowering) — Hydraulics, chapter 7
- [Maximum obstruction height without upstream rise](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aobstruction-max-height-no-upstream-rise) — Hydraulics, chapter 7
- [Contraction and bed rise, 3.5 m channel (1.2 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1-2m-depth) — Hydraulics, chapter 7
- [Slope break: jump depth and jump length by step](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aslope-break-jump-downstream-depth-and-length) — Hydraulics, chapter 9

## Related in other subjects

- [Vertical drop in 10 m wide canal, Q 20](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Avertical-drop-20cumec-10m-canal) — Irrigation and Drainage Engineering, chapter 7
- [Flood discharge in 10-m rectangular channel](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aslope-area-rectangular-10m) — Engineering Hydrology, chapter 4
- [Uniform flow in rectangular channel (Chezy)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Arectangular-channel-chezy-flow) — Hydro Power, chapter 1
- [Reservoir level change, water balance, 1600 ha](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Areservoir-level-change-1600ha) — Engineering Hydrology, chapter 1, asked 2×