---
title: "Slope break 0.01 to 0.0064: profile and length"
subject: "Hydraulics"
chapter: 8
key: "semester_4/hydraulics:slope-break-125-m3s-nature-and-length"
source: https://www.bctengineer.com/q/semester_4/hydraulics/slope-break-125-m3s-nature-and-length
site: BCT Engineering
---

# Slope break 0.01 to 0.0064: profile and length

Hydraulics · Chapter 8: Non-uniform gradually varied flow (GVF)

- Asked in: 2069 Poush, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/slope-break-125-m3s-nature-and-length
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-8

## Question

A rectangular channel 10 m wide is laid with a break in its bottom slope from 0.01 to 0.0064. If it carries 125 m³/s, determine the nature of the surface profile and compute its length. Take n = 0.015.

## Short answer

The downstream profile is S3 (supercritical, depth rising). Computed by the direct step method, its length to 99% of the new normal depth is about 370 m.

## Answer

**Given:** $b = 10$ m, $Q = 125\ \text{m}^3/\text{s}$ ($q = 12.5\ \text{m}^2/\text{s}$), $n = 0.015$, slope changes from $S_{01} = 0.01$ to $S_{02} = 0.0064$.

**Depths**

$$
y_c = \left(\frac{12.5^2}{9.81}\right)^{1/3} = 2.516\ \text{m}
$$

By Manning's equation (trial, $R = A/P$):

| Reach | $S_0$ | $y_n$ (m) | Class |
|---|---|---|---|
| 1 | 0.0100 | 1.633 | steep |
| 2 | 0.0064 | 1.896 | steep |

Both slopes are steep, and the flow is supercritical throughout. The flow arrives at the break with $y = 1.633$ m, which is below the new normal depth 1.896 m, so on the downstream reach $y < y_{n2} < y_c$. The profile is **S3** (rising depth, asymptotic to $y_{n2}$). No jump occurs.

**Length of S3 profile.** The curve reaches $y_{n2}$ only at infinity, so its length is computed up to $0.99\,y_{n2} = 1.877$ m, using the direct step method with 4 equal steps and $S_0 = 0.0064$.

$$
E = y + \frac{V^2}{2g},\quad S_f = \frac{n^2V^2}{R^{4/3}},\quad \Delta x = \frac{E_2 - E_1}{S_0 - \bar S_f}
$$

| $y$ (m) | $R$ (m) | $V$ (m/s) | $E$ (m) | $S_f$ | $\bar S_f$ | $\Delta x$ (m) |
|---|---|---|---|---|---|---|
| 1.6326 | 1.231 | 7.656 | 4.6204 | 0.01000 | | |
| 1.6937 | 1.265 | 7.380 | 4.4699 | 0.00896 | 0.00948 | 48.9 |
| 1.7547 | 1.299 | 7.124 | 4.3412 | 0.00806 | 0.00851 | 61.1 |
| 1.8158 | 1.332 | 6.884 | 4.2312 | 0.00728 | 0.00767 | 86.9 |
| 1.8768 | 1.365 | 6.660 | 4.1377 | 0.00659 | 0.00693 | 175.0 |

$$
L = 48.9 + 61.1 + 86.9 + 175.0 \approx 372\ \text{m}
$$

A finer calculation gives about 407 m.

**Answer: S3 profile on the downstream slope; length about 370 m (to within 1% of the new normal depth).**

## Check yourself

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

1. With yc = 2.516 m, what are both reaches?
   A. Mild
   B. Steep, since yn (1.633, 1.896) are below yc
   C. Critical
   D. One mild, one steep
   Answer: B. Both normal depths are below critical depth, so both are steep. (Key point: Slope classes)

2. Why is the downstream profile S3?
   A. The flow arrives at 1.633 m, below the new yn2 = 1.896 m, which is below yc
   B. Depth exceeds yc
   C. Depth lies between yn and yc
   D. The slope is mild
   Answer: A. Depth below yn on a steep slope is S3, rising toward yn2. (Key point: Profile)

3. Is a hydraulic jump formed?
   A. Yes, at the break
   B. Yes, downstream
   C. No, flow stays supercritical throughout
   D. Yes, upstream
   Answer: C. Flow remains supercritical (steep to steep), so no jump is needed. (Key point: Jump)

4. Why is the length computed to 0.99 yn2?
   A. The profile reaches yn2 only at infinity
   B. To avoid critical depth
   C. Because yn2 is unknown
   D. Because depth falls
   Answer: A. S3 is asymptotic to yn2, so 0.99 yn2 = 1.877 m gives a finite length of about 370 m. (Key point: End depth)

## Related in Hydraulics

- [Slope change 1/120 to 1/1500, jump and profile length](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awide-channel-slope-break-jump-two-steps) — Hydraulics, chapter 8
- [Bresse's method for 20 m wide channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Abresse-method-20m-channel-65-m3s) — Hydraulics, chapter 8
- [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
- [Force on sluice gate, 10 m wide channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asluice-gate-force-10m-channel-7-m3s) — Hydraulics, chapter 9
- [Slope break and distance to depth 0.4 m](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awide-channel-slope-break-direct-step-3-steps) — Hydraulics, chapter 8
- [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

## Related in other subjects

- [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
- [Economical rectangular condition and trapezoidal bed slope](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atrapezoidal-channel-bed-slope) — Hydro Power, chapter 1
- [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
- [Length of bottom intake rack](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aintake-bottom-rack-length-8cumec) — Hydropower Engineering, chapter 5