---
title: "Manning's equation, terms and slope S"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:mannings-equation-terms-and-slope"
source: https://www.bctengineer.com/q/semester_4/hydraulics/mannings-equation-terms-and-slope
site: BCT Engineering
---

# Manning's equation, terms and slope S

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2078 Chaitra, 1+1+1+1+1 marks
- Video: [Manning's Equation](https://www.youtube.com/watch?v=JQ0FIluECP8) — HydrologyVideos
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/mannings-equation-terms-and-slope
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

Describe the Manning's equation and the various terms that make it up. In particular define the slope S, used in the Manning's equation and show how it relates to the energy diagram. Explain also why the uniform flow assumption is usually made for most application of the Manning's equation.

## Short answer

Manning equation: V=(1/n)R^{2/3}S^{1/2}, where n is roughness, R hydraulic radius, S energy slope. S is the slope of the energy grade line; in uniform flow S equals bed slope S_0 because depth and velocity are constant. Uniform flow is usually assumed because channels are prismatic, bed slope is directly measurable, and the formula was derived for such conditions.

## Answer

#### Manning's equation
Manning's equation is an empirical formula for mean velocity in open channels, for uniform (turbulent, rough) flow:
$$
V=\frac{1}{n}R^{2/3}S^{1/2}\qquad\Rightarrow\qquad Q=\frac{1}{n}AR^{2/3}S^{1/2}\quad(\text{SI units})
$$
| Term | Meaning | Unit |
|---|---|---|
| $V$ | Mean velocity of flow | m/s |
| $Q$ | Discharge | m³/s |
| $n$ | Manning's roughness coefficient (e.g. 0.012 for smooth concrete, 0.03 for earth canal, 0.035-0.05 for natural rivers) | s/m$^{1/3}$ |
| $A$ | Flow area | m² |
| $P$ | Wetted perimeter | m |
| $R=A/P$ | Hydraulic radius | m |
| $S$ | Slope of the energy grade line | m/m |

It is related to Chezy's formula $V=C\sqrt{RS}$ by $C=R^{1/6}/n$.

#### The slope $S$ and the energy diagram
$S$ is the **energy slope** $S_f$, the loss of total energy head per unit length:
$$
S=S_f=-\frac{dH}{dx},\qquad H=z+y+\frac{\alpha V^2}{2g}
$$
On the energy diagram it is the slope of the energy grade line (EGL), measured relative to the horizontal. It is not necessarily the bed slope.

```text
 EGL -----\
 WSL  ------\  \  S = slope of EGL (S_f)
 Bed  -------\---\--
```
For uniform flow, $S_f=S_w=S_0$, since depth and velocity do not change, so the EGL, water surface and bed are parallel. Then $S$ may be taken as the bed slope. For non-uniform flow, $S_f$ is used at each section (the local friction slope), found from the Manning equation: $S_f=\dfrac{n^2V^2}{R^{4/3}}$.

#### Why uniform flow is assumed in most applications
1. Most canals, flumes and drains are prismatic with constant slope and roughness, and long reaches reach equilibrium (normal depth).
2. The bed slope can be measured, but the energy slope cannot be measured directly. With uniform flow, $S_f=S_0$ so only the bed slope is needed.
3. In gradually varied flow, depth changes slowly, so the uniform-flow formula is valid for local friction loss at each section.
4. It gives a simple design method: for a given $Q$, $n$ and $S_0$, find the normal depth.
5. Manning's equation was itself derived from data of (nearly) uniform flow.

Hence, Manning's equation gives acceptable accuracy when the flow is turbulent, steady and nearly uniform.

## Check yourself

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

1. In Manning's equation, what is S in general?
   A. Bed slope always
   B. Water surface slope
   C. Energy slope Sf, the slope of the EGL
   D. Slope of the banks
   Answer: C. S is the energy slope, S = Sf = -dH/dx; it equals the bed slope only in uniform flow. (Key point: Slope S)

2. How is Chezy's C related to Manning's n?
   A. C = n R^(1/6)
   B. C = R^(1/6)/n
   C. C = n/R^(1/6)
   D. C = R^(2/3)/n
   Answer: B. Equating Chezy and Manning velocities gives C = R^(1/6)/n. (Key point: Chezy relation)

3. Why is uniform flow usually assumed when applying Manning's equation?
   A. Because energy slope cannot be measured directly, but with uniform flow Sf = S0, which is measurable
   B. Because natural rivers are always uniform
   C. Because Manning's equation holds only for laminar flow
   D. Because n is zero in uniform flow
   Answer: A. Under uniform flow Sf = S0, so the measurable bed slope can be used. (Key point: Why uniform flow)

4. For gradually varied flow, how is Manning's equation used?
   A. It cannot be used
   B. Only the bed slope is used
   C. Sf = n^2 V^2 / R^(4/3) is used as the local friction slope
   D. Sf is set to zero
   Answer: C. Depth changes slowly, so the uniform-flow formula gives the local friction slope. (Key point: Non-uniform flow use)

## Related in Hydraulics

- [Steady uniform, non-uniform and spatially varied flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aflow-types-uniform-nonuniform-svf-examples) — Hydraulics, chapter 5, asked 4×
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