Open channel flow
Manning’s equation is the workhorse of open-channel hydraulics: discharge from the area, the hydraulic radius, the bed slope and a roughness coefficient. The geometry is exact, the roughness is an estimate, and the Froude number tells you whether the flow is tranquil or about to jump.
| Quantity | Value | Note |
|---|---|---|
| Shape | trapezoidal | 3.00 m bottom width |
| Flow depth | 1.200 m | |
| Flow area | 5.7600 m² | |
| Wetted perimeter | 7.327 m | |
| Hydraulic radius | 0.7862 m | A / P |
| Top width | 6.600 m | |
| Hydraulic depth | 0.8727 m | A / T |
| Bed slope | 0.5000 m/km | 0.0500% |
| Manning n | 0.025 | Chow / FHWA values |
| Discharge | 4.3885 m³/s | Q = (1/n) A R^⅔ S^½ |
| Velocity | 0.762 m/s | Q / A |
| Froude number | 0.260 | V / √(gD) |
| Regime | subcritical | tranquil |
| Critical depth | 0.547 m | where Fr = 1 |
| Normal depth | 1.200 m | for the discharge shown |
| Critical slope | 0.000 | the slope that holds the flow at critical depth |
Normal depth is the depth a long uniform channel settles at for a given discharge; critical depth is where the specific energy is minimum. When normal depth is below critical the channel is steep and the flow is supercritical.
Q = (1/n)·A·R^⅔·S^½ · SI; use 1.486 in US customary units.
R = A/P · Area over wetted perimeter.
Fr = V/√(gD) · D is the hydraulic depth, A/T.
Q²T/(gA³) = 1 · Minimum specific energy.
A clear path from inputs to results.
Manning's equation is the workhorse of open-channel hydraulics: discharge from the area, the hydraulic radius, the bed slope and a roughness coefficient. The geometry is exact, the roughness is an estimate, and the Froude number tells you whether the flow is tranquil or about to jump.
How to use
- Choose the section shape and enter its dimensions.
- Set the flow depth, the bed slope and the Manning roughness.
- Read the discharge, velocity and Froude number.
- Use the critical depth to see whether the channel is steep or mild at this flow.
The formula
Reference: Manning's equation; roughness values from Chow, Open-Channel Hydraulics, and FHWA HDS-3.
Worked example
Trapezoidal channel, 3 m bottom, 1.2 m deep
- Bottom width 3 m, side slopes 1.5H:1V.
- Flow depth 1.2 m, bed slope 0.5 m per 1000 m.
- Manning roughness 0.025 (clean earth).
- 1
The flow area is (3 + 1.5 × 1.2) × 1.2 = 5.760 m².
- 2
The wetted perimeter is 3 + 2 × 1.2 × √(1 + 1.5²) = 7.327 m.
- 3
The hydraulic radius is 5.760 / 7.327 = 0.786 m.
- 4
Manning gives Q = (1/0.025) × 5.760 × 0.786^(2/3) × 0.0005^0.5 = 4.388 m³/s.
- 5
Velocity is 4.388 / 5.760 = 0.762 m/s.
- 6
The Froude number is 0.260, so the flow is well subcritical — a hydraulic jump is not possible here.
- 7
Critical depth for this discharge is 0.547 m, so the channel is mild at this flow.
4.388 m³/s · 0.762 m/s · Fr 0.260 · critical depth 0.547 m.
Interpreting the result
The Froude number is the number that decides how the channel behaves: below one the flow is tranquil and downstream conditions propagate upstream, above one it is rapid and a jump is possible. The comparison between normal depth and critical depth tells you whether the channel is steep or mild, which is the first thing a hydraulic designer establishes.
Assumptions & limitations
- Uniform, steady, turbulent flow in a prismatic channel.
- Manning's n is constant over the depth; in reality it varies with vegetation and stage.
- The energy slope equals the bed slope, which is the normal-depth assumption.
- No backwater, no bridge or culvert constriction, and no sediment transport.
- The circular section is computed part-full, with the angle from the water surface geometry.
Frequently asked questions
What Manning n should I use?
0.011 to 0.013 for concrete, 0.025 for clean earth, 0.035 for earth with some grass, and 0.05 or more for a natural stream. The coefficient is the dominant uncertainty in the calculation — a natural channel is quoted to two significant figures at best.
Why does the Froude number matter?
It is the ratio of inertial to gravitational forces, and it decides whether the flow is subcritical, critical or supercritical. Subcritical flow is controlled from downstream; supercritical flow is controlled from upstream, and the transition between them is a hydraulic jump.
Can Manning be used for a steep channel?
It can, but the assumption that the energy slope equals the bed slope becomes poor when the flow is accelerating, and air entrainment changes the effective density. For chutes and spillways a step or energy method is preferred.
What is normal depth?
The depth a long uniform channel settles at for a given discharge, where the energy slope equals the bed slope. It is what you solve for when you know the flow and the geometry rather than the depth.
Why does a circular pipe's capacity peak before it is full?
Because the wetted perimeter rises faster than the area as the pipe approaches full, so the hydraulic radius falls. Maximum discharge occurs at about 94% of the diameter, and maximum velocity at about 81%.
Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.