Governing Formula
The Manning equation predicts the mean velocity of uniform open-channel flow from the channel geometry, bed slope, and a roughness coefficient n. It is the standard design tool for stormwater drains, irrigation channels, culverts, and sewers flowing partly full.
v = (1/n) × R_h^(2/3) × S^(1/2) Q = A·v Where:
-
Q= Channel discharge / capacity [m³/s, L/s] -
A= Wetted cross-sectional area [m²] -
R_h= Hydraulic radius (A / wetted perimeter) [m] -
S= Channel bed slope (energy gradient ≈ bed slope, uniform flow) [m/m] -
n= Manning's roughness coefficient [—] -
v= Mean channel velocity [m/s]
Derived Equations:
R_h = A / P v = (1/n)·R_h^(2/3)·S^(1/2) Q = A·v How the Calculation Works
For the active channel shape, the engine computes the wetted area A and wetted perimeter P, then the hydraulic radius R_h = A/P:
Rectangle: A = b·y, P = b + 2y
Trapezoid: A = y·(b + z·y), P = b + 2y·√(1+z²)
For a circular section the engine solves the partly-full geometry (via the central-angle formulation) and returns the wetted area and perimeter for the given fill depth. The mean velocity follows from the Manning relation and the discharge from Q = A·v.
Worked Engineering Example
Design Scenario: Rectangular Drain, 1.0 m Wide × 0.5 m Deep, Slope 0.001, n = 0.013
- Wetted geometry:
A = 1.0 × 0.5 = 0.50 m²
P = 1.0 + 2 × 0.5 = 2.00 m
R_h = 0.50 / 2.00 = 0.25 m - Mean velocity (Manning):
v = (1/0.013) × 0.25^(2/3) × √0.001 = 0.965 m/s - Discharge:
Q = 0.965 × 0.50 = 0.483 m³/s
Q = 482.7 L/s = 1,738 m³/h - Check velocity:
0.965 m/s is within the typical 0.6–1.8 m/s non-scouring range for lined drains.
Engineering Notes & Design Benchmarks
| Channel Lining | Manning n | Typical Max Velocity |
|---|---|---|
| Concrete (trowelled) | 0.013 | 1.8 – 2.5 m/s |
| Concrete (form, uncoated) | 0.014 – 0.017 | 1.5 – 2.0 m/s |
| Earth, uniform clean | 0.022 – 0.025 | 0.6 – 1.2 m/s (erosion-limited) |
| Corrugated metal culvert | 0.021 – 0.025 | 1.2 – 1.8 m/s |
| Vegetated lined | 0.030 – 0.050 | 0.3 – 0.9 m/s |
The trapezoid shape preselects the standard z = 1.5 (34°) side slope. Note that the most hydraulically efficient rectangle is 2:1 (depth = half width), while a trapezoid carrying the same area can be more efficient still — the section you enter should reflect constructability, not only hydraulics.
Assumptions & Limitations
- Uniform (normal-depth) flow with bed slope equal to the energy gradient — backwater and gradually varied flow are out of scope.
- Manning’s n is applied to water at usual temperatures; temperature has only a mild effect on this empirical relation.
- Circular sections are solved partly full; the wetted geometry and hydraulic radius should be verified against a conduit hydraulics chart for extreme fills.
- Vegetation growth, sediment, and bends can raise the effective roughness above the tabulated clean-ditch values.
Frequently Asked Questions
What is the most hydraulically efficient channel shape?
The semi-circular section is theoretically the most efficient because it maximises area for a given wetted perimeter. Among open-top rectangles the 2:1 (depth = half width) section is best; the trapezoid optimum is a half-hexagon with side slopes of 60° from horizontal.
How do I choose a Manning n for design?
Start from published tables (e.g. Chow’s Open-Channel Hydraulics or the USGS “Barnes” table) that match your lining, then apply a conservative lift for aging, sediment, or vegetation. Never size an earth channel with a pristine first-day value.
Can this predict flooding or backwater conditions?
No — Manning describes uniform flow at normal depth. Pressure-flow culverts, road-crossing backwater, and rapid transitions must be analysed with gradually varied flow or a full hydraulic model.
Engineering Disclaimer
Engineering Note: This calculator provides normal-depth, uniform-flow estimates for sizing and education. Real designs must check scour, freeboard, supercritical transitions, and backwater, and prefer a routed hydraulic model for sensitive catchments.
Technical References
- Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, 1959.
- Barnes, H. H., Roughness Characteristics of Natural Channels, USGS Water-Supply Paper 1849, 1967.
- US Bureau of Reclamation, Design of Small Dams, 3rd ed., 1987.
- Akan, A. O. & Houghtalen, R. J., Urban Hydrology, Hydraulics, and Stormwater Quality, Wiley.