What it does
Manning’s equation is the everyday formula for water flowing steadily down a ditch, canal, river reach, culvert or part-full sewer:
Q = (k / n) × A × R^(2/3) × √S
where A is the flow area, R the hydraulic radius (area ÷ wetted perimeter), S the bed slope, n the roughness and k = 1 in SI units or 1.486 in US units. This tool solves it either way: the discharge for a known depth, or the normal depth that carries a given discharge. It also gives the velocity, the Froude number and flow regime, the critical depth, specific energy and the average bed shear stress, draws the cross-section and a rating curve.
The second card handles weirs, the simple structures used to measure flow in channels and streams: give the head over the crest to get the discharge, or the discharge to find the head.
How to use it
- Choose SI (metres, m³/s) or US (feet, ft³/s).
- Pick the shape and its sizes. Side slope z is horizontal distance per 1 vertical (2 means 2 : 1). For a pipe give the diameter.
- Give the bed slope as a fraction (0.001 is 1 in 1,000, or 1 m drop per km) and Manning’s n. The typical values in the list are rounded starting points: clean concrete about 0.013, a clean earth channel about 0.022, a natural stream 0.035 or more.
- Choose whether to find the discharge or the normal depth.
- For a weir, pick the type, size and coefficient, and type the head or the discharge.
What the results mean
- Froude number below 1 is calm (subcritical) flow, typical of rivers and gentle channels; above 1 is fast, shallow (supercritical) flow, as on spillways. Near 1 the surface is wavy and unstable.
- Critical depth is where the flow has the least specific energy for its discharge. If the normal depth is above it, the slope is “mild”; below it, “steep”.
- Part-full pipes: a circular pipe carries the most at about 94 % full, a little more than when flowing full, because the last few percent adds lots of wetted perimeter and little area.
- Bed shear stress helps check whether a lining or soil will scour.
Worked example
A made-up earth canal: trapezoid with a 2 m bottom and 2 : 1 side slopes, n = 0.025, slope 0.0005, flowing 1.2 m deep. Area = 2 × 1.2 + 2 × 1.2² = 5.28 m², wetted perimeter = 2 + 2 × 1.2 × √5 = 7.37 m, R = 0.717 m. Manning gives Q = 3.78 m³/s at 0.716 m/s. The Froude number is 0.26 (subcritical) and the critical depth 0.584 m, so the slope is mild. A sharp-crested rectangular weir 2 m long with two end contractions and Cd = 0.62 passes 0.889 m³/s at a 0.4 m head.
FAQ
Is the flow really uniform? Manning assumes steady, uniform flow, with depth parallel to the bed. Near bridges, bends, gates and changes in slope the depth differs; that needs a backwater calculation.
Can I use it for a full pipe under pressure? No. A pressurised pipe needs a pipe friction formula such as Darcy-Weisbach or Hazen-Williams.
Can I design a culvert or sewer with it? Use it for a first estimate and a sense check. Real designs need inlet and outlet losses, backwater, local rules and a check by a qualified engineer with a full hydraulic model.
Where the numbers come from
- Manning's equation, critical flow, specific energy and bed shear stress: standard open-channel hydraulics, for example Chow, V. T. (1959), Open-Channel Hydraulics, and Chanson, H. (2004), The Hydraulics of Open Channel Flow (equations only).
- Weir equations: the sharp-crested rectangular weir with Francis's (1883) end contraction, the triangular (V-notch) weir, the broad-crested weir with critical depth on the crest, and the Cipolletti weir, as given in hydraulics textbooks and the US Bureau of Reclamation Water Measurement Manual (public domain).
- Typical Manning's n values are rounded, general figures chosen by fibub for a first estimate, not copied from any table.
- Checked against an independent SciPy calculation written for the tests. The example channel and weir are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-08.