How this instrument works
Manning's equation predicts how fast water moves through an open channel — a ditch, culvert, canal, or river reach — from nothing more than the channel's shape, its slope, and a roughness coefficient describing what the wetted surface is made of. It is empirical, not derived from fluid mechanics first principles: Robert Manning proposed the form in 1889 after comparing several existing discharge formulas against measured river data, and the roughness coefficient n is itself a tabulated number, calibrated by matching the equation to real rivers and lined channels rather than computed from surface texture. Smooth trowelled concrete gets n around 0.013; a channel choked with weeds and cobbles can run past 0.06.
The shape of the formula follows from balancing gravity against friction along the wetted boundary. Flow area A = b·y is width times depth for this rectangular channel; hydraulic radius R_h = A ⁄ (b + 2y) divides that area by the wetted perimeter — the bed plus both submerged walls, but not the open top, since the free surface contributes no friction. A wide, shallow channel has a smaller hydraulic radius relative to its area, because the walls do proportionally more retarding; a deep, narrow one behaves differently again. The R_h^(2⁄3) exponent and the square-root slope term both trace back to Manning fitting the older Chézy resistance formula into a single discharge equation.
The equation assumes steady, uniform flow in a prismatic channel — the same cross-section and slope from end to end, with depth constant along its length. It does not hold near a hydraulic jump, a sudden width change, a culvert entrance, or anywhere the water surface is curving quickly; those situations call for varied-flow methods layered on top of Manning's equation, not this formula alone. It also assumes fully rough turbulent flow: at very shallow depths or very smooth linings, viscous effects the tabulated n values were never calibrated for start to matter, and the formula quietly loses accuracy.
- Set Manning's roughness coefficient n for the channel lining — 0.013 for smooth concrete, 0.022 for a clean earth channel, higher for anything rougher.
- Enter Channel width and Flow depth in metres or feet; together they define the wetted rectangular cross-section.
- Enter Channel slope, m ⁄ m — the bed's drop per metre of channel length, not a percent grade.
- Read Flow area and Hydraulic radius, the two geometric results the discharge formula runs on.
- Read Flow rate (Manning's equation) for the resulting discharge, switchable between m³/s and l/s.
Worked example — a 3 m concrete channel at 1 m depth
A concrete drainage channel is 3 m wide (b = 3 m) and carries water 1 m deep (y = 1 m) on a mild 0.1 percent slope (S = 0.001), with a trowelled-concrete roughness of n = 0.013. Flow area is simply width times depth: A = 3 × 1 = 3.0 m². The wetted perimeter is the bed plus the two submerged side walls, b + 2y = 3 + 2 = 5 m, so the hydraulic radius is R_h = 3.0 ⁄ 5 = 0.6 m.
Manning's equation then gives Q = (1 ⁄ 0.013) × 3.0 × 0.6^(2⁄3) × √0.001, which works out to about 5.191 m³ ⁄ s. That's roughly 5,191 litres every second — enough to fill a 2,500 m³ Olympic-sized pool in a little over eight minutes — moving down a channel most people would walk past without a second look.
Swap in a rougher lining instead — n = 0.025, more like aged or lightly weathered concrete — and, with the geometry unchanged, the discharge falls to about 2.699 m³ ⁄ s, since Q is exactly proportional to 1 ⁄ n. Double the flow depth to 2 m instead and the same channel carries about 13.170 m³ ⁄ s, more than double the original rate, because hydraulic radius also grows, from 0.6 m to about 0.857 m, as the depth increases.
Questions
What does the hydraulic radius actually measure?
It's the flow area divided by the wetted perimeter — the length of channel boundary actually touching the water, not the open top. For this channel, R_h = 3.0 m² ⁄ 5 m = 0.6 m. It isn't the physical radius of anything; the name is a holdover from applying pipe-flow reasoning to open channels. A larger hydraulic radius means less wetted boundary per unit of area, so friction has less surface to act on and the channel carries more flow at the same slope and roughness.
Why does flow rate depend on the square root of the slope?
Because slope is the force driving water downhill against friction, and in Manning's equation — as in the older Chézy formula it's built from — that driving term enters as a square root rather than linearly. Slope matters, but with diminishing returns: quadrupling it only doubles the discharge, all else equal. This channel's 0.1 percent grade (S = 0.001) is typical of a storm drain or irrigation lateral; steepen it to 0.4 percent, four times as steep, and the flow rate exactly doubles, to about 10.38 m³ ⁄ s.
How is Manning's n chosen for a real channel?
From tables built by matching the formula to real, measured channels, not computed from a formula of its own. Smooth trowelled concrete, used in this example, sits near n = 0.013; riveted steel or unfinished concrete runs around 0.015 to 0.017; a clean, straight earth channel is roughly 0.022; one choked with vegetation or debris can exceed 0.06. Picking n correctly is the biggest source of error in a Manning's-equation estimate — since Q is exactly proportional to 1 ⁄ n, overestimating n by 20 percent does not merely shift the answer by 20 percent, it understates the true flow rate by about 17 percent, because the two quantities are reciprocals, not equals.
Does this formula work for a river with an irregular cross-section?
Not directly — this instrument models a rectangular channel, where flow area is simply width times depth. A natural river cross-section is usually split into subsections, such as a main channel plus overbank areas, each with its own area, wetted perimeter, and sometimes its own n, and Manning's equation is applied to each before the discharges are summed. The physics stays identical, friction against a wetted boundary driven by slope, but the geometry bookkeeping changes for a non-rectangular shape.
What happens if I enter zero for Manning's n?
The calculator rejects it: Manning's roughness coefficient must be greater than zero, because it sits in the denominator of the flow-rate formula. A channel with n approaching zero would carry an unbounded amount of water for any slope, which describes no real surface — even glass-smooth concrete has measurable roughness, with n around 0.011. If you're unsure what to enter, 0.013 for smooth concrete or 0.030 for a natural earth channel are reasonable starting points.
Is channel slope entered as a percent or a ratio?
As a ratio: metres of bed drop per metre of channel length, not a percentage and not degrees. A slope of 0.001 means the bed falls 1 millimetre for every metre travelled, equivalent to 0.1 percent. If your survey data is in percent grade, divide by 100 before entering it here — a 0.5 percent grade becomes S = 0.005, not 0.5.