Manning's equation, first published in its modern form by Robert Manning (1891) and refined into engineering practice through the early 20th century, is the workhorse formula for steady uniform flow in open channels. It expresses the mean velocity as a power-law function of hydraulic radius and bed slope, with all geometric, fluid, and surface effects rolled into a single empirical roughness coefficient n.
Why two formulas — and why 1.486?
In SI metric units, the equation is
$$ v = \frac{1}{n}\, R^{2/3}\, S^{1/2} \qquad\Longrightarrow\qquad Q = \frac{1}{n}\, A\, R^{2/3}\, S^{1/2} $$
with v in m/s, R in m, A in m², Q in m³/s.
In US customary units the same formula needs an additional factor of 1.486:
$$ Q = \frac{1.486}{n}\, A\, R^{2/3}\, S^{1/2} $$
with v in ft/s, R in ft, A in ft², Q in ft³/s (cfs). The factor arises because $1\text{ m}^{1/3} = 3.2808^{1/3}\text{ ft}^{1/3} = 1.486\text{ ft}^{1/3}$, and Manning's n is treated as the same numerical value in both systems even though strictly it has units of $\text{s}/\text{m}^{1/3}$. This n-as-dimensionless convention is universal in practice and explains why the same n table (Chow, USGS, USDA) works for both unit systems — the factor 1.486 absorbs the dimensional mismatch.
Choosing Manning's n
The roughness coefficient n is the largest source of uncertainty in any Manning calculation. Tabulated values from Chow's Open Channel Hydraulics (1959) remain the standard reference, but real channels exhibit 20 – 40 % variation depending on stage, season, and sediment load. Composite cross-sections (a paved low-flow channel with vegetated overbanks) require compound n calculations using one of the area-weighting or perimeter-weighting methods — this calculator assumes a single n applies to the entire wetted perimeter.
For natural channels, field calibration against measured flows beats any tabulated value. If a stream gauge is available nearby, back-calculate n from a known Q and compare with handbook values.
Uniform vs. non-uniform flow
Manning's equation describes normal depth — the depth at which gravity exactly balances boundary friction in a prismatic channel of constant slope. Real channels rarely run at normal depth: backwater from a downstream control, contractions or expansions, or any change in slope produce gradually varied flow that requires a step-by-step solution of the energy equation, not Manning's equation alone. Use this calculator's normal depth as the boundary condition, then run a separate water-surface profile (M1, M2, S1, S2 etc.) for the reach as a whole.
What this calculator does not include
(1) Sediment transport. Bed armoring, bedform drag, and bedload all change effective n with stage. This calculator assumes a fixed bed and clear water.
(2) Compound channels. Cross-sections with main channel + floodplains have multiple roughness zones; results from a single-n Manning calculation become inaccurate for stages above bank-full.
(3) Air entrainment and bulking. At very high velocities (typically $v > 6$ m/s or $\mathrm{Fr} > 4$), the flow entrains air, increasing apparent depth and reducing density. Manning underestimates depth in such flows.
(4) Surcharged pipes. For circular sections, the calculator restricts normal-depth solutions to the open-channel branch ($y/D < 0.94$, peak-Q point). Beyond that, the pipe runs pressurized and Manning no longer applies — use the Darcy-Weisbach pipe-friction equation instead.
For preliminary design, teaching, and quick checks. For final design, calibrate n to local data and verify against a 1-D water-surface profile model.