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.
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.
Cross-section geometry — what the calculator computes under the hood
Manning's equation only knows about three properties of a cross-section: the flow area $A$, the wetted perimeter $P$, and (through their ratio) the hydraulic radius $R = A/P$. The top width $T$ enters separately when we evaluate the Froude number and critical depth. Each of the four supported shapes has a closed-form expression for these quantities in terms of the depth $y$:
| Shape |
Area $A$ |
Wetted perim. $P$ |
Top width $T$ |
Notes |
| Rectangular |
$by$ |
$b + 2y$ |
$b$ |
$b$ = bottom width |
| Trapezoidal |
$(b + zy)\,y$ |
$b + 2y\sqrt{1+z^2}$ |
$b + 2zy$ |
$z$ = horizontal run per unit rise |
| Triangular |
$zy^2$ |
$2y\sqrt{1+z^2}$ |
$2zy$ |
V-shape, both side slopes equal to $z$ |
| Circular |
$\dfrac{D^2}{8}(\theta - \sin\theta)$ |
$\dfrac{D\theta}{2}$ |
$D\sin\tfrac{\theta}{2}$ |
$\theta = 2\cos^{-1}\!\left(1 - \tfrac{2y}{D}\right)$ |
For the circular section, $\theta$ is the central angle subtended at the pipe centre by the wetted arc, measured in radians: it runs from $0$ (empty), through $\pi$ (half full), to $2\pi$ (just full). The depth-to-angle relation $\theta = 2\cos^{-1}(1 - 2y/D)$ comes directly from the chord geometry of a circle and is the key step that turns the circular case from "no closed form" into "closed form, evaluated through one inverse cosine."
These formulas appear in Chow (1959), Henderson (1966), and every subsequent open-channel textbook in exactly this form. The hydraulic radius then follows mechanically as $R = A/P$. The calculator evaluates each formula every time you change a parameter — there is no interpolation table behind the scenes.
Critical depth and the Froude number — what the result is telling you
The Froude number $\mathrm{Fr}$ at the bottom right of the results panel is the single most important indicator of flow regime:
$$ \mathrm{Fr} = \frac{v}{\sqrt{g\,D_h}}, \qquad D_h = \frac{A}{T} $$
where $D_h$ is the hydraulic depth (area divided by surface width). $\mathrm{Fr}$ is the ratio of flow velocity to the speed of a small gravity wave on the water surface, and it distinguishes three physically very different regimes:
Subcritical ($\mathrm{Fr} < 1$). Slow, deep flow. Disturbances can travel upstream — a downstream obstruction (a culvert headwall, a weir, a tidal estuary) controls the upstream water surface. Typical of mild-sloped natural rivers and most stormwater drains.
Critical ($\mathrm{Fr} = 1$). A knife-edge condition. The specific energy is at its minimum for the given discharge, and the water surface tends to be unstable, often forming undular waves. The flow passes through critical depth at any section where a subcritical reach transitions to a supercritical one — typically the crest of a weir, the brink of a free overfall, or the entrance to a steep chute.
Supercritical ($\mathrm{Fr} > 1$). Fast, shallow flow. Disturbances cannot travel upstream; the upstream condition controls everything downstream. Hydraulic jumps form where a supercritical reach meets a subcritical control — chutes, spillway aprons, and steep storm sewers.
The critical depth $y_c$ in the results grid is the depth at which $\mathrm{Fr} = 1$ for the given $Q$. Comparing $y_n$ (normal depth, from Manning) with $y_c$ tells you the slope category at a glance: $y_n > y_c$ is a mild slope (subcritical normal flow), $y_n < y_c$ is a steep slope (supercritical normal flow), and $y_n = y_c$ is the critical slope, an unstable corner case that designers actively avoid.
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. A circular pipe running partially full has the curious property that the discharge peaks before the pipe runs full: $Q_\text{peak}$ occurs at $y/D \approx 0.938$ and is about $\mathbf{7.6\%}$ higher than the just-full discharge $Q_\text{full}$. (At half-full, by symmetry, $Q = 0.5\,Q_\text{full}$.) Above $y/D = 0.938$ a second normal-depth branch exists, but the flow there is unstable — small perturbations cause the pipe to slug back and forth between open and pressurized flow. The calculator restricts normal-depth solutions to the stable open-channel branch ($y/D < 0.938$). For sustained full-pipe flow, switch to the Darcy-Weisbach pipe-friction equation.
(5) Depth-varying roughness in pipes. Manning's n is treated as a single constant here, but for circular pipes the effective roughness rises by 20 – 30 % as the depth approaches the crown (Camp, 1946 — caused by the curvature of the wetted boundary near the top). This is a known limitation of plain Manning and not unique to this tool, but it means computed discharges near full flow are slightly optimistic.
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.