The open channel flow calculator above applies Manning's equation to water with a free surface: a ditch, a swale, a culvert running part full, a lined canal or a natural stream. The distinguishing feature of these problems is that gravity drives the flow and the water surface is at atmospheric pressure, so the driving force comes from the slope of the bed rather than from any pressure applied at one end.
Arb Digital builds free calculators that are explicit about their range of validity. This one reports the Froude number alongside the discharge, because a channel carrying the same flow can behave in two completely different ways depending on which side of critical it sits, and the discharge figure on its own does not tell you which.
What This Open Channel Flow Calculator Does
It computes the cross-sectional area and wetted perimeter of the shape you choose at the depth you enter, forms the hydraulic radius from them, and applies Manning's equation to get the mean velocity. Multiplying velocity by area gives the discharge, reported in cubic metres per second and in cubic feet per second.
Four shapes are covered. Rectangular and trapezoidal handle most constructed channels, triangular handles roadside V-ditches, and the circular option handles a pipe running part full — which is a genuinely different problem from a pipe running full, because the wetted perimeter and area both change with depth.
The Froude number in the grid divides the mean velocity by the speed of a shallow water wave at that depth. Below one the flow is subcritical: disturbances travel upstream, the flow is controlled from downstream, and the surface is generally tranquil. Above one it is supercritical: nothing propagates upstream, control comes from upstream, and any obstacle produces standing waves or a hydraulic jump.
How to Use It
- Pick the shape and enter its dimensions. Bed width for rectangular and trapezoidal channels, internal diameter for a circular pipe. The side slope field applies only to trapezoidal and triangular sections.
- Enter the depth of flow, not the depth of the channel. This is the single most common error. A ditch one metre deep carrying 300 mm of water is a 300 mm problem.
- Set the bed slope as a decimal. A fall of 1 m over 1,000 m is 0.001. If your figure is a percentage, divide by 100 first.
- Choose a roughness value carefully. The menu offers common nominal figures, but the field stays editable. Roughness dominates the uncertainty here, and doubling n halves the discharge exactly.
- Read the Froude number before acting on the discharge. A result close to one indicates near-critical flow, where the water surface becomes unstable and small changes in depth produce large changes in behaviour.
The Formula: How Manning's Equation Works
Manning's equation gives the mean velocity of steady uniform flow as V = (1 ÷ n) × R2/3 × S1/2 in metric units, where n is the roughness coefficient, R is the hydraulic radius in metres and S is the energy slope. Discharge is then Q = V × A.
The hydraulic radius is the flow area divided by the wetted perimeter, R = A ÷ P. It is a measure of how efficiently a shape carries water: a large area touching a small length of boundary loses little to friction. This is why a deep narrow channel outperforms a wide shallow one of the same area, and why a half-full pipe is close to the most efficient circular section there is.
Work the defaults. A rectangular channel 3 m wide carrying 1 m of water has A = 3 m² and P = 3 + 2 = 5 m, so R = 0.600 m. Raised to the two-thirds power that is 0.711. The slope of 0.001 has a square root of 0.03162. With n = 0.030 the velocity is (1 ÷ 0.030) × 0.711 × 0.03162 = 0.750 m/s, and the discharge is 0.750 × 3 = 2.250 m³/s, or 79.4 ft³/s.
The Froude number uses the hydraulic depth, which is area divided by top width — here 3 ÷ 3 = 1 m. A shallow wave travels at √(9.80665 × 1) = 3.13 m/s, so the Froude number is 0.750 ÷ 3.13 = 0.240. That is firmly subcritical, which is what you want in a drainage channel.
Why This Is Not the Pipe Flow Calculation
A full pressurised pipe and a part-full one are different problems and they need different equations. In a full pipe the driving force is a pressure difference between the ends, the cross-section is fixed, and the friction loss is normally handled with the Darcy-Weisbach relation and a friction factor. That is what our pipe flow calculator does, and what the friction factor calculator supplies the coefficient for.
In an open channel the water surface is free and at atmospheric pressure, the cross-section changes with depth, and the driving force is the component of gravity along the bed. The depth is not given to you; it is part of the answer, and it adjusts itself until the slope, the roughness and the discharge are consistent.
The circular option here spans the boundary between the two. Increase the depth toward the diameter and this calculator will tell you when the pipe has filled and hand the problem over, rather than returning a number from an equation that no longer applies. Curiously, the discharge of a circular pipe peaks slightly before it is completely full, at around 93 per cent of the diameter, because the last of the wetted perimeter adds friction faster than it adds area.
Roughness Is Where the Error Lives
Every other input here can be measured to within a few per cent. Roughness cannot. It is an empirical lumping-together of surface texture, channel irregularity, vegetation, obstructions, meandering and sediment load, and published values for a single description of a channel routinely span a factor of two.
Because discharge is inversely proportional to n, that factor of two in roughness is a factor of two in the answer. No amount of care with the geometry compensates for it. This is why practising hydrologists calibrate n against a measured flow wherever one exists, and why a design is usually checked at both ends of a plausible roughness range rather than at a single value.
Roughness also changes with the season and with the depth. A grassy channel is far rougher in summer than in winter. A stream that is smooth at low flow becomes rough when the water rises into vegetation on the banks, and rougher still once it spills out of the main channel entirely. The National Weather Service describes bankfull stage as the level at which water begins to overflow the natural banks, and above that point a single-channel calculation like this one stops representing the situation at all.
Subcritical, Supercritical and Why the Jump Matters
The Froude number decides which of two regimes the flow is in, and the two behave so differently that treating them as one is a serious error.
Subcritical flow is deep and slow. Information travels both ways, so a downstream obstruction backs water up. This is the normal state of drainage channels and of most natural rivers, and it is stable and forgiving.
Supercritical flow is shallow and fast, as found on a steep spillway or in a lined chute. Nothing travels upstream, so a downstream feature has no effect until the flow reaches it — at which point the water must return to subcritical through a hydraulic jump, a violent, turbulent, energy-shedding transition. Jumps are useful when designed deliberately and destructive when they land where nothing was built to take them.
Near a Froude number of one, the specific energy curve is almost flat, so a tiny change in energy produces a large change in depth. The surface becomes wavy and unstable. The Bureau of Reclamation's Water Measurement Manual sets out the energy balance and specific energy relationships that underlie this behaviour. Designing a channel to sit at critical is generally avoided for exactly that reason.
Where This Calculation Stops Being Valid
Manning's equation assumes steady uniform turbulent flow of water in a prismatic channel. Each of those words excludes something real.
Steady rules out a flood wave passing through, where the discharge at a point changes with time. Uniform rules out the approach to a weir, a culvert entrance, a bend or a change in section, where depth varies along the reach. Prismatic rules out a channel whose shape changes as you travel down it. And the turbulence assumption fails at very small depths and very low velocities, where viscosity begins to dominate — the same transition described in OpenStax University Physics Volume 1, section 14.7 on viscosity and turbulence.
Nothing here addresses sediment, ice, debris blockage, or the structural adequacy of a culvert or bank. A drainage design that people or property depend on is signed off by a qualified civil or hydraulic engineer working to the applicable standard, and this tool publishes no allowable-capacity figures of its own.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering the channel depth instead of the water depth — the calculation only sees the water, and a half-full ditch carries far less than a full one.
- Entering the slope as a percentage — a 0.1 per cent grade is 0.001, not 0.1. Getting this wrong scales the answer by a factor of about 32.
- Using a single roughness value with confidence — n is the dominant uncertainty and it changes with season, depth and vegetation. Check the answer across a plausible range.
- Applying it near a structure — culvert entrances, weirs, bends and section changes all produce non-uniform flow that this equation does not describe.
- Using it on a full pipe — once there is no free surface the problem becomes a pressurised one and needs the Darcy-Weisbach approach instead.
Related Free Tools From Arb Digital
For a full pressurised line use the pipe flow calculator with the friction factor calculator. To get a discharge from a measured velocity and area, use the flow rate calculator, and to move between litres per second, gallons per minute and cubic metres per hour use the flow rate converter. For the energy relationship between pressure, elevation and velocity, see the Bernoulli equation calculator, and for fluid property units the viscosity converter. Everything Arb Digital publishes is listed at the free online tools hub.
Frequently Asked Questions
It is the flow area divided by the wetted perimeter. It captures how much water is being carried relative to how much boundary is producing friction, which is what actually governs the velocity. Depth alone cannot do that, because a wide shallow channel and a deep narrow one can share a depth and behave very differently.
No. Once there is no free water surface the flow is pressurised and driven by a pressure difference rather than by gravity along the bed, which is a different calculation. This tool detects that case and says so instead of returning a number.
Because near the top the wetted perimeter grows faster than the flow area. Friction increases more than capacity does, so the discharge reaches a maximum at roughly 93 per cent of the diameter and then falls slightly as the pipe fills.
Whether the flow is subcritical or supercritical. Below one, disturbances can travel upstream and the flow is controlled from downstream, which is the usual tranquil state. Above one they cannot, the flow is fast and shallow, and returning to subcritical requires a hydraulic jump.
It is only as accurate as the roughness coefficient, which is empirical and often uncertain by a factor of two. Since discharge is inversely proportional to that coefficient, the same uncertainty carries straight through to the answer. The geometry and slope contribute far less error.
Only in steady uniform flow, which is what this equation assumes. Where the depth changes along the reach — approaching a weir, a culvert or a bend — the energy slope differs from the bed slope and a backwater calculation is needed instead.
It is a study and estimating tool. A real drainage design depends on design storms, headwater limits, sediment, blockage risk and the applicable local standard, and must be signed off by a qualified civil or hydraulic engineer. This page publishes no allowable-capacity figures.
This tool is provided for educational and study use. It models steady uniform flow in a prismatic channel and does not represent flood routing, backwater effects, sediment transport or the adequacy of any real drainage structure.
