Open Channel Flow Calculator

Background.

Water in a roadside ditch, a canal, a storm drain running part-full or a river moves under gravity alone, with its surface open to the air. How much it carries comes down to a tug-of-war between the slope pulling it downhill and the friction of the bed and banks holding it back. When the two balance and the depth stays the same along the reach, the flow is called uniform, and Manning's equation gives its discharge.

Manning's formula dates from the late nineteenth century and remains the workhorse of drainage and channel design because it folds all of that friction into one empirical number, the roughness coefficient n. Finished concrete sits around 0.012 to 0.013, a clean, straight natural stream around 0.03, and a weedy or brushy channel 0.05 and upwards. Discharge is divided by n, so choosing it is usually the largest source of error: a channel taken as 0.03 that is really 0.036 carries a sixth less water than calculated.

Geometry enters through two numbers: the flow area A, and the hydraulic radius R, which is that area divided by the wetted perimeter, the length of bed and banks actually touching water. R measures how much water there is for each metre of friction-producing boundary. A wide, shallow sheet of water has a small R and drags heavily; a compact, deep section has a larger one and flows more freely. Slope enters as a square root, so quadrupling the gradient only doubles the flow.

The equation describes steady, uniform flow and nothing more. Backwater from a culvert or confluence downstream, hydraulic jumps, bends, sediment, different roughness across a main channel and its floodplain, and the freeboard and safety margins a real design needs all lie outside it.

What is open channel flow calculator?

Open channel flow is flow with a free surface, driven by gravity rather than pressure, as in rivers, canals, gutters and partly full pipes. The quantity computed here is the uniform-flow discharge: the volume of water passing a cross-section each second, in cubic metres per second, when depth, velocity and cross-section stay constant along the channel so that the bed, the water surface and the energy line all share one slope. Manning's equation links that discharge to the channel's roughness, flow area, hydraulic radius and slope.

The equation grew out of an older one. Chézy's formula gives the mean velocity as V = C √(R S), with C a friction factor that had to be found for each channel. Manning's contribution was to notice that C varies roughly as the sixth root of the hydraulic radius, C = R^(1/6) ÷ n; substituting gives V = R^(2/3) S^(1/2) ÷ n, and multiplying by the flow area gives the discharge. For the worked example, with R = 0.8 m and n = 0.03, the equivalent Chézy C is about 32.12.

The idea of uniform flow also defines the normal depth: the one depth at which a long, prismatic channel carries a given discharge with gravity and friction exactly in balance. Designers use it to size ditches and culverts and to check whether a proposed lining keeps velocities within limits. Real flows approach it in long, straight reaches away from structures; near a weir, a culvert inlet or a change of slope the depth departs from it, and a different calculation takes over.

How to use this calculator.

1. 01Choose Manning's n for the channel lining from a published table such as the USGS guide cited below. For natural channels, raise it for vegetation, irregular banks and obstructions.
2. 02Enter the flow area in square metres at the depth you are checking, and the hydraulic radius in metres: that area divided by the wetted perimeter of the same cross-section. For a rectangular channel b wide and y deep, A = by and R = by ÷ (b + 2y).
3. 03Enter the energy slope as a ratio, metres of fall per metre of length. In uniform flow it equals the bed slope, so a drop of 2 m over 1 km is 0.002, not 0.2 or 2.
4. 04Read the discharge in m³/s. Divide it by the flow area for the mean velocity, or multiply by 1,000 for litres per second. Check the velocity against the lining as well as the capacity: water that is too fast scours bare earth and grass, and water that is too slow lets silt settle and the channel choke.

The formula.

Q = (1/n) A R^(2/3) S^(1/2) in SI units. Q is the discharge in m³/s, n is Manning's roughness coefficient, A is the flow area in m², R = A/P is the hydraulic radius in metres (P being the wetted perimeter) and S is the energy slope in metres per metre. Dividing Q by A gives Manning's velocity, V = (1/n) R^(2/3) S^(1/2).

The exponents are empirical, which is why n carries awkward units of s/m^(1/3) and why the number you use depends on the unit system. Convert the same channel to feet and the coefficient in front changes from 1 to the cube root of 3.28084 feet per metre, about 1.486, which is the familiar 1.49 of the US customary form. Only the SI form is used here, so enter metres.

Discharge varies inversely with n, with the square root of slope and with the two-thirds power of hydraulic radius. A 10% error in n therefore shifts Q by about 10%, while a 10% error in slope shifts it by under 5%. Area and hydraulic radius are entered separately and nothing ties them to the same cross-section, so take both from one drawing at one depth.

The model assumes fully turbulent flow over a rough boundary, a channel that keeps the same shape and slope along the reach, and a single n that does not change with depth. Each assumption can fail. In very shallow sheet flow the friction behaves differently, which is why drainage manuals such as HEC-22 treat overland flow and gutter flow with their own methods. Grass and brush flatten as flow deepens, so their effective n falls with depth. And wherever a downstream structure or confluence backs water up, the energy slope is no longer the bed slope, and a gradually varied flow calculation is needed instead.

A worked example.

Frequently asked questions.

References& sources.

1. [1]FHWA HEC-22, Urban Drainage Design Manual, 4th ed.
2. [2]USACE EM 1110-2-1601, Hydraulic Design of Flood Control Channels.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]USGS Water-Supply Paper 2339, Guide for Selecting Manning's Roughness Coefficients.