Open Channel Flow Calculator

Purpose and model boundary

This calculator estimates uniform flow in an open channel whose cross-section you draw. You draw the section on a grid as one closed outline, type the depth of water above its lowest point, the channel's slope and Manning's n, and the spreadsheet returns the flow area, the wetted perimeter, the hydraulic radius, the top width, the flow rate by Manning's equation and the mean velocity, with a curve of the flow rate at twenty water depths. It treats the whole section as one channel carrying uniform flow at the depth you type. It is not a hydraulic design or a flood study: it does not find the depth that carries a given flow, compute a Froude number, a critical depth or the flow regime, route a flood, model a culvert, a bridge pier or any other obstruction, divide a compound section into a main channel and overbanks, or choose a roughness for you.

Inputs and units

The drawing is the first input. Its outline is sent as an ordered list of up to 100 corner points, exactly as on the Area Calculator, which measures the same drawing, and the US/Metric switch on the drawing chooses the unit the points are written in, feet for US and metres for Metric. Y runs down the page, as the drawing tool draws it: the channel bottom is the drawing's lowest point, the one with the largest Y, and the top of the drawing is the smallest Y. Draw the bed and the banks as the outline and close it across the top with a level edge at or above the water. The page opens with Keep walls square on, which keeps every wall level or upright; switch it off to drag a bank to a slope. The drawing tool also offers exclusions, but this calculator reads the outline only: a drawn exclusion is refused (see Validation and status logic).

The inputs beside the drawing are:

• Water depth: the depth of water above the channel bottom, in feet (US) or metres (Metric). The water line is level at this height, which may equal the drawn section's height but not exceed it.
• Channel slope: the channel's longitudinal slope as fall over length, taken as the energy slope of uniform flow; 0.001 is 1 ft per 1,000 ft, or 1 m per km. The same number in both systems.
• Manning's n: the roughness coefficient of the channel lining, a plain number, the same in both systems.

The water depth has one field for each system, each with its own default: 3 ft in US and 1 m in Metric, the figures of worked examples 11.6.3.1 and 11.6.3.5 of the requirement this calculator was built against. They are not conversions of each other: 3 ft is 0.9144 m. The slope of 0.001 and the n of 0.013 the page opens with are those examples' figures too; the worked examples use n of 0.013 and 0.025 (an earth channel), so choose n for your channel's lining. They are the worked-example figures this calculator was built against, not a code, a standard or a manufacturer's figure, so review them before use. Switching between US and Metric shows the other system's water depth with its own value and never converts a number you typed; the value you typed for the system you left is kept for when you switch back, and it is not sent while it is hidden. The spreadsheet reads only the water depth of the selected system.

Nothing is converted. The drawing's points and the water depth are read in the unit of the system chosen, so the figures come back in feet, square feet, cubic feet per second and feet per second for US, and in metres, square metres, cubic metres per second and metres per second for Metric.

Governing relationships

The water line is level at the channel bottom less the water depth: with Y running down the page, its Y is the bottom's Y minus the depth. The spreadsheet walks the outline's edges in order, the last point closing back to the first, and takes from each the part below the water line: the whole edge when both of its ends are below it, the part from the end that is below it to the point where the edge meets the water line when only one end is, and nothing when neither is. A point exactly on the water line is not below it, so an edge lying on the water line is not wetted, and neither is the level edge that closes the outline at the top. So every edge, or part of an edge, below the water line is wetted; an edge lying on the water line is not.

Flow area = the area between the wetted pieces and the water line, one trapezoid term per piece, added and taken as a positive number, so the outline may run either way round

Wetted perimeter = the lengths of the wetted pieces, added; the water surface itself is not included

Top width = the horizontal runs of the wetted pieces, added, which is the length of the water line inside the section, added over separate surfaces

Hydraulic radius R = flow area ÷ wetted perimeter

Flow rate Q = (k ÷ n) × A × R^(2/3) × √S, Manning's equation, with A the flow area, S the channel slope and n Manning's n; k = 1.486 in US units (feet, and Q in cubic feet per second) and k = 1.0 in Metric (metres, and Q in cubic metres per second)

Mean velocity = flow rate ÷ flow area

1.486 is the customary US figure, the rounding of the cube root of 3.2808 feet per metre (1.4859); worked examples 11.6.3.1, 11.6.3.2 and 11.6.3.4 reproduce at their printed decimals only with 1.486. Both constants are cells of the spreadsheet's Data sheet. The flow is uniform: the slope is taken as the energy slope, and nothing else about the flow is computed.

Further figures from the same walk decide the status line. The outline's orientation is the sign of its shoelace sum, and a wetted piece faces down when the section's inside lies below it on the page, as it does under a pipe crown or an overhang. A level bench is an edge wholly under the water that is level, lies above the channel bottom and faces up: a floor. And an edge pair crosses when each edge's two ends lie strictly on opposite sides of the other edge, by more than 0.000001 (a length squared, the same number in square feet and square metres); edges that only touch are not counted. The edges that cross the water line are counted too: more than two means ground standing above the water splits its surface.

The curve, "Flow rate by water depth", is computed the same way with the water at twenty depths, from a twentieth of the drawn section's height to its full height, each labelled with its depth to two decimals when the drawn section is at least 0.2 ft (or m) high and to three when it is lower. The dashed line is the flow rate result, the same at every depth. No returned figure is rounded in the spreadsheet: its six-place rounding decides only its comparisons (what lies under the water, what is level, what crosses), and every figure is computed from unrounded values.

Calculation sequence

1. The spreadsheet counts the outline's points and finds the channel bottom (the largest Y), the top of the drawing (the smallest Y) and the drawn section's height.
2. It reads the selected system's water depth, sets the water line, and reads the slope, Manning's n and the constant k of that system.
3. For every edge it finds the part below the water line, where the edge meets the line, and that piece's length, area term and horizontal run, with the tests for a piece facing down, a level bench, an edge crossing the water line and an edge crossing another edge.
4. It adds them into the flow area, the wetted perimeter and the top width, then the hydraulic radius, the flow rate and the mean velocity.
5. It computes the curve's twenty points.
6. The status line is chosen, as described below. When it begins NOT VALID:, every hydraulic figure and every point of the curve reads zero; the drawn section's area and the gross and exclusion areas are still measured from the drawing.

For the drawing the page opens with, the canvas's 20 by 15 ft rectangle read as a rectangular channel 20 ft wide, with 3 ft of water: 20 × 3 = 60 sq ft of flow area, 20 + 3 + 3 = 26 ft of wetted perimeter, a hydraulic radius of 60 ÷ 26 = 2.3077 ft and a top width of 20 ft, so Q = 1.486 ÷ 0.013 × 60 × 2.3077^(2/3) × √0.001 = 378.74 cfs, at 378.74 ÷ 60 = 6.31 ft/s. It is a starting drawing, not a worked example. Worked example 11.6.3.1, a rectangular channel 10 ft wide drawn 6 ft deep, with 3 ft of water, a slope of 0.001 and n of 0.013: 30 sq ft, 16 ft wetted, a hydraulic radius of 1.875 ft, 10 ft across the top, 164.89 cfs and 5.50 ft/s. Worked example 11.6.3.2, a trapezoidal channel 10 ft across the bottom with 1:1 sides, drawn 22 ft across the top and 6 ft deep, with 4 ft of water, a slope of 0.002 and n of 0.025 (earth): (10 + 18) ÷ 2 × 4 = 56 sq ft, 10 + 2 × 4√2 = 21.31 ft wetted, 2.6274 ft, 18 ft across, 283.44 cfs and 5.06 ft/s. Worked example 11.6.3.3, a V channel with 1:1 sides drawn 16 ft across and 8 ft deep, with 3 ft of water, a slope of 0.005 and n of 0.015: 9 sq ft, 2 × 3√2 = 8.49 ft wetted, 1.0607 ft, 6 ft across, 65.57 cfs and 7.29 ft/s. Worked example 11.6.3.4, a compound channel, a floodway 40 ft wide and 3 ft deep with a main channel 10 ft wide and 5 ft deeper in its middle, with 6 ft of water, 1 ft over the benches, a slope of 0.001 and n of 0.03: 10 × 5 + 40 × 1 = 90 sq ft, 10 + 5 + 5 + 15 + 15 + 1 + 1 = 52 ft wetted, 1.7308 ft, 40 ft across, 203.22 cfs and 2.26 ft/s, and its status reads CHECK: because the water covers the benches. Worked example 11.6.3.5, in Metric, a trapezoidal channel 3 m across the bottom with 1:1 sides, drawn 6 m across the top and 1.5 m deep, with 1 m of water, a slope of 0.001 and n of 0.013: (3 + 5) ÷ 2 × 1 = 4 m², 3 + 2√2 = 5.83 m wetted, 0.6863 m, 5 m across, Q = 1.0 ÷ 0.013 × 4 × 0.6863^(2/3) × √0.001 = 7.57 m³/s, at 1.89 m/s.

Outputs and interpretation

Flow rate, Flow area, Wetted perimeter and Hydraulic radius are the four headline results. The page groups every result: under "Flow" the flow rate, in cubic feet per second (US) or cubic metres per second (Metric), and the mean velocity, in feet or metres per second; under "Wetted section" the flow area, in square feet or square metres, and the wetted perimeter, the hydraulic radius and the top width, in feet or metres. Under "From the drawing" the page shows what the spreadsheet measures from the drawing: Drawn section area, the area inside the outline, the whole section and not only the part under water; Gross area, the same area before any exclusion is taken out, equal to it here because this calculator takes no exclusion; and Exclusion 1 area to Exclusion 5 area, 0 unless an exclusion was drawn, which the status line refuses. The page shows every figure to at most two decimals, so the hydraulic radius, which the spreadsheet formats to four, reads 2.31 ft for the opening drawing (2.3077 ft in the spreadsheet) and 0.69 m for worked example 11.6.3.5 (0.6863 m). Every figure is computed from unrounded values, so a figure worked out by hand from the ones shown on the page can differ in its last place.

The curve beside the results shows how much each extra foot or metre of water buys in this section, and the last point is the full section's flow. The dashed line is the flow rate at the depth you typed: the curve crosses it at your depth and, where the curve dips, at other depths too. The curve can dip where the water first spreads over a bench, because a single channel's hydraulic radius falls when a wide, shallow bench adds much more wetted perimeter than area. On worked example 11.6.3.4's section, with n of 0.03 and a slope of 0.001, the flow rate is 144.26 cfs with the water exactly at the benches, 5 ft deep, and 135.42 cfs half a foot over them, less flow for more water; its curve falls from 136.60 cfs at 4.8 ft to 99.77 cfs at 5.2 ft. That is the limit of treating a compound channel as one channel, which the status line flags for a level bench. The page draws the water line neither on the drawing nor anywhere else: it shows only what the spreadsheet returns.

A result belongs to the drawing and the inputs it was calculated for. When you change either after pressing Calculate, the result and the curve are kept but marked stale until you press Calculate again; changing the value back makes them current again. Switching between US and Metric changes what is sent, so it marks the result stale too.

Validation and status logic

The drawing tool checks every edit to the drawing before it is kept, with the Area Calculator's rules: at least three points per shape, no two consecutive points in the same place, no wall crossing another wall of its shape, every exclusion inside the outline, no two exclusions overlapping, and at most five exclusions. A refused edit leaves the drawing as it was, and an imported drawing is refused as a whole if any rule fails. The drawing tool always writes a shape's points into consecutive rows, each point complete; the spreadsheet refuses a request that leaves a point half written or a blank row between points.

The status line is the spreadsheet's own verdict, the first of these that applies:

• NOT VALID: choose US or Metric
• NOT VALID: draw at least three outline points
• NOT VALID: every outline point needs both coordinates
• NOT VALID: every point of exclusion 1 needs both coordinates
• NOT VALID: every point of exclusion 2 needs both coordinates
• NOT VALID: every point of exclusion 3 needs both coordinates
• NOT VALID: every point of exclusion 4 needs both coordinates
• NOT VALID: every point of exclusion 5 needs both coordinates
• NOT VALID: this calculator reads the outline only; remove the exclusions
• NOT VALID: outline and exclusion points must fill consecutive rows, with no blank row between them
• NOT VALID: the channel outline encloses no area
• NOT VALID: the outline crosses itself; list its points in order around the section
• NOT VALID: water depth must be greater than zero
• NOT VALID: the water depth is more than the drawn section's height; draw the banks higher or type a smaller depth
• NOT VALID: channel slope must be greater than zero
• NOT VALID: Manning's n must be greater than zero
• NOT VALID: no part of the outline lies below the water line
• CHECK: part of the outline under the water faces down (an overhang, a pipe crown, or a closing edge below the water line); if the rows were written with Y running up, send them with Y running down the page
• CHECK: the water covers a level bench above the channel bottom, so this is a compound channel; Manning's Q for the whole section as one channel can understate its flow
• CHECK: ground standing above the water splits its surface into separate flows; Q treats them as one channel
• OK

Any point in any exclusion is refused, because an open-channel section is one outline. That refusal comes before the row check, so an exclusion whose points leave a blank row between them reads it too, and only an exclusion point with one coordinate missing reads its own line first. An outline that crosses itself is refused: list its points in order around the section. The water depth (the selected system's), the slope and Manning's n must each be greater than zero, and a water depth equal to the drawn section's height is allowed (the water is then up to the closing edge, which is not wetted). A water depth so small that the flow area or the wetted perimeter is zero at six decimal places is refused rather than returned as a zero flow.

The CHECK: lines come with the figures. A part of the outline under the water that faces down is a pipe crown, an overhang or a closing edge drawn below the water line, and it is also how an outline written with Y running up shows, since every floor then reads as a ceiling; a pipe running more than half full reads this line, and so does a full pipe. The compound-channel line applies when the water covers a level bench above the channel bottom: worked example 11.6.3.4 reads it, and so does its section half a foot over the benches; with the water exactly at the benches, the benches lie on the water line and are not wetted, and the status reads OK. A sloped overbank is not flagged. The split-surface line applies when ground standing above the water divides its surface; a peak exactly at the water line is counted as dividing it. A NOT VALID: verdict is shown in place of the figures and the curve.

Assumptions and limitations

• Uniform flow at the depth you type, with the slope taken as the energy slope; no Froude number, critical depth, normal depth or flow regime.
• Compound channels are treated as one channel; the single-channel method can understate the flow once the water spreads over a bench, and the status says CHECK when the bench is level; a sloped overbank is not flagged.
• One outline and no exclusions: piers and other obstructions in the flow are not modelled, and a drawn exclusion is refused.
• Manning's n is the one roughness of the whole wetted boundary; the worked examples use 0.013 and 0.025 (an earth channel), and the figure for your lining is yours to choose.
• The defaults are the worked-example figures this calculator was built against, not a code, a standard or a manufacturer's figure; the US and Metric water depths are not conversions of each other.
• Nothing is converted: the drawing's rows are read in the unit of the system chosen, so a section drawn in metres and sent under US is read as feet.
• The spreadsheet refuses edges that cross each other, but it does not detect edges that only touch or overlap along a line; the drawing tool prevents those on the page.
• Drawing needs a mouse, a pen or a touch screen; a drawing prepared elsewhere can be loaded with Import.

Trust boundary. This calculator is served on its web page and over the Model Context Protocol, where an agent sends the outline's rows itself. Its spreadsheet states the boundary: this workbook refuses an outline with fewer than three complete points, a blank row between points, any exclusion, no enclosed area, or edges that cross each other, and a water depth above the drawing. It marks with CHECK any part of the outline under the water that faces down, which is also how rows sent with Y running up show. It does not detect edges that only touch or overlap along a line, and it cannot tell whether the rows are in the unit system chosen: metres sent under US are read as feet. FloorCanvas prevents touching and overlapping edges on the page; a direct API or MCP caller bypasses it.

Restrictions and non-computing states

This calculator accepts a water depth, a channel slope and a Manning's n of zero or more, of which zero is refused by the status line, and a water depth no greater than the drawn section's height. The water depth of the system that is not selected is not read. The drawing tool refuses any edit that would leave a shape outside its limits. Nothing is sent until you press Calculate, and pressing it again on an unchanged drawing with unchanged inputs sends nothing.

Errors and warnings

A blank required input keeps Calculate from sending, and pressing it names the input and moves to it. If the calculation service refuses an input, the status line names it and the input is marked until you change it. If the service does not answer, the page says so, keeps your drawing, your inputs and the last result, and lets you press Calculate again. A CHECK: status means the figures are returned with a reason to look again; a NOT VALID: status means they cannot be computed and none are shown.

Sources and further reading

• National Institute of Standards and Technology, Guide for the Use of the International System of Units (SI), NIST Special Publication 811, Appendix B.8. nist.gov. The exact foot (0.3048 m) and square foot (0.09290304 m²), and the cubic foot per second (0.02831685 m³/s), behind the drawing's two systems. This calculator converts nothing: the US figures are in feet and cubic feet per second and the Metric ones in metres and cubic metres per second, and its US and Metric water depths are separate example figures, 3 ft and 1 m, not conversions of each other (3 ft is 0.9144 m).
• Bureau International des Poids et Mesures, "SI base units." bipm.org. The metre (m), the SI base unit of length, and the second (s), the base unit of time, in which the Metric results are given: square metres, metres, metres per second and cubic metres per second.
• Eric W. Weisstein, "Polygon Area," MathWorld, Wolfram Research. mathworld.wolfram.com/PolygonArea.html. The shoelace formula for the area of a polygon from the points of its corners, with the condition it needs, a flat polygon whose sides do not cross, which the spreadsheet uses to measure the drawn section, as on the Area Calculator, and whose sign gives the outline's direction round.

Manning's equation, its constant 1.486 for US units and 1.0 for Metric, the roughness coefficients and the treatment of compound channels are stated here as the spreadsheet uses them, and the worked examples this calculator was built against reproduce with them; no textbook or design manual is cited for them, because none was read for this page. These sources describe the units and the measurement; they do not certify a particular channel or its result. Your lining's roughness, your channel's slope and the flows you design for govern: take n and the slope from your own survey and references, and check the figures before relying on them.