Weir Flow Calculator — discharge over sharp-crested, V-notch and broad-crested weirs

The weir flow calculator above turns a head measurement into a discharge. A weir is a deliberate obstruction placed across an open channel so that the water has to pass over a crest of known shape; once the shape is fixed, the depth of water above that crest is a repeatable measure of how much is flowing. That is the whole idea, and it is why weirs are still the standard primary device for gauging irrigation deliveries, treatment plant flows and small streams, decades after electronic meters became cheap.

Arb Digital publishes a companion page that is often confused with this one, and the boundary matters. The open channel flow calculator applies Manning's equation to a prismatic channel, gridding discharge, mean velocity, hydraulic radius and Froude number from a bed slope and a roughness coefficient. That describes gradually varied flow along a reach. A weir does the opposite: it forces the flow through critical depth at a control section, and the relation between head and discharge comes from that control, not from the channel's slope or roughness. Manning's equation does not describe what happens over a weir, and this page does not describe what happens along a channel.

What This Weir Flow Calculator Does

Three weir types are covered, each with its own published head-discharge relation. A sharp-crested rectangular weir is a thin plate with a horizontal crest; the nappe springs clear and the discharge goes as the head to the power 1.5. A V-notch weir replaces the rectangle with a triangular notch, so the flow area grows with the head as well as the depth and the discharge goes as the head to the power 2.5. A broad-crested weir is thick enough in the flow direction for the streamlines to become parallel over the crest, which puts a genuine critical-depth control on the structure and again gives a 1.5-power law with its own coefficient.

The tool reports discharge in three unit systems, shows the coefficient it used, and reports the geometry ratio that governs whether the relation is being applied inside its published range. It also shows what a ten-millimetre error in the head reading costs, because that is usually the dominant uncertainty in a real installation.

How to Use It

1. Pick the weir type that matches the structure. If you can see daylight under the nappe and the plate is a few millimetres thick, it is sharp-crested. If the crest is a concrete sill the water runs along, it is broad-crested.
2. Measure the head properly. The head is the vertical distance from the crest to the undisturbed upstream water surface, taken far enough back that the drawdown near the weir has not started. Four to five times the maximum head upstream is the usual guidance.
3. Enter the geometry. Crest width and crest height for a rectangular weir, notch angle for a V-notch, and crest length in the flow direction for a broad-crested one.
4. Leave the coefficient on automatic to start. The tool applies a published relation for it and shows you the value. Switch to manual the moment you have a field calibration.
5. Check the geometry ratio. It is reported in the grid, and the note tells you when the structure has moved outside the range the published coefficient was established over.

The Formula and the Coefficients

For a sharp-crested rectangular weir the relation is Q = (2/3) C_d √(2g) b H^3/2. The coefficient is not a constant: the widely used Rehbock form makes it depend on how tall the head is relative to the crest height, C_d = 0.602 + 0.083 H/P, which is the tool's automatic value. Work the default through: with b = 1 m, H = 0.3 m and P = 0.5 m, H/P is 0.6, so C_d = 0.602 + 0.0498 = 0.6518. Then Q = 0.6667 × 0.6518 × 4.4287 × 1 × 0.3^1.5 = 0.4345 × 4.4287 × 0.16432 = 0.3162 m³/s, or 316.2 litres per second.

For a V-notch the relation is Q = (8/15) C_e √(2g) tan(θ/2) H^5/2, with C_e close to 0.578 for a fully contracted 90-degree notch and rising a little as the notch angle narrows. A 90-degree notch at a head of 0.2 m gives 0.5333 × 0.578 × 4.4287 × 1 × 0.2^2.5 = 0.02442 m³/s, about 24.4 litres per second, which agrees with the published rating tables for that notch to within the width of the table.

For a broad-crested weir the flow passes through critical depth on the crest, and setting the specific energy accordingly gives Q = C_d (2/3)^3/2 √g b H^3/2. The coefficient absorbs the boundary layer on the crest and the shape of the upstream corner, and published values for a well-made rectangular broad-crested weir sit around 0.85 to 0.95, with rounded upstream edges at the upper end and square edges lower. The tool uses 0.90 on automatic.

The most important thing about all three coefficients is that they are empirical. They were established by tank calibration over specific ranges of geometry, and the USBR Water Measurement Manual chapter on weirs traces how the modern Kindsvater–Carter treatment replaced a family of separate "standard" weir formulas by accounting for approach velocity and the variation of the coefficient directly.

A Real Weir Is Rated, Not Calculated

This is the sentence that matters most on the page. The relations above describe an idealised laboratory structure: a sharp square crest, a fully ventilated nappe, a still and symmetrical approach, no sediment, no algae, and free discharge into air. Every one of those conditions degrades in the field, and every one of them shifts the coefficient.

Anything installed to measure water that someone pays for, or that a permit is written against, is normally rated in place: a series of simultaneous head readings and independent discharge measurements are taken across the working range, and a site-specific rating curve is fitted. Agencies that operate stream gauges do exactly this and re-check the rating after floods, because a single scour event can move the control. Treat the numbers this page produces as a design estimate and a sanity check on a rating, not as a substitute for one. That is what the manual coefficient input is for.

Nappe Aeration Is Not a Detail

A sharp-crested weir only obeys its published relation while the underside of the nappe is at atmospheric pressure. If the sheet of water seals against the downstream face and entrains the air trapped beneath it, the pressure under the nappe falls, the sheet is pulled down and the weir passes more water at the same head than the formula says. A clinging nappe can push the discharge well above the free-nappe value, and the error is silent because the head reading looks entirely normal.

The fix is designed in: the weir plate must be arranged so that air can reach the space under the nappe, either because the channel is wider than the notch or because a vent pipe is fitted through the downstream wall. If you are looking at a weir where the water clings to the face rather than springing clear, the head-discharge relation on this page does not apply to it. The same physics governs the energy loss on the downstream side, which is where the hydraulic jump calculator takes over if the fast sheet lands in a slower tailwater.

Submergence Ends the Measurement

Everything here assumes free discharge: the tailwater is below the crest and does not influence the flow over it. Once the downstream level rises above the crest, the weir becomes submerged, discharge falls at the same upstream head, and the single-variable relation breaks down entirely. Sharp-crested weirs start to be affected as soon as the tailwater touches the crest. Broad-crested weirs tolerate more, typically staying modular until the downstream head reaches roughly two-thirds to four-fifths of the upstream head, depending on the crest profile.

A submerged weir can still be used to estimate flow, but it needs both upstream and downstream heads and a submergence correction, which is outside what this page computes. If your tailwater is anywhere near the crest, check it before you trust the number. Downstream channel capacity is what sets that tailwater, and the hydraulic radius calculator and the open channel flow calculator are the pages for working it out.

Choosing Between the Three Types

The exponent decides it. A V-notch discharge goes as H^2.5, so at low flows a small change in discharge produces a relatively large change in head, and the notch resolves small flows far better than a rectangular weir of any width. That makes it the standard choice for gauging a small stream, a spring, or a treatment plant on low load. The same exponent works against it at high flow: the head climbs quickly and the required freeboard becomes large.

A rectangular sharp-crested weir handles a wider range at the cost of resolution at the bottom end, and its accuracy is good when the plate is clean and the approach is well behaved. A broad-crested weir is the rugged option. It is a concrete sill rather than a plate, so it tolerates sediment, debris and cattle, needs less head loss to work, and stays modular under higher submergence. Its coefficient is less tightly defined, which is the price. If you want to compare against a completely different primary device, the orifice flow calculator covers the submerged-opening alternative, and the flow rate calculator and flow rate converter handle volumes and units once you have the discharge.

Common Mistakes to Avoid

• Measuring the head at the weir — the surface is drawn down there. Take the reading several heads upstream, where the water is still level.
• Using a textbook coefficient on a rated structure — if the weir has a field rating curve, that curve wins. The published coefficient is a starting point for design, not a measurement.
• Ignoring the nappe — a sheet that clings to the downstream face passes more water than the formula predicts, and the head reading gives no clue that it is happening.
• Applying a free-discharge relation to a submerged weir — once the tailwater rises above the crest, head alone no longer determines discharge.
• Letting sediment build up upstream — it reduces the crest height, raises the approach velocity, and quietly biases every reading high.

Related Free Tools From Arb Digital

For flow along a channel rather than over a structure, use the open channel flow calculator, which applies Manning's equation, and the hydraulic radius calculator for the geometry it needs. The hydraulic jump calculator covers the energy loss where fast flow meets a slower tailwater downstream of the crest. The hydraulic gradient calculator handles the slope of the energy line, and the Bernoulli equation calculator is the underlying energy statement all of these rest on. Once you have a discharge, the flow rate converter puts it in whatever unit your report wants. Everything Arb Digital publishes is indexed on the free online tools hub. The critical-flow theory behind the broad-crested case is developed in MIT OpenCourseWare's 2.25 Advanced Fluid Mechanics, and the practical rating procedures are set out across the USBR Water Measurement Manual.

Frequently Asked Questions

The relations here describe idealised laboratory weirs with free, aerated discharge. Any structure used for allocation, compliance or billing should be rated in place by a qualified hydrologist or engineer, and that rating governs.