Weir Flow Calculator (rectangular & V-notch)

Understanding the weir flow calculator — concept and principle

A weir is a wall or plate across a channel that water flows over. Because the flow over a sharp-edged crest depends only on the height of water above it, a weir doubles as a simple flow meter for streams, canals and treatment plants. This calculator gives the discharge for a rectangular sharp-crested weir and for a triangular V-notch weir from the measured head.

The blue flow window is much smaller in the V-notch, so the same head means far less flow.

How it is calculated

Rectangular (Francis): Q = 1.84·b·H^1.5, with b the crest width (m) and H the head over the crest (m). It applies to a sharp crest without end contractions.
V-notch: Q = (8/15)·C_d·√(2g)·tan(θ/2)·H^2.5, with notch angle θ and discharge coefficient C_d ≈ 0.58.

Both results are converted from m³/s to L/s (× 1000).

The principle

Treating each thin horizontal strip of the nappe like a small orifice, its velocity is about √(2gh), where h is its depth below the upstream water surface. Integrating velocity times strip width over the head gives (2/3)√(2g)·b·H^1.5 for a rectangle. For a V-notch the width grows with depth, 2h·tan(θ/2), and the integral gives the 8/15 factor and H^2.5. Real flow contracts and loses a little energy, which the coefficient absorbs; Francis's 1.84 corresponds to C_d ≈ 0.62 for the rectangle. The steeper power for the V-notch makes it far more sensitive at low flows.

Worked example

Inputs: H = 0.2 m, b = 1 m, θ = 90°.
H^1.5 = 0.08944, so rectangular Q = 1.84 × 1 × 0.08944 = 0.16457 m³/s = 164.57 L/s.
For the V-notch, (8/15) × 0.58 × √19.62 = 1.3702 and tan 45° = 1.
H^2.5 = 0.017889, so V-notch Q = 1.3702 × 0.017889 = 0.024510 m³/s = 24.51 L/s.

Each point is the calculator re-run with only “Head over crest (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Head over crest (m)	Rectangular (L/s)
0.05	20.57182539
0.1	58.18590895
0.15	106.8943404
0.2	164.5746031
0.25	230
0.3	302.3428517
0.4	465.4872716

How to read the result

The same 0.2 m head passes almost seven times more water over a 1 m rectangular crest than through a 90° notch, which is why V-notches are chosen for small or highly variable flows and rectangular weirs for larger ones. Since Q depends on H^1.5 or H^2.5, a 1% error in the head gives a 1.5% or 2.5% error in flow, so head measurement is the main accuracy issue.

Common mistakes

Measuring the head right at the crest, where the surface has already dropped, instead of well upstream.
Using the Francis coefficient for a broad-crested or rounded weir, which needs a different formula.
Ignoring end contractions or a submerged (drowned) nappe, both of which reduce discharge.

Keep in mind: The coefficients assume a sharp crest, a ventilated free-falling nappe, and calm approach flow. For flow measurement that must meet a standard, use calibrated weirs installed to ISO 1438 or similar, and have a qualified engineer check the installation.

Frequently asked questions

Where should the head be measured?

Upstream of the crest, typically about four or more times the maximum head away, in a stilling well or with a staff gauge.

Why is the V-notch angle usually 90°?

It gives a convenient range and well-documented coefficients; narrower notches suit even smaller flows.

What is a contracted weir?

One narrower than the channel, so the jet contracts at the ends; Francis reduced the width by 0.1H per end contraction.

Sources: J. B. Francis, Lowell Hydraulic Experiments, 1855 · ISO 1438, Hydrometry, Open channel flow measurement using thin-plate weirs · US Bureau of Reclamation, Water Measurement Manual

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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