How this instrument works
A broad-crested weir is a wide, flat, horizontal sill set across a channel — long enough, in the direction of flow, that the water surface can settle onto it and pass through critical depth before spilling off the downstream edge. That is the physical event this instrument is built around: unlike a sharp-crested weir, where the stream springs clear of a thin plate as a falling nappe, a broad crest lets the flow itself find critical depth while still touching the structure, and that critical section is what fixes the relationship between upstream head and discharge.
The two-thirds and the square root in the formula are not arbitrary. At critical flow the Froude number equals one, and for a rectangular channel that forces the depth on the crest, yc, to sit at exactly two-thirds of the upstream head, h — the classic result yc = (2/3)h. Combine that with the critical discharge per unit width, q = √(g·yc³), and the constants collapse into (2/3)·√(2g/3): the formula is critical-depth theory rearranged, not an empirical curve fit. The discharge coefficient, Cd, is the one part borrowed from measurement — it folds in boundary-layer friction along the crest and the small energy loss real corners cause that the ideal derivation ignores.
Two conditions have to hold for that arithmetic to survive contact with a real channel. The crest has to be long enough, relative to the head, for critical depth to actually establish on its flat top — a short sill behaves more like a sharp-crested weir and this formula overstates its flow. And the tailwater downstream has to stay low enough that the crest is not drowned; once submergence pushes the control point off the crest and back into the approach channel, the free-flow assumption behind Cd no longer holds, and the reading needs a submergence correction this formula alone cannot supply.
- Enter Discharge coefficient — 0.85 is the usual figure for a sharp, square upstream edge on a concrete crest; rounded edges run higher.
- Enter Weir width, the crest's dimension measured across the channel, in millimetres, centimetres or metres.
- Enter Head over the weir crest, read on a gauge upstream of the drawdown curve — not on top of the crest itself.
- Read Flow rate in cubic metres per second, or switch its unit menu to litres per second for smaller channels.
- Check that the weir is running free: downstream water should sit well below the crest, since a drowned crest breaks this formula.
Worked example — a 2 m broad-crested weir at 0.3 m head
Take an irrigation control structure with a broad concrete sill 2 m wide, a discharge coefficient of 0.85 rated for its sharp square edge, and 0.3 m of head read on a staff gauge well upstream of the crest. The two building blocks are h^1.5 = 0.3^1.5 ≈ 0.16432 and √(2g/3) ≈ 2.55691; multiply Cd, 2/3, b, that root and h^1.5 together and the weir passes Q = 0.476161 m³/s, or about 476 l/s.
That figure shows exactly why head is the number to watch on a structure like this. Raise the upstream stage by 30 mm, to 0.33 m — a 10% rise in head — and discharge climbs to about 0.549 m³/s, roughly 15% more, because Q scales with h to the power 1.5 rather than with h itself. Double the crest width instead, to 4 m, holding head at 0.3 m, and discharge exactly doubles to 0.952 m³/s — width alone is the one variable in this formula that scales the flow linearly.
Questions
What makes a weir 'broad-crested' rather than sharp-crested?
The crest is a flat, horizontal top long enough, relative to the head flowing over it, for the water to settle onto it and pass through critical depth before running off the downstream edge. A sharp-crested weir instead has a thin edge the stream springs clear of as a falling nappe with air underneath. A crest that is too short behaves more like the sharp-crested case, and this critical-depth formula overstates its flow; one that is far too long adds enough friction to eat into the discharge coefficient.
Why is there a factor of two-thirds in the formula?
It comes from the critical-depth condition, not from curve-fitting. At critical flow the Froude number is one, and for a rectangular channel that forces the depth on the crest, yc, to equal exactly two-thirds of the upstream head: yc = (2/3)h. Substitute that into the critical discharge relation q = √(g·yc³) per unit width, bring in width b and coefficient Cd, and the constants regroup into (2/3)·√(2g/3) — the formula is that derivation written out, not a fitted constant.
What discharge coefficient should I use?
0.85 is the standard figure for a broad crest with a sharp, square upstream edge, which is why it is the default here. Rounding or chamfering that edge reduces the contraction and can push Cd toward 0.90 or higher; a poorly finished or eroded crest can pull it below 0.85. For a structure relied on for billing or compliance, a field calibration against a known volume beats any tabulated figure.
Does the formula still work if the weir is submerged downstream?
No. Everything here assumes modular, free flow, where the critical section on the crest controls discharge regardless of what the tailwater is doing. Once the downstream level rises enough to drown that critical section, the control shifts, and the same head produces less flow than this formula predicts. Submerged operation needs a separate correction factor applied to the free-flow discharge, not this equation on its own.
Where should the head be measured?
Upstream of the drawdown curve, not on top of the crest itself. As flow approaches the crest and accelerates toward critical depth, the water surface curves downward, so a gauge on the crest reads low. Standard practice places the gauge three to four times the maximum expected head upstream of the crest's leading edge, where the approach velocity is still small enough to ignore.
Why does doubling the width double the flow but doubling the head do more?
Width, b, enters the formula to the first power — it is simply how much of the channel is carrying flow, so twice the width carries exactly twice the discharge at the same head. Head, h, enters to the power of 1.5 because it governs both the depth of flow on the crest and, through critical-flow theory, the speed at that depth; both grow with h, so their product grows faster than h itself.