Broad-Crested Weir Discharge

A broad-crested weir is long enough in the direction of flow that the streamlines go parallel over the crest, and once they do, the flow passes through critical depth. That is the whole derivation. Substituting $y_c = \tfrac23 H$ into the critical-flow relation gives $Q = C_d(\tfrac23)^{3/2}\sqrt{g}\,bH^{3/2}$, and in SI the constant group is $(2/3)^{1.5}\sqrt{9.807} = 1.705$. A 3 metre crest with a discharge coefficient of 0.9 under 0.4 m of head passes $1.705 \times 0.9 \times 3 \times 0.253 = 1.164$ m³/s.

The theoretical coefficient is 1.0 and real weirs come in below it, typically 0.85 to 0.95, with the shortfall going to boundary friction on the crest and to the contraction at the upstream edge. Rounding that upstream edge is worth several percent of capacity and is the cheapest improvement available on an existing structure. The head in the equation is the total energy head, so on a fast approach you must add the approach velocity head to the measured depth, which people routinely forget on a narrow channel.

What decides whether a weir is broad-crested at all is the ratio of head to crest length, and it is a narrow window: roughly $0.08 < H/L < 0.50$. Below 0.08 the crest is long enough that friction dominates and the coefficient falls away. Above 0.50 the streamlines curve, critical depth never establishes, and the structure behaves as a sharp-crested weir with completely different coefficients. Do not reach for the Francis constant on a broad crest or this one on a thin plate. The other quiet failure is submergence: a broad-crested weir tolerates a downstream level up to about 0.66 of the upstream head with no correction, and rather more forgivingly than a sharp-crested weir, but past that point the single upstream reading no longer determines the flow.