About

Miscalculating flow over a broad-crested weir leads to undersized spillways, upstream flooding, or failed irrigation schedules. This calculator applies the energy-based discharge equation Q = C_d L √2g H^3/2 with empirical discharge coefficients derived from Bos (1989) laboratory data correlated to the H/P ratio. It handles both free and submerged flow conditions via the Villemonte correction and optionally iterates to include approach velocity head. The tool assumes hydrostatic pressure distribution, a rectangular horizontal crest, and negligible friction losses across the crest length.

A broad-crested weir is defined by a crest length L_w long enough that critical flow establishes on top, typically when 0.08 < H/L_w < 0.50. Outside this range the structure behaves as a sharp-crested weir or a long culvert, and the formula loses accuracy. Pro tip: field measurements of H should be taken at least 3H to 4H upstream of the weir face to avoid drawdown effects.

Formulas

The discharge over a broad-crested weir under free-flow (non-submerged) conditions is derived from the energy equation assuming critical depth occurs on the crest:

Q = C_d ⋅ L ⋅ √2g ⋅ H^3/2

Where Q = volumetric discharge (m³/s), C_d = dimensionless discharge coefficient (typically 0.85 - 1.08), L = crest width perpendicular to flow (m), g = gravitational acceleration (9.80665 m/s²), and H = upstream head above crest (m).

When the tailwater submerges the weir, the Villemonte correction is applied:

Q_s = Q ⋅ [1 − (H₂H₁)ⁿ]^0.385

Where H₂ = downstream head above crest (m), H₁ = upstream head (= H), and n = 1.5 for a broad-crested weir (the exponent in the free-flow equation).

The approach velocity correction replaces H with the total energy head:

H_e = H + V_a²2g

Where V_a = approach velocity = Q ÷ [L ⋅ (P + H)], iterated until convergence. P = weir height above channel bed (m).

Reference Data

H/P Ratio	C_d (Empirical)	Flow Regime Note
0.02	0.848	Very low head, surface tension effects possible
0.05	0.855	Low head, stable critical flow
0.10	0.870	Standard operating range begins
0.15	0.885	Well-established nappe
0.20	0.900	Optimal measurement accuracy
0.25	0.916	Approach velocity becoming significant
0.30	0.932	Correct for velocity head recommended
0.35	0.948	Upper standard range
0.40	0.965	Approaching non-modular limit
0.50	0.995	Risk of transition to sharp-crest behavior
0.60	1.025	Beyond standard range, use with caution
0.70	1.050	Weir nearly drowned, verify downstream
0.80	1.065	High submergence likely
1.00	1.080	Transitional regime, formula unreliable
Common Crest Length (L_w) Guidelines
H/L_w < 0.08	Long crested - friction losses significant, acts as open channel
0.08 ≤ H/L_w ≤ 0.50	Broad-crested weir - valid range for this calculator
H/L_w > 0.50	Approaching sharp-crested weir - use Kindsvater-Carter
Material Roughness Corrections (Approximate)
Machined metal	1.00	Reference surface (no correction)
Smooth concrete	0.98	Multiply C_d by this factor
Rough concrete	0.95	Multiply C_d by this factor
Timber (planed)	0.97	Multiply C_d by this factor
Stone masonry	0.93	Multiply C_d by this factor
Gabion / riprap	0.85 - 0.90	Highly variable, calibrate in situ

Frequently Asked Questions

The equation assumes critical flow establishes on the crest, which requires the ratio H/L_w to fall between 0.08 and 0.50. Below 0.08, friction over the long crest dominates and the structure behaves as an open channel. Above 0.50, the nappe separates from the crest and the weir acts more like a sharp-crested type, requiring the Kindsvater-Carter or Rehbock equation instead. This calculator warns you when your inputs fall outside the valid range.

When the downstream tailwater rises above the weir crest, it creates a backwater effect that reduces discharge. The Villemonte equation corrects for this by applying a reduction factor based on the ratio H₂/H₁. At a submergence ratio of 0.70, discharge drops roughly 15%. At 0.90, it drops about 35%. The correction is empirical and validated for rectangular weirs; compound or irregular shapes require physical model calibration.

Include it when the ratio H/P exceeds approximately 0.25. At low H/P values, the channel cross-section is large relative to the head so approach velocity is negligible (< 1% effect). At H/P = 0.50, ignoring approach velocity underestimates discharge by roughly 3 - 5%. This calculator iterates the velocity head until convergence within 0.0001 m.

The calculator uses a polynomial regression fitted to the empirical data published by Bos (1989, ISO 3846) relating C_d to the H/P ratio. The relationship is approximately C_d ≈ 0.848 + 0.325 ⋅ (H/P) − 0.12 ⋅ (H/P)². You may override this with a manual value if you have site-specific calibration data.

Head measurement error dominates. Because discharge scales with H^3/2, a 5% error in head produces approximately 7.5% error in discharge. The gauge station must be positioned 3H to 4H upstream of the weir face to avoid the drawdown zone. Sediment buildup on the crest effectively reduces P and changes the H/P ratio, silently drifting C_d. Regular surveying of the crest elevation is essential.

No. This tool is specifically for rectangular broad-crested weirs with a level, horizontal crest. Trapezoidal (Cipoletti) weirs use a different width integration, and V-notch (Thomson) weirs use H^5/2 rather than H^3/2. Using this calculator for non-rectangular profiles will produce incorrect results.