Engineering & Science
Broad-Crested Weir Discharge & Critical Crest Depth (y_c = 2/3 H_0) Formula
Calculate dam spillway and levee overflow discharge Q = C_d * (2/3)^1.5 * SQRT(g) * b * H_0^1.5 and critical crest flow depth y_c = (2/3) * H_0.
How Froude number Fr = 1 over a flat weir sill locks crest water depth to exactly 2/3 of total energy head!
When water flows freely over a broad horizontal sill (crest length L_crest >= 2.5 * H_0 so streamlines run parallel with hydrostatic pressure), specific energy E = y + v^2 / (2*g) reaches its minimum-energy critical state Fr = 1 right atop the sill! Differentiating Q = b * y * SQRT(2*g*(H_0 - y)) with respect to y sets critical crest depth to y_c = (2 / 3) * H_0 and ideal maximum discharge to (2 / 3)^(3/2) * SQRT(g) * b * H_0^(3/2) ≈ 1.7049 * b * H_0^1.5 (SI)!
Multiplying by boundary-layer discharge coefficient C_d (typically 0.93–0.98 for rounded-nose broad-crested spillways) yields Q = C_d * (2/3)^1.5 * SQRT(9.80665) * b * H_0^1.5! Across a b = 2.40 m floodway weir at upstream energy head H_0 = 0.45 m and C_d = 0.96, critical sill depth is y_c = 0.3000 m and discharge is Q = 1.1856 m^3/s (climbing to 1.8253 m^3/s at H_0 = 0.60 m)!
| Upstream Total Head H_0 (Width b = 2.4 m, C_d = 0.96) | Critical Crest Depth y_c = 2/3 H_0 | Volumetric Discharge Q (m³/s) | Hydraulic Control |
|---|---|---|---|
| H_0 = 0.45 m (Design Flood Stage) | 0.3000 m | 1.1856 m³/s | Critical Fr = 1 control over sill |
| H_0 = 0.60 m (100-Yr Peak Stage) | 0.4000 m | 1.8253 m³/s | 1.5-power head scaling |
| Invalid b <= 0, H_0 <= 0, or C_d Outside (0, 1] | Error Guard | "Invalid Weir Geometry" | Hydraulic Bounds Guard |
Broad-Crested Weir Volumetric Discharge Q (m³/s)
Evaluates C_d * (2/3)^1.5 * SQRT(9.80665) * b * H_0^1.5 from weir width B2 (m), total upstream energy head B3 (m), and discharge coefficient B4.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 > 1), "Invalid Weir Geometry", ROUND(B4 * ((2 / 3) ^ 1.5) * SQRT(9.80665) * B2 * (B3 ^ 1.5), 4))Critical Flow Depth Atop Sill Crest y_c = (2 / 3) * H_0 (m)
Computes the theoretical critical water depth where local Froude number equals 1.0.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B4 > 1), "Invalid Weir Geometry", ROUND((2 / 3) * B3, 4))Try the example
Use the original values or change them to test your understanding.