Engineering & Science
Bélanger Equation: Hydraulic Jump Sequent Depth & Specific Energy Head Loss
Calculate rectangular stilling-basin sequent depth y_2 = 0.5*y_1*(SQRT(1 + 8*Fr_1^2) - 1) and turbulent energy dissipation ΔE = (y_2 - y_1)^3 / (4*y_1*y_2).
Why dam spillways conservation of momentum—not Bernoulli energy—governs a hydraulic jump!
Jean-Baptiste Charles Joseph Bélanger (1838) corrected his own 1828 Bernoulli attempt by applying hydrostatic-plus-momentum conservation across a supercritical-to-subcritical open-channel shock! Whenever supercritical spillway flow (upstream Froude number Fr_1 = v_1 / SQRT(g * y_1) > 1) transitions into subcritical tailwater, intense turbulent rollers dissipate mechanical energy while preserving specific momentum, forcing sequent (conjugate) depth to y_2 = 0.5 * y_1 * (SQRT(1 + 8 * Fr_1^2) - 1)!
Once y_2 is known, exact specific-energy head loss dissipated as heat and turbulence inside the stilling basin is ΔE = (y_2 - y_1)^3 / (4 * y_1 * y_2)! For upstream toe depth y_1 = 0.45 m at Fr_1 = 3.8 (oscillating/steady transition jump), tailwater depth jumps to y_2 = 2.204 m and dissipates ΔE = 1.360 m of head—climbing to y_2 = 3.600 m in a strong Fr_1 = 6.0 jump!
| Upstream Toe Flow (y_1 = 0.45 m) | Sequent Depth y_2 (m) | Energy Head Loss ΔE (m) | USBR Stilling Basin Regime |
|---|---|---|---|
| Fr_1 = 3.8 (Spillway Chute Toe) | 2.204 m (4.90× y_1) | 1.360 m | Oscillating-to-Steady Jump (2.5–4.5) |
| Fr_1 = 6.0 (High-Head Outlet) | 3.600 m (8.00× y_1) | 3.876 m | Steady Jump (4.5–9.0 — Best Dissipation) |
| Subcritical Fr_1 <= 1 or y_1 <= 0 | No Jump Possible | "Requires Supercritical Fr1 > 1" | Hydraulic Regime Guard |
Bélanger Hydraulic Jump Sequent (Conjugate) Depth y_2 (m)
Computes subcritical tailwater depth y_2 from supercritical toe depth y_1 (B2, m) and upstream Froude number Fr_1 > 1 (B3).
=IF(OR(B2 <= 0, B3 <= 1), "Requires Supercritical Fr1 > 1", ROUND(0.5 * B2 * (SQRT(1 + 8 * (B3 ^ 2)) - 1), 3))Hydraulic Jump Turbulent Specific-Energy Head Loss ΔE (m)
Evaluates (y_2 - y_1)^3 / (4 * y_1 * y_2) using unrounded conjugate helper B4.
=IF(OR(B2 <= 0, B3 <= 1), "Requires Supercritical Fr1 > 1", ROUND(((B4 - B2) ^ 3) / (4 * B2 * B4), 3))Try the example
Use the original values or change them to test your understanding.