Formula and Method for the Hydraulic Jump Calculator
A hydraulic jump is a sudden, turbulent rise in water depth that occurs when a fast, shallow (supercritical) open-channel flow is forced to slow down into a slower, deeper (subcritical) flow — for example, just downstream of a sluice gate, spillway apron, or weir. Rather than a smooth transition, the flow forms a standing, breaking wave that dissipates a large fraction of its kinetic energy as turbulence and heat. This calculator applies conservation of momentum (the Belanger equation) to your upstream depth and velocity to find the sequent (downstream) depth, the downstream velocity, the head lost through the jump, and which of the five standard jump types it falls into.
How the calculation works
The starting point is the upstream Froude number, Fr₁ = V₁ / √(g·y₁), which compares the flow speed to the speed of a shallow-water surface wave. A hydraulic jump can only exist when Fr₁ > 1, i.e. the approach flow is supercritical. Applying the momentum equation to a control volume spanning the jump in a horizontal, rectangular, frictionless channel (the "specific force" balance) and simplifying gives the Belanger sequent-depth equation: y₂/y₁ = ½(√(1 + 8Fr₁²) − 1). Because the channel width is constant, continuity gives the downstream velocity directly from the upstream discharge per unit width: V₂ = V₁y₁/y₂. The energy (head) lost across the jump follows from the difference in specific energy before and after: ΔE = (y₂ − y₁)³ / (4y₁y₂). The downstream Froude number, Fr₂ = V₂/√(g·y₂), should come out below 1, confirming the flow really has become subcritical.
Jump classification and common mistakes
- Jump type depends only on Fr₁: undular (1.0–1.7), weak (1.7–2.5), oscillating (2.5–4.5), steady (4.5–9.0), and strong (>9.0). Steady jumps are the design target for most stilling basins because they dissipate energy efficiently (roughly 45–70%) without the wave action of oscillating jumps.
- Entering the wrong depth: y₁ must be the upstream (pre-jump, shallow, fast) depth. Entering the downstream depth instead will give a Froude number below 1 and the calculator will report that no jump forms.
- Forgetting the channel assumption: the Belanger equation assumes a horizontal, prismatic, rectangular channel with negligible boundary friction across the (short) length of the jump — it does not directly apply to sloped, trapezoidal, or circular channels without modification.
- Mixing unit systems: keep depth and velocity in the same system (metric m and m/s, or US ft and ft/s); the calculator uses g = 9.81 m/s² or g = 32.2 ft/s² to match your selection.
Real-world applications
- Stilling basins below dam spillways and sluice gates use hydraulic jumps to safely dissipate energy and prevent erosive scour downstream.
- Canal and irrigation check structures rely on jump behavior to control water levels and measure discharge.
- Storm-drain and culvert outlets are sized so any jump forms in a protected, armored channel section rather than on an unlined streambed.
- Open-channel flow measurement flumes (e.g. Parshall flumes) use the transition through critical flow — closely related to jump theory — to relate depth to discharge.