3.5696
0.9146
m
1.0934
m/s
0.4987
m
0.34%
4.5729
0.2187
1
m²/s
3.5696
0.9146
m
1.0934
m/s
0.4987
m
0.34%
4.5729
0.2187
1
m²/s
The Hydraulic Jump Calculator analyzes the sudden transition from supercritical (fast, shallow) to subcritical (slow, deep) open-channel flow known as a hydraulic jump. This phenomenon is commonly observed at the base of spillways, below sluice gates, and in stilling basins, where it serves as a natural energy dissipator that protects downstream channels from erosion.
A hydraulic jump occurs when the upstream Froude number exceeds unity (supercritical flow). The calculator determines the conjugate depth h₂ downstream of the jump using the Belanger equation: $$\frac{h_2}{h_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$$ It also computes the downstream velocity by continuity and the energy dissipated across the jump.
Understanding hydraulic jumps is critical in dam engineering, irrigation design, flood control, and wastewater treatment. Engineers deliberately induce jumps in stilling basins to convert destructive kinetic energy into turbulent heat, protecting infrastructure downstream.
The analysis applies conservation of momentum and continuity to a horizontal rectangular channel (the classical Belanger approach):
Froude number: $$Fr_1 = \frac{v_1}{\sqrt{g h_1}}$$ A value Fr₁ > 1 indicates supercritical flow and a jump will form.
Conjugate (sequent) depth: $$h_2 = \frac{h_1}{2}\left(\sqrt{1 + 8 Fr_1^2} - 1\right)$$
Downstream velocity by continuity (constant discharge per unit width q = v₁h₁ = v₂h₂): $$v_2 = \frac{v_1 h_1}{h_2}$$
Energy loss: $$\Delta E = E_1 - E_2 = \left(h_1 + \frac{v_1^2}{2g}\right) - \left(h_2 + \frac{v_2^2}{2g}\right)$$ This energy is dissipated as turbulent heat within the jump roller.
A higher upstream Froude number produces a stronger jump with a larger depth ratio and greater energy dissipation. Weak jumps (Fr₁ = 1.0–1.7) show only surface undulations. Oscillating jumps (Fr₁ = 2.5–4.5) generate irregular waves. Steady, well-defined jumps (Fr₁ = 4.5–9.0) are ideal for stilling basins. Choppy, strong jumps (Fr₁ > 9) dissipate the most energy but can be destructive. The energy loss ratio indicates what fraction of the upstream specific energy is consumed by the jump.
Inputs
Results
Fast flow at 8 m/s and 0.3 m depth yields Fr₁ ≈ 4.66 — a steady jump. The depth increases to ~1.72 m and about 48% of the upstream energy is dissipated.
Inputs
Results
At Fr₁ ≈ 3.57, the depth ratio is about 4.5:1 and roughly 34% of the energy is dissipated in the jump.
A hydraulic jump is an abrupt transition in open-channel flow from supercritical (fast, shallow, Fr > 1) to subcritical (slow, deep, Fr < 1) conditions. It is characterized by intense turbulence, surface rollers, air entrainment, and significant energy dissipation.
The Froude number Fr = v/√(gh) is a dimensionless ratio of flow velocity to the speed of shallow-water gravity waves. Fr < 1 is subcritical (tranquil), Fr = 1 is critical, and Fr > 1 is supercritical (rapid). A hydraulic jump transitions from Fr > 1 to Fr < 1.
Across the jump, the momentum equation (including hydrostatic pressure forces) holds because no external horizontal forces act on the control volume. However, the intense turbulence inside the jump roller converts mechanical energy into heat, so the energy equation shows a loss.
Hydraulic jumps occur in open-channel flow (free surface). In partially filled pipes or culverts flowing with a free surface, jumps can and do form. In pressurized (full) pipe flow, the concept does not apply — instead, shock-like water hammer phenomena may occur.
A stilling basin is an engineered structure at the base of a spillway or dam designed to contain and optimize the hydraulic jump. Features like chute blocks, baffle piers, and end sills help stabilize the jump position and maximize energy dissipation to protect the downstream channel.
The Belanger equation used here is specific to wide rectangular channels. For trapezoidal, circular, or irregular cross-sections, the momentum equation must be solved numerically using the actual area and centroid relationships for the given geometry.
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