Hydraulic Jump Depth Calculator
Calculates the sequent (conjugate) depth downstream of a hydraulic jump in open channel flow. Used by hydraulic engineers to design energy dissipators below spillways, sluice gates, and stilling basins.
About this calculator
A hydraulic jump occurs when supercritical open-channel flow (Froude number > 1) abruptly transitions to subcritical flow (Froude number < 1), dissipating kinetic energy as turbulence. The sequent depth y₂ is found from the Belanger equation: y₂ = (y₁ / 2) × (−1 + √(1 + 8 × v₁² / (g × y₁))), where y₁ is the upstream depth (m), v₁ is the upstream velocity (m/s), and g is gravitational acceleration (9.81 m/s²). The term v₁² / (g × y₁) is the square of the upstream Froude number (Fr₁²). A hydraulic jump only forms when Fr₁ > 1. The energy loss across the jump is ΔE = (y₂ − y₁)³ / (4 × y₁ × y₂), and this dissipated energy is converted to heat and sound, protecting downstream channel beds from erosion. Stronger jumps (higher Fr₁) produce greater sequent depth ratios and higher energy losses.
How to use
Supercritical water flows through a rectangular channel at an upstream depth of y₁ = 0.3 m and velocity v₁ = 4 m/s. Use g = 9.81 m/s². Apply y₂ = (0.3 / 2) × (−1 + √(1 + 8 × 4² / (9.81 × 0.3))). Step 1: 8 × 16 / (9.81 × 0.3) = 128 / 2.943 ≈ 43.49. Step 2: √(1 + 43.49) = √44.49 ≈ 6.670. Step 3: y₂ = 0.15 × (−1 + 6.670) = 0.15 × 5.670 ≈ 0.851 m. The flow depth increases from 0.3 m to 0.851 m across the jump. Enter y₁ = 0.3 m, v₁ = 4 m/s, and g = 9.81 m/s² to verify instantly.
Frequently asked questions
What is a hydraulic jump and why does it occur in open channel flow?
A hydraulic jump is a sudden, standing wave that forms when fast-moving (supercritical) water decelerates rapidly to slow-moving (subcritical) conditions, analogous to a shockwave in compressible gas flow. It occurs because the flow cannot smoothly transition between the two regimes — instead it releases excess kinetic energy violently as turbulence, foam, and heat. Common triggers include a channel slope change from steep to mild, a raised sill, or the outflow from a sluice gate meeting a tailwater condition that forces subcritical depth. Engineers deliberately induce hydraulic jumps in stilling basins to prevent erosion of the downstream channel bed.
How do I calculate energy loss across a hydraulic jump?
Once you know the conjugate depths y₁ and y₂, the specific energy loss is: ΔE = (y₂ − y₁)³ / (4 × y₁ × y₂). This formula comes from applying the energy equation between sections 1 and 2 after using momentum conservation to relate the depths. For the worked example above (y₁ = 0.3 m, y₂ = 0.851 m): (0.851 − 0.3)³ = 0.551³ ≈ 0.167 m³; 4 × 0.3 × 0.851 = 1.021 m²; ΔE ≈ 0.167 / 1.021 ≈ 0.164 m. This 0.164 m of head is permanently lost, which can represent a significant fraction of total upstream energy in high-Froude-number jumps.
What Froude number is needed for a hydraulic jump to form, and how is it classified?
A hydraulic jump requires an upstream Froude number Fr₁ = v₁ / √(g × y₁) strictly greater than 1.0. The intensity of the jump is classified by Fr₁: undular jumps (1.0–1.7) show gentle surface waves and low energy dissipation; weak jumps (1.7–2.5) produce a smooth rise with some turbulence; oscillating jumps (2.5–4.5) are unstable and undesirable in design; steady jumps (4.5–9.0) are most efficient for stilling basins; and strong jumps (Fr₁ > 9.0) dissipate up to 85% of the incoming energy but generate extreme turbulence. Engineers typically target the steady jump range when designing energy dissipators.