Hydraulic Jump — Reference
The rapid transition from supercritical to subcritical flow used to dissipate energy below spillways, chutes, and outlets. Subscript 1 = upstream (supercritical) section, 2 = downstream (subcritical) section.
Core Equations (Rectangular Channel)
The relation is reversible: y&sub1;/y&sub2; = ½(√(1 + 8·Fr&sub2;²) − 1). The two depths are conjugate (equal specific force), not equal specific energy — the difference is the dissipated ΔE.
Jump Classification by Upstream Froude Number
| Fr&sub1; | Type | Character & energy dissipation |
|---|---|---|
| 1.0–1.7 | Undular | Standing waves, minimal loss (< 5%) |
| 1.7–2.5 | Weak | Smooth surface, low loss (5–15%) |
| 2.5–4.5 | Oscillating | Avoid — jet oscillates, sends damaging waves downstream (15–45%) |
| 4.5–9.0 | Steady | Stable, well-balanced, best performance (45–70%) |
| > 9.0 | Strong | Rough, intense turbulence, very effective (up to ~85%) |
Sources: Chow, V.T. (1959), Open-Channel Hydraulics. USBR, Hydraulic Design of Stilling Basins and Energy Dissipators (Engineering Monograph 25). FHWA HEC-14.
Worked example: jump below a spillway chute
Rectangular stilling basin, b = 10 ft, Q = 400 cfs, entering depth y₁ = 1.2 ft
The number that decides whether this works is not on the card: the tailwater. The jump only forms in the basin if the downstream channel holds about 8.5 ft of depth at 400 cfs. If tailwater is lower, the jump sweeps downstream off the apron. If it is much higher, the jump drowns and the jet dives along the floor. Check the tailwater rating curve across the full range of flows, not just the design flow. The 10-year and the 100-year flow usually fall in different regions of that curve.
Common mistakes
- Treating y₁ and y₂ as alternate depths. They are conjugate depths: equal specific force, not equal energy. Reading y₂ off the specific-energy curve instead gives about 18.4 ft in the example above, more than double the 8.52 ft sequent depth.
- Using the rectangular sequent-depth equation in a trapezoidal or circular section. The ½(√(1+8Fr²)−1) form is rectangular only. Other shapes need the momentum function solved numerically.
- Taking y₁ at the toe of the chute without checking it against the chute’s own water-surface profile. A long, steep chute may still be accelerating at the toe, and Fr₁ grows with it.
- Ignoring the oscillating range (Fr₁ 2.5–4.5) at partial flows. A basin designed into the steady range at peak flow can fall into the oscillating range at the flows it sees most often.
When this card doesn’t apply
The relations assume a horizontal, smooth, prismatic rectangular channel with hydrostatic pressure at both sections. A jump on a sloping apron (USBR Type B through D jumps) has a longer, different sequent depth. Basins with chute blocks, baffle piers or end sills follow the USBR Monograph 25 design curves, not the free-jump equation. Air entrainment at very high Froude numbers also changes y₂ slightly. At dam spillways, the hydraulic model or the agency’s design criteria govern.
Related cheat sheets and tools
Get the approach depth and velocity from Manning's with the channel geometry card, design the basin in the stilling basin tool, and protect the downstream channel with riprap sizing. For routing the spillway flow that feeds the dissipator, see HydroComplete.