Hydraulic Jump

What a hydraulic jump is

Open the kitchen tap and watch the water hit the sink. Around the point of impact there's a bright, glassy disc where the water races outward in a thin sheet you can barely see. Then, at some radius, the surface abruptly jumps up into a thicker, churning ring. That ring is a hydraulic jump — and the exact same phenomenon, scaled up a million times, is what protects the foundation of every large dam on Earth.

The jump is what happens when water flowing fast and shallow is forced to become slow and deep. The fast, shallow state is called supercritical flow; the slow, deep state is subcritical. The dividing line between them is the Froude number, Fr = V / √(g·y), the ratio of the flow speed to the speed of a small gravity surface wave. When Fr > 1 the water moves faster than its own waves, so downstream disturbances cannot travel upstream — the flow is "blind" to what's ahead. When the flow must slow down to match a deeper downstream condition, it can't ease into it gradually. It collides with itself. The collision is the jump: depth rises abruptly over a short reach, velocity collapses, and the excess kinetic energy is shredded into turbulence and heat in a stationary, foaming roller.

This is the open-channel twin of a normal shock wave in supersonic gas flow. There, flow faster than sound (Mach > 1) cannot smoothly decelerate to subsonic, so it slams through a thin shock where pressure and density jump and stagnation energy is lost. Swap "speed of sound" for "speed of a surface wave" and "Mach number" for "Froude number" and the mathematics is nearly identical. Both are governed by momentum, not energy, because both destroy energy in the transition.

How the jump works: momentum, not energy

The single most important thing to understand about a hydraulic jump is why you cannot use Bernoulli's equation across it. Bernoulli conserves energy. A jump destroys energy — that's its entire purpose. The conserved quantity across the jump is the momentum flux plus pressure force, a combination engineers package into the specific force (or momentum function):

Specific force per unit width (rectangular channel):

  M = (q² / (g·y)) + (y² / 2)        [units: m²]

where  q = discharge per unit width (m²/s)
       y = flow depth (m)
       g = 9.81 m/s²

Across a jump the specific force is equal on both sides:  M₁ = M₂.
Setting M₁ = M₂ and using Fr₁ = q / √(g·y₁³) gives the Bélanger equation:

  y₂ / y₁ = ½ · ( √(1 + 8·Fr₁²) − 1 )

y₁ = upstream (supercritical) depth, y₂ = sequent / conjugate depth (subcritical).

The two depths y₁ and y₂ that share the same specific force are called sequent depths (or conjugate depths). Note they do not share the same specific energy — that's the whole point. Once you know y₂, the energy actually lost in the jump comes out of a clean algebraic identity:

Energy loss across the jump:

  ΔE = E₁ − E₂ = (y₂ − y₁)³ / (4·y₁·y₂)        [units: m of head]

Relative loss ΔE / E₁ grows fast with Fr₁:

  Fr₁ = 2   →  ΔE/E₁ ≈  9%       (weak jump, low rolling surface)
  Fr₁ = 5   →  ΔE/E₁ ≈  50%      (steady, well-formed jump)
  Fr₁ = 9   →  ΔE/E₁ ≈  70%
  Fr₁ = 15  →  ΔE/E₁ ≈  85%      (strong jump, very rough)

That cubic in (y₂ − y₁) is the magic. Because the loss scales with the cube of the depth jump, doubling the height of the standing wave roughly octuples the energy you destroy. A dam designer who needs to kill a huge amount of energy doesn't fight the flow with brute mass — they engineer the geometry so the jump forms at the highest practical Froude number and let the cubic do the work.

Classifying jumps by Froude number

The US Bureau of Reclamation classified jumps by their upstream Froude number because the character of the jump — smooth, oscillating, stable, or rough — changes dramatically across the range, and the right energy-dissipation structure depends on which type you'll get.

Upstream Fr₁	Jump type	Behavior	Energy loss
1.0	(critical)	No jump — flow is exactly critical	0%
1.0 – 1.7	Undular jump	Standing surface ripples, no real roller	< 5%
1.7 – 2.5	Weak jump	Smooth surface rises, small surface rollers	5 – 18%
2.5 – 4.5	Oscillating jump	Unstable jet oscillates, throws surface waves downstream — avoid in design	18 – 45%
4.5 – 9.0	Steady jump	Stable, well-defined, insensitive to tailwater — best for design	45 – 70%
> 9.0	Strong jump	Rough, intermittent slugs of water, very effective dissipation	70 – 85%

The villain of the table is the oscillating jump at Fr₁ = 2.5 to 4.5. The high-velocity jet entering the basin flips erratically between the bed and the surface, launching irregular waves far downstream that can erode banks, rock barge moorings, and damage riprap. This range is precisely why the USBR Type IV stilling basin exists — it has large, widely-spaced chute blocks specifically to suppress the oscillation. Whenever a designer has a choice, they shape the approach so the jump lands in the steady 4.5 to 9 range, where it sits put and shrugs off changes in tailwater.

Worked example: a spillway toe

Take a medium concrete gravity dam. Water leaves the spillway chute at the toe with a unit discharge q = 12 m²/s at a supercritical depth y₁ = 0.80 m. Find the jump.

Approach velocity:   V₁ = q / y₁ = 12 / 0.80 = 15.0 m/s
Approach Froude:     Fr₁ = V₁ / √(g·y₁) = 15.0 / √(9.81 × 0.80)
                         = 15.0 / 2.80 = 5.36   → steady jump

Sequent depth:       y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1)
                         = 0.40 × (√(1 + 8×28.7) − 1)
                         = 0.40 × (√230.6 − 1)
                         = 0.40 × 14.18 = 5.67 m

Downstream velocity: V₂ = q / y₂ = 12 / 5.67 = 2.12 m/s   (Fr₂ = 0.28 ✓ subcritical)

Energy loss:         ΔE = (y₂ − y₁)³ / (4·y₁·y₂)
                         = (5.67 − 0.80)³ / (4 × 0.80 × 5.67)
                         = 115.5 / 18.1 = 6.38 m of head

Power dissipated per metre of width:
  P = ρ·g·q·ΔE = 1000 × 9.81 × 12 × 6.38 ≈ 751 kW per metre

For a 40 m wide spillway that's ~30 MW of erosive power
destroyed inside the basin instead of out in the river.

The flow leaves the chute at 15 m/s — fast enough to tear a plunge pool 20 m deep into solid rock over a few flood seasons. The jump turns that velocity into a 2 m/s amble, converting roughly 6.4 m of head into turbulence. Sizing the concrete apron to hold this jump (length roughly 6·y₂ ≈ 34 m for a steady jump) is far cheaper than rebuilding a scoured foundation.

Engineered jumps: stilling basins

A bare jump on a flat apron works, but it's long and its position wanders as the discharge and tailwater change. The USBR's research at the mid-20th century produced standardized basins with chute blocks, baffle (impact) blocks, and end sills that anchor the jump, shorten it, and stabilize it against off-design flows.

Basin	Design Fr₁ / velocity	Internal features	Typical use
USBR Type I	Fr₁ < ~1.7	Plain horizontal apron, no blocks	Low-head structures, canal drops
USBR Type II	Fr₁ > 4.5, V₁ > ~18 m/s	Chute blocks + dentated end sill (no baffle blocks — too fast, would cavitate)	High dams, large spillways
USBR Type III	Fr₁ > 4.5, V₁ < ~18 m/s	Chute blocks + baffle blocks + solid end sill — shortest basin	Small dams, weirs, outlet works
USBR Type IV	Fr₁ = 2.5 – 4.5	Large deflector chute blocks to fight wave-making	Canal structures, low-head diversion dams
SAF basin	Fr₁ = 1.7 – 17	Compact economical layout (St. Anthony Falls)	Small drainage / soil-conservation works

Chute blocks at the basin entrance split and aerate the incoming jet, lifting part of it and shortening the jump. Baffle blocks mid-basin take a direct hit from the high-velocity flow — the dynamic pressure on the upstream face scales as p ≈ ½·ρ·V₁² (so a 15 m/s jet loads the face at on the order of 100 kPa), so they're avoided above about 18 m/s where the negative pressure on their downstream face would cavitate and pit the concrete. The end sill (solid or dentated) lifts the exiting flow off the bed and directs the bottom roller back into the basin, killing the scour-causing jet before it reaches the natural channel. A correctly tuned Type III basin can be 30–40% shorter than the equivalent bare apron, which on a large project is millions of cubic metres of saved concrete and excavation.

The tailwater problem: why jumps drift

A jump only sits at the toe of the structure if the downstream water level — the tailwater — exactly matches the sequent depth y₂ that the approach flow demands. This is the chronic headache of dissipator design, because tailwater is set by the downstream channel, not by the dam:

• Tailwater too low (y_t < y₂). The jump can't form against insufficient downstream depth, so it sweeps out — the supercritical jet shoots straight across the apron and forms the jump out in the unlined river, where it scours a hole at the worst possible place. The classic failure. Designers add an end sill or depress the basin floor below the river bed to guarantee enough depth.
• Tailwater too high (y_t > y₂). The jump is submerged (drowned) — it's pushed back upstream onto the chute and becomes a deep, sluggish surface roller. It dissipates much less energy and can carry high-velocity flow far downstream along the bed, again risking scour.
• The basin is tuned to a single discharge, but a real spillway runs from a trickle to a design flood. The jump's required y₂ changes with discharge while the rating curve of the river sets a different tailwater for each flow. The design must hold an acceptable jump across the whole operating band, usually checked by overlaying the sequent-depth curve on the tailwater rating curve and reading off where they cross.

Where hydraulic jumps appear

• Dam spillway toes. The headline application. Every large overflow or chute spillway ends in a stilling basin or flip bucket whose job is to trigger and contain a jump (or throw the flow clear and let a jump form in a plunge pool). The Bonneville, Grand Coulee, and countless smaller dams rely on engineered jumps.
• Below sluice and radial (Tainter) gates. Flow shoots out under a partially-open gate as a thin supercritical jet; a downstream apron forces the jump that protects the channel.
• Canal drops and chutes. Irrigation networks lose elevation in steps; each drop ends in a small basin that jumps the flow back to a navigable, non-erosive subcritical state.
• Culvert and storm-drain outlets. Pipes discharging fast onto soft ground need an outlet structure that jumps the flow before it undermines the headwall.
• Aeration and mixing. The intense turbulence entrains huge volumes of air, so jumps are deliberately used in wastewater treatment and chemical dosing to mix and oxygenate — a jump can raise dissolved oxygen and blend chlorine in metres of channel.
• Tidal bores. A river-mouth bore (the Severn in England, the Qiantang in China, the Amazon's pororoca) is a hydraulic jump that moves — a translating jump where the incoming tide overruns the seaward river flow.
• The kitchen sink. The circular ridge around the tap stream is the everyday teacher's version, and a genuine radial hydraulic jump.

Common misconceptions and pitfalls

• "Use Bernoulli across the jump." The most common student error and a real design trap. Energy is not conserved through a jump; momentum is. Apply the specific-force / momentum equation to find y₂, then compute the energy loss as a result — never assume E₁ = E₂.
• "A jump always saves the structure." Only if it lands where you want it. A swept-out jump moves the scour into the unprotected river; a drowned jump leaks bed velocity downstream. Matching sequent depth to tailwater across the full discharge range is the actual engineering job.
• "Bigger jump, more turbulence, always better." Strong jumps (Fr₁ > 9) are excellent dissipators but extremely rough — they fling spray, vibrate the structure, and the high local velocities around baffle blocks invite cavitation damage. Above ~18 m/s designers drop baffle blocks entirely (Type II) to avoid pitting the concrete.
• "The undular jump dissipates energy." Near Fr₁ = 1 the jump is just a train of standing surface waves with almost no roller and under 5% loss. It's not a useful dissipator; it's a sign your Froude number is too low for the job.
• "Jump length is fixed." There's no exact analytical jump length — it's an empirical fit, roughly L ≈ 6·y₂ for steady jumps (USBR curves), and it shifts with basin geometry. Sizing the apron from a single textbook multiplier without checking the USBR/SAF design curves for your Froude range under-builds the basin.
• "Aeration is a side effect to ignore." The air a jump entrains bulks the flow (the visible white water is mostly air), changes the effective density, and can drive the surface higher than the clear-water sequent depth — basin walls are freeboarded for it.