Hydraulic Jump Interactive Calculator

When you're designing a spillway stilling basin or drop structure, it's critical to figure out exactly where supercritical flow crashes into subcritical conditions. This happens abruptly—with a hydraulic jump. The jump doesn't ease the transition: it instantly dissipates a lot of energy over a short distance. The Hydraulic Jump Interactive Calculator here lets you work out sequent depths, energy loss, Froude numbers, jump lengths, and jump types using your upstream depth, flow velocity, and channel data. Nailing these calculations is key. If you guess low on the sequent depth, the channel can scour. Too high and you might end up drowning the jump, moving it upstream where you don't want it. Below, you'll find the core equations, a real-world worked example, design theory for jump classification, and an FAQ on practical points like sloped chutes, air entrainment, and scale model effects.

What is a hydraulic jump?

A hydraulic jump is an abrupt spot where fast shallow flow drops its speed and the water suddenly gets deeper. This isn't a smooth transition—it's highly turbulent, giving up its kinetic energy to turbulence and heat. The point is to lose energy through mixing and churning, not by eroding the channel or riverbed further downstream.

Simple Explanation

Picture water shooting across a kitchen sink: it starts as a thin, fast sheet, then pile-drives into a ring where it stacks up, forming a visible jump. That's the hydraulic jump. Engineers use this effect deliberately after dams, spillways, or drops, so the flow slows down before it hits areas that can be damaged. The bigger the change in depth, the more energy gets taken out of the flow.

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Hydraulic Jump Diagram

Hydraulic Jump Calculator

How to Use This Calculator

1. Pick the calculation mode you need—for sequent depth, energy loss, jump length, Froude numbers, or upstream depth.
2. Enter upstream depth (y₁, meters) and velocity (V₁, m/s), or downstream depth (y₂) if solving for y₁.
3. Adjust gravity (g) if you’re dealing with unusual site conditions; otherwise leave it at 9.81 m/s².
4. Click Calculate to see what you get.

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Governing Equations

Simple Example

Upstream depth y₁ = 0.5 m, upstream velocity V₁ = 6 m/s, g = 9.81 m/s².

Fr₁ = 6 / √(9.81 × 0.5) = 6 / 2.215 = 2.709 — supercritical, jump will form.

Sequent depth: y₂ = (0.5/2) × [√(1 + 8 × 2.709²) − 1] = 0.25 × [√59.77 − 1] = 0.25 × 6.732 = 1.683 m

Energy loss: E₁ = 0.5 + 36/19.62 = 2.335 m; V₂ = (0.5 × 6)/1.683 = 1.783 m/s; E₂ = 1.683 + 3.178/19.62 = 1.845 m; ΔE = 0.490 m (21% dissipated).

Use the formula below to calculate the sequent depth ratio.

Use the formula below to calculate the Froude number.

Use the formula below to calculate energy loss across the jump.

Use the formula below to calculate jump length.

Use the formula below to calculate volumetric flow rate using the continuity equation.

Theory & Practical Applications

A hydraulic jump is one of the more abrupt and energy-heavy changes you’ll get in open channel flow. If you have high-velocity, shallow flow with Froude number greater than 1 (supercritical), and downstream conditions force it to slow to Fr < 1 (subcritical), the change happens immediately at the jump. Think of it as a stationary shock: the water piles up, loses speed, and most of the incoming energy is dumped into turbulence and churning. This isn't just a math exercise—this is the straightforward way to dissipate energy in real engineering.

Momentum Analysis and the Bélanger Equation

All the key relationships for hydraulic jumps in rectangular channels come straight from momentum balance—not energy. Ignore channel slope and friction over the short length of the jump and assume the pressures are hydrostatic before and after; you end up with a constant specific force across the jump. For a rectangular channel, you eventually get:

M₁ = M₂ → (ρgA₁²/2Q²) + Q²/(gA₁) = (ρgA₂²/2Q²) + Q²/(gA₂)

After simplifying with Froude number and continuity equations, you have Bélanger's formula for sequent depths. There's always a unique y₂ for any upstream Fr₁ > 1. Note: this depth ratio jumps up sharply as Fr₁ rises—at Fr₁ = 2, y₂ is about 2.3 times y₁; at Fr₁ = 10, it's about 13 times y₁. The water doesn't mix over a long distance—a strong jump forms a deep, sudden roller in a few meters.

Energy Dissipation Characteristics

Unlike gentle transitions where energy loss is minor, a jump chews up a lot of energy as turbulence. The higher Fr₁ goes, the less efficient the jump: at Fr₁ = 1.7, you lose about 5% of the specific energy. At Fr₁ = 5.0 (a typical steady jump), it’s closer to 45%. Strong jumps (Fr₁ > 9.0) can destroy 70% or more. This is good for energy dissipation but rough on structures: these are high-vibration, high-wear areas if not designed with that in mind.

The energy goes into big vortices, then smaller turbulent eddies, and a lot of air gets mixed in—forming two-phase bubbly flow. The surface roller is the main mixing zone: surface velocities nearly stop while the bottom continues moving along. If you’ve ever watched high-speed video, you’ll see periodic vortex 'bursts' rolling downstream, which can put cyclic stresses on concrete if you don’t reinforce the basin properly.

Jump Classification and Flow Regimes

In practical design, jumps are grouped by upstream Froude number, since their behavior changes a lot with Fr₁. Here's how they play out in real channels:

Undular Jump (1.0 < Fr₁ < 1.7): Surface just forms standing ripples, not a real roller. Little energy loss—under 5%. You’ll see these after some weirs, but they're no good for energy dissipation if you really need it. The problem can be downstream waves sneaking into unprotected zones.

Weak Jump (1.7 < Fr₁ < 2.5): You get a small roller, smooth downstream. Energy loss is 5-15%. Handy for small drop structures but can fall apart with flow variations.

Oscillating Jump (2.5 < Fr₁ < 4.5): The flow jet swings up and down, bed to surface. Unpredictable pressure pulses and 15-45% energy loss. Most awkward for design—can cause structure vibration. Try to avoid this region by changing tailwater or approach depth.

Steady Jump (4.5 < Fr₁ < 9.0): Well-behaved jump with a good roller, 45-70% energy loss. This is where you want to be for most energy dissipation basins. Jump length is predictable and designs are straightforward.

Strong Jump (Fr₁ > 9.0): Very turbulent, big waves and spray, over 70% energy loss. Now you may need extra baffle and end sills—otherwise, you risk the jump moving or eroding downstream.

Stilling Basin Design Applications

The go-to use for hydraulic jumps is eating up kinetic energy after things like spillways, sluice gates, or irrigation drops. The U.S. Bureau of Reclamation (USBR) sorts stilling basins by their target Froude numbers:

Type I Basin: Basic rectangular basin for Fr₁ < 1.7. No fancy blocks, length about four times the downstream depth.

Type II Basin: For Fr₁ > 4.5. Add chute blocks at the entrance, baffle blocks, and an end sill—cuts the required basin length by a third or more. Chute blocks split the jet, baffle blocks chop up the roller, end sill boosts tailwater locally to hold the jump in place.

Type III Basin: For Fr₁ = 4.5 to 17 when you’d otherwise need a huge basin. Use closely spaced baffle blocks and a sloping apron. Compact, but you’ll be repairing these more often if fluids cavitate.

The crucial design step is matching tailwater to y₂. If tailwater is too low, the jump shoots downstream—no energy dissipation where you want it. If too high, the jump drowns and shifts upstream—possibly right onto the spillway face, which is bad for both cavitation and concrete. Tailwater has to be checked for seasonal and operation variations, not just one design flood number.

Advanced Considerations: Conjugate Depths and Jump Location

Conjugate depths y₁ and y₂ mean depths upstream and downstream of the jump for the same momentum. However, the jump doesn’t necessarily form the instant those depths are present. The actual jump sets up where the rapid flow profile meets the slower, subcritical profile, given real-world boundary conditions. You often have to force the jump to stay put—with sills, walls, or by expanding the channel abruptly. If you don’t, the jump might drift or shift with changing flows, making your protection useless. For critical projects, CFD modeling can pick up on three-dimensional effects like sidewall flows or corners, which basic theory can’t handle.

Worked Example: Spillway Stilling Basin Design

A spillway chute discharges onto a rectangular basin 12 m wide. At design flow, depth y₁ = 0.85 m, velocity V₁ = 8.5 m/s. Downstream, you need at least 3.2 m depth to avoid scouring the bed. Will a jump form, how does it behave, and what’s the energy loss?

Step 1: Calculate upstream Froude number

Fr₁ = V₁ / √(g·y₁) = 8.5 / √(9.81 × 0.85) = 2.942

So, Fr₁ > 1; jump can occur.

Step 2: Calculate sequent depth y₂

y₂ = (0.85/2) × [√(1 + 8(2.942)²) - 1] = 0.425 × (8.381 - 1) = 3.137 m

Step 3: Compare to tailwater

Tailwater = 3.2 m, just above required y₂. The jump will form, slightly submerged, and position shifts upstream a little—that’s fine in practice. If this excess gets much bigger you’d want to tune your basin or controls.

Step 4: Downstream flow conditions

V₂ = V₁·y₁/y₂ = 8.5 × 0.85 / 3.137 = 2.303 m/s. Fr₂ = V₂ / √(g·y₂) = 2.303 / √(9.81 × 3.137) = 0.415. That’s safely subcritical.

Step 5: Energy loss

E₁ = y₁ + V₁²/(2g) = 0.85 + 3.682 = 4.532 m

E₂ = y₂ + V₂²/(2g) = 3.137 + 0.270 = 3.407 m

ΔE = 1.125 m (about 25% relative loss)

Step 6: Power dissipated

Q = V₁·b·y₁ = 8.5 × 12.0 × 0.85 = 86.7 m³/s

P = 1000 × 9.81 × 86.7 × 1.125 ≈ 956 kW

Nearly a megawatt of energy is neutralized in the roller—energy that can’t rip out your channel after the stilling basin.

Step 7: Jump length and type

Lⱼ ≈ 6.9 × (3.137 - 0.85) = 15.8 m

Fr₁ = 2.942 is in the oscillating jump range. Standard fix is the Type II basin with blocks to steady things—jump length shrinks, and it won’t vibrate as badly. Keep an eye on vibration and run pressure monitoring if needed. Another approach is to adjust spillway geometry to push Fr₁ into the ‘steady jump’ range, or at least stay clear of the oscillating band.

This example shows: if you’re in the oscillating region, use baffles and appurtenances to keep the jump in place and cut down on irregular surging. You can also adjust geometry to move Fr₁ into a more stable range if the structure allows.

Industry-Specific Applications

Hydroelectric Power: High-flow stilling basins (like at Grand Coulee or Hoover Dam) need big jumps. Here, designers use aeration slots and buckets to prevent vapor bubbles and minimize damage from violent jumps.

Irrigation Networks: Canals use weak jumps for energy dissipation at grade drops. These systems have to stay stable with big flow swings; design tuning is often needed.

Wastewater Treatment: Jumps can help mix chemicals and boost aeration by sucking in air, improving oxygen for downstream processes. Be careful: excess turbulence can break up delicate material in the flow.

Stormwater Management: Jumps slow flow at culvert outlets before it hits natural channels. Urban retrofit jobs often use compact, high-maintenance basins due to space limits.

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Frequently Asked Questions

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