Hydraulic Jump Calculator

Hydraulic Jump Inputs, Ratios, and Equations

The hydraulic jump calculator is driven by three linked quantities: the upstream Froude number, the sequent-depth relationship, and the drop in specific energy across the roller. All of the formulas shown here are for unit width, which is the usual way to treat a rectangular channel when the width is constant.

Upstream Froude Number

The Froude number compares inertial forces to gravitational forces. For flow of depth y and mean velocity v in a rectangular channel, the upstream Froude number is

In this hydraulic jump calculator, the Froude number is the quickest way to tell whether the approach flow is energetic enough for a jump. The upstream depth and velocity are combined into F₁, and that value determines whether the output is a genuine jump prediction or a warning that the flow is already subcritical.

where:

• F₁ = upstream Froude number (dimensionless)
• v₁ = upstream mean velocity (m/s)
• y₁ = upstream flow depth (m)
• g = gravitational acceleration (≈ 9.81 m/s²)

When F₁ > 1, the flow is supercritical and a hydraulic jump can form if downstream conditions force a transition to subcritical flow. When F₁ < 1, the flow is already subcritical and a classical hydraulic jump does not occur.

Sequent (Downstream) Depth

For a rectangular hydraulic jump, conservation of momentum links the incoming depth to the deeper sequent depth that appears immediately after the roller. That is the depth this calculator labels y₂, and it is the key number to compare with tailwater depth or stilling basin water level.

In non-dimensional form:

y₂ / y₁ = 0.5 × [√(1 + 8 F₁²) − 1]

or, equivalently,

y₂ = 0.5 y₁ [√(1 + 8 F₁²) − 1]

The calculator uses this relationship directly, so y₂ grows rapidly as F₁ increases. A modest increase in approach velocity can therefore produce a much deeper downstream section.

Specific Energy and Energy Loss

Specific energy in this hydraulic jump calculator is the depth plus the velocity head at a given section. Across the jump, the available specific energy drops because the roller dissipates part of the flow's mechanical energy.

E = y + v² / (2g)

The idealized loss formula below estimates that head reduction between the two sequent depths. It is a compact way to express how much of the incoming motion is converted into turbulence rather than carried downstream.

ΔE = (y₂ − y₁)³ / (4 y₁ y₂)

where ΔE is expressed in meters of head. The calculator reports this value to show how effectively the hydraulic jump dissipates energy.

Reading the Hydraulic Jump Results

After you enter the upstream depth y₁ (in meters) and upstream velocity v₁ (in m/s), the calculator reports the quantities that matter most for a rectangular hydraulic jump.

• Upstream Froude number F₁: Shows whether the incoming flow is supercritical and whether a classical jump is expected. If F₁ ≤ 1, the calculator is effectively telling you that the upstream reach is not fast enough for a textbook jump.
• Sequent depth y₂: The predicted depth immediately downstream of the roller. Compare y₂ with the available tailwater depth to judge whether the jump can sit where you want it.
• Depth ratio y₂ / y₁: A compact measure of jump strength. Larger ratios mean a deeper pool after the jump and usually a more vigorous roller.
• Specific energy loss ΔE: The idealized reduction in specific energy, expressed in meters of head. Higher values mean the jump is dissipating more of the incoming energy in the basin.

In preliminary basin checks, engineers often look first at y₂. If the downstream water level is close to the predicted sequent depth, the jump is more likely to remain contained and stable. If the tailwater is too low, the roller can sweep out; if it is too high, the jump may submerge and behave differently from the ideal theory used here.

Hydraulic Jump Regimes by Upstream Froude Number

The upstream Froude number also gives a quick sense of how the hydraulic jump will look and how violently it will dissipate energy. Lower values produce milder transitions, while higher values create stronger rollers and more splash.

Use the calculated Froude number to place the jump in the nearest regime. For example, an F₁ near 5 usually falls in the steady jump range, which is why that part of the scale is often favored when a stilling basin needs controlled dissipation rather than random turbulence.

Worked Example: a spillway jump in a rectangular basin

To see the hydraulic jump calculator in a realistic setting, imagine a rectangular stilling basin below a spillway where the incoming flow is shallow but fast.

• Upstream depth: y₁ = 0.5 m
• Upstream mean velocity: v₁ = 8.0 m/s
• Gravitational acceleration: g = 9.81 m/s²

1. Compute the upstream Froude number for the jump

First calculate the denominator:

√(g y₁) = √(9.81 × 0.5) ≈ √4.905 ≈ 2.21 m^1/2/s

Then the Froude number:

F₁ = v₁ / √(g y₁) ≈ 8.0 / 2.21 ≈ 3.6

With F₁ ≈ 3.6, the hydraulic jump sits in the oscillating to steady range, so a fairly vigorous roller is expected.

2. Compute the sequent depth predicted by the jump calculator

Use the depth ratio:

y₂ / y₁ = 0.5 [√(1 + 8F₁²) − 1]

First, evaluate the expression inside the square root:

1 + 8F₁² = 1 + 8 × 3.6² ≈ 1 + 8 × 12.96 ≈ 1 + 103.68 = 104.68

Then take the square root:

√(104.68) ≈ 10.23

Now compute the bracketed term and multiply:

y₂ / y₁ ≈ 0.5 × (10.23 − 1) = 0.5 × 9.23 ≈ 4.615

Therefore,

y₂ ≈ 4.615 × y₁ = 4.615 × 0.5 ≈ 2.31 m

The water depth increases from 0.5 m upstream to about 2.3 m immediately downstream of the jump, which is the kind of rise you would expect when a strong supercritical stream slows down in a basin.

3. Estimate the specific energy loss ΔE for the hydraulic jump

Use the idealized formula:

ΔE = (y₂ − y₁)³ / (4 y₁ y₂)

First compute the difference and its cube:

y₂ − y₁ ≈ 2.31 − 0.5 = 1.81 m

(1.81)³ ≈ 5.93 m³

Then compute the denominator:

4 y₁ y₂ ≈ 4 × 0.5 × 2.31 = 4.62 m²

Finally,

ΔE ≈ 5.93 / 4.62 ≈ 1.28 m

The hydraulic jump dissipates the equivalent of about 1.3 m of head, which is a substantial drop in specific energy for a basin of this size.

In practice, you would compare the predicted sequent depth (≈ 2.3 m) with the expected tailwater depth. If the existing tailwater is much lower, the jump can sweep downstream; if it is much higher, the jump may become submerged and depart from the ideal rectangular-channel theory used by the calculator.

Assumptions and Limitations for Rectangular Hydraulic Jumps

The hydraulic jump calculator uses the classical one-dimensional relations that are normally applied to a rectangular jump in a horizontal channel. Those relations are very useful for quick checks, but they are only as good as the assumptions behind them.

• Rectangular, prismatic, horizontal channel: The formulas assume a constant-width rectangular section with a nearly level bed and no abrupt changes in geometry at the jump location.
• Steady, one-dimensional flow: Flow depth and velocity are treated as steady in time and uniform across the width, apart from the jump itself.
• Supercritical upstream conditions: The sequent-depth equations are only valid when the upstream flow is supercritical, meaning F₁ > 1. If the calculator shows F₁ below 1, the result is not a real hydraulic-jump depth.
• Neglect of bed friction and air entrainment within the jump: Detailed wall friction, aeration, and the complex structure of the roller are not modeled explicitly.
• Uniform velocity distribution: The mean velocity is used as a representative value, so non-uniform profiles and secondary currents are ignored.
• Sufficient basin length: The jump is assumed to have enough room to form and settle to the predicted sequent depth within the reach being studied.

Because of these simplifications, the outputs should be treated as idealized estimates rather than final design values. Real channels can differ because of sloping beds, sidewall effects, baffle blocks, end sills, sediment transport, or changes in tailwater depth.

For spillways, outlet works, flood channels, and other critical structures, use the calculator as a screening tool and then confirm the design with more detailed hydraulic analysis, physical modeling, or the standards that apply to your project.

Practical Use and Safety Note for Stilling Basin Checks

This hydraulic jump calculator uses the standard momentum and specific-energy relations that appear in open-channel hydraulics textbooks. It is well suited to classroom demonstrations, rough sizing, and quick checks of whether a stilling basin is in the right range.

It is not a substitute for full hydraulic design. Before relying on the predicted sequent depth or energy loss for apron sizing, basin length, baffle blocks, or outlet protection, review the site conditions, local design manuals, and tailwater data with an experienced hydraulic engineer.