Hydraulic Jump Calculator

Understanding Hydraulic Jumps

A hydraulic jump is a sudden transition from fast, shallow, supercritical flow to slower, deeper, subcritical flow in an open channel. It commonly appears where high-velocity water from a spillway, sluice gate, or steep chute enters a flatter, calmer downstream reach. The water surface rises abruptly, and a highly turbulent roller forms, dissipating a significant amount of energy.

Engineers intentionally create hydraulic jumps in stilling basins to protect the channel bed and nearby structures from scour. By converting kinetic energy into turbulence and heat over a relatively short distance, the jump reduces downstream velocities and shear stresses.

This calculator estimates the sequent (downstream) depth and the associated energy loss of a hydraulic jump in a rectangular, horizontal, prismatic channel, using upstream depth and velocity as inputs. It is based on standard one-dimensional hydraulic theory and is intended for preliminary analysis and educational use.

Key Parameters and Formulas

The behavior of a hydraulic jump is governed primarily by the upstream Froude number, the sequent depth relationship, and the change in specific energy across the jump. The calculator implements these relationships for unit channel width (or equivalently per meter of channel width).

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

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 channel, conservation of momentum across the jump leads to a relationship between the upstream depth y₁ and the downstream, or sequent, depth y₂. 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 formula to compute the depth immediately after the jump, assuming a fully developed jump in a horizontal, rectangular channel.

Specific Energy and Energy Loss

The specific energy at a cross-section is the sum of the depth and the velocity head:

E = y + v² / (2g)

Across a hydraulic jump, specific energy decreases, and the lost energy is converted into turbulence. For rectangular channels, the idealized energy loss between sequent depths y₁ and y₂ is

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.

How to Interpret the Calculator Results

After you enter the upstream depth y₁ (in meters) and upstream velocity v₁ (in m/s), the tool reports key hydraulic quantities for a potential jump.

• Upstream Froude number F₁: Indicates whether the approach flow is supercritical. If F₁ ≤ 1, a classical hydraulic jump is not expected, and the sequent depth formula is not physically applicable.
• Sequent depth y₂: The predicted depth immediately downstream of the jump. Comparing y₂ to y₁ shows how much the water surface rises through the jump.
• Depth ratio y₂ / y₁: A convenient non-dimensional measure of jump strength. Larger ratios correspond to stronger jumps and higher energy dissipation.
• Specific energy loss ΔE: The reduction in specific energy across the jump, expressed in meters of head. Higher ΔE indicates more effective energy dissipation.

In preliminary design, engineers often compare the sequent depth y₂ with the available tailwater depth. If the tailwater depth is close to the sequent depth, the jump can be contained within the stilling basin and will function efficiently. Significant mismatches may lead to swept-out or submerged jumps, which are less predictable and may require design adjustments.

Hydraulic Jump Regimes by Froude Number

Different ranges of upstream Froude number correspond to qualitatively different jump types. These regimes guide expectations about turbulence intensity, roller size, and the efficiency of energy dissipation.

Use the calculated Froude number to identify the approximate regime of the jump. For example, if F₁ ≈ 5, the jump is typically steady and well formed, which is favorable for controlled dissipation within a stilling basin.

Worked Example

Consider a rectangular stilling basin downstream of a spillway. The measured or estimated upstream conditions at the start of the basin are:

• 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

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 jump falls into the oscillating to steady regime, indicating a relatively strong and energetic jump.

2. Compute the sequent depth y₂

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.

3. Estimate the energy loss ΔE

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, substantially reducing the kinetic energy of the flow entering the downstream reach.

In practice, you would compare the predicted sequent depth (≈ 2.3 m) with the expected tailwater depth. If the existing tailwater is much lower, you may expect an unstable or swept-out jump; if it is much higher, the jump may be submerged and behave differently from the ideal theory.

Assumptions and Limitations

The hydraulic jump calculator relies on simplified, classical open-channel flow theory. Its predictions are most reliable when the following assumptions are reasonably satisfied:

• Rectangular, prismatic, horizontal channel: The formulas assume a constant-width rectangular cross-section with a nearly horizontal bed and no abrupt changes in geometry at the jump location.
• Steady, one-dimensional flow: Flow depth and velocity are assumed constant in time and uniform across the width, except through the jump itself.
• Supercritical upstream conditions: The sequent depth relationships are valid only when the upstream flow is supercritical (F₁ > 1). For subcritical flow, the computed "sequent" depth has no physical meaning as a hydraulic jump outcome.
• Neglect of bed friction and air entrainment within the jump: Momentum-based formulas typically neglect detailed frictional losses and complex air–water interactions in the roller region.
• Uniform velocity distribution: The mean velocity is used to characterize the flow. Non-uniform velocity profiles and secondary currents are not explicitly modeled.
• Sufficient basin length: The jump is assumed to be fully contained within the channel segment of interest, with enough distance to develop its characteristic roller and reach the sequent depth.

As a result, the outputs should be interpreted as idealized estimates. Real-world jumps may differ due to sloping beds, non-rectangular sections, baffle blocks, end sills, sidewall effects, tailwater variability, and sediment transport.

For critical infrastructure such as dam spillways, flood control works, or major irrigation structures, always verify preliminary calculations using more detailed hydraulic modeling tools, physical model studies, or relevant design standards and guidelines.

Practical Use and Safety Note

The hydraulic jump calculator implements widely used textbook relationships based on conservation of momentum and specific energy concepts for open-channel flow. It is suitable for teaching, quick checks, and early-stage design scoping.

It is not a substitute for comprehensive engineering design. Before relying on the results for sizing stilling basins, aprons, or protective linings, consult local codes, design manuals, and an experienced hydraulic engineer. Always consider site-specific data, including tailwater curves, sediment conditions, and structural constraints, when interpreting the results.