Hydraulic Jump Calculator
Enter channel width, discharge and upstream depth to solve a hydraulic jump in a horizontal rectangular channel. The calculator applies the Belanger equation to find conjugate (sequent) depth, then outputs upstream and downstream velocities, both Froude numbers, head loss, jump height, jump length, jump efficiency, and the Chow (1959) jump type. Switch between metric and imperial units below.
Formula
Worked example
Channel: B = 2 m, Q = 5 m³/s, y₁ = 0.3 m. V₁ = 5 / (2 × 0.3) = 8.333 m/s. Fr₁ = 8.333 / √(9.81 × 0.3) = 4.86 (steady jump). y₂ = 0.5 × 0.3 × (√(1 + 8 × 4.86²) - 1) = 1.888 m. Head loss ΔE = (1.888 - 0.3)³ / (4 × 0.3 × 1.888) = 3.48 m.
What is a hydraulic jump?
A hydraulic jump is the abrupt transition from supercritical flow (fast, shallow) to subcritical flow (slow, deep) in an open channel. It occurs when the upstream Froude number exceeds 1.0 and the downstream channel cannot sustain supercritical conditions. The jump is marked by a steep, turbulent roller region where kinetic energy is violently converted to potential energy and heat. Engineers deliberately induce hydraulic jumps at the base of spillways, sluice gates and weirs to dissipate excess energy before it can erode the channel bed or damage structures. Understanding jump location, conjugate depth and energy loss is therefore central to hydraulic design.
The Belanger equation and sequent depth
The key formula for a hydraulic jump in a horizontal rectangular channel is the Belanger equation: y₂/y₁ = 0.5 × (√(1 + 8Fr₁²) - 1), where y₁ is the upstream (supercritical) depth and y₂ is the conjugate or sequent depth downstream. It derives from simultaneous application of the momentum equation and continuity equation while ignoring wall friction - a valid assumption over the short length of the jump. The ratio grows rapidly with Froude number: at Fr₁ = 2 the downstream depth is roughly 2.6 times the upstream depth, while at Fr₁ = 9 it reaches about 12 times. The head loss formula ΔE = (y₂ - y₁)³ / (4y₁y₂) follows from the difference in specific energy upstream and downstream. Head loss is always positive for a real jump (Fr₁ > 1), confirming that energy is dissipated, never created.
Jump type, efficiency and engineering application
Chow (1959) divides hydraulic jumps into five types based on upstream Froude number. Undular jumps (Fr₁ 1.0-1.7) dissipate very little energy and produce oscillating waves that persist far downstream. Weak jumps (Fr₁ 1.7-2.5) are well-formed but low-energy; oscillating jumps (Fr₁ 2.5-4.5) produce erratic standing waves and are avoided in design. Steady jumps (Fr₁ 4.5-9) are the hydraulic engineer’s workhorse: highly stable with 45-70% energy dissipation, they form the basis of USBR stilling basin designs. Strong jumps (Fr₁ > 9) dissipate 70% or more of specific energy but are extremely turbulent and can cause violent scour if the apron is not protected. Jump efficiency (η = E₂/E₁) quantifies what fraction of upstream specific energy survives the jump, complementing head loss as a design parameter.
Using the calculator in practice
Enter the channel width (B), total discharge (Q) and upstream depth (y₁) to solve all jump parameters. Choose metric (metres and m³/s) or imperial (feet and ft³/s) units. The gravitational acceleration field defaults to 9.81 m/s² and only needs changing for tidal or planetary applications. If the upstream Froude number is less than or equal to 1, flow is already subcritical and no jump will form; you may need to adjust Q or y₁. The jump length formula (L = 220 y₁ tanh[(Fr₁ - 1)/22]) by Hager gives the estimated roller length, useful for sizing an apron or stilling basin. Always verify that tailwater depth matches the conjugate depth; if tailwater is too low the jump will be swept out, and if it is too high the jump will submerge and lose effectiveness. The chart plots how head loss and conjugate depth vary with upstream depth at constant Q and B, helping you visualize sensitivity to operating conditions.
Hydraulic Jump Classification (Chow, 1959)
| Upstream Fr₁ | Jump type | Characteristics | Energy dissipation |
|---|---|---|---|
| < 1.0 | No jump | Subcritical flow - no transition possible | None |
| 1.0 - 1.7 | Undular jump | Surface undulations, low turbulence | Very low (< 5%) |
| 1.7 - 2.5 | Weak jump | Small rollers, smooth surface downstream | Low (5-15%) |
| 2.5 - 4.5 | Oscillating jump | Irregular waves, not for design use | Moderate (15-45%) |
| 4.5 - 9.0 | Steady jump | Stable, high dissipation, preferred | High (45-70%) |
| > 9.0 | Strong jump | Intense turbulence, potential scour | Very high (> 70%) |
Classification by upstream Froude number for horizontal rectangular channels. Steady jumps (Fr₁ 4.5-9) are preferred for stilling basin design.
Frequently asked questions
What does the upstream Froude number need to be for a hydraulic jump to occur?
The upstream Froude number (Fr₁) must be greater than 1.0. A value exactly equal to 1.0 is critical flow, and below 1.0 the flow is subcritical, so no jump is possible. In practice, a minimum of about Fr₁ = 1.5 to 2.0 is needed for a well-formed jump; below this threshold only an undular surface disturbance appears.
What is the conjugate (sequent) depth and how is it different from critical depth?
Conjugate depth (also called sequent depth) is the downstream water depth that satisfies both the momentum equation and continuity equation simultaneously with the given upstream depth. It is the only depth at which the jump is in equilibrium. Critical depth is the depth at which specific energy is minimised for a given discharge, corresponding to Fr = 1. The two are completely different values: conjugate depth is always greater than critical depth downstream of a jump, while critical depth falls between y₁ and y₂.
Why do steady hydraulic jumps (Fr₁ 4.5-9) give the best energy dissipation?
Steady jumps strike the best balance between energy dissipation and predictability. Their roller is compact and stable, so the hydraulic action stays within the designed apron. Below Fr₁ = 4.5, oscillating jumps produce unpredictable wave trains; above Fr₁ = 9, strong jumps are so violent that they risk scour even with apron protection. Most dam spillway stilling basins are designed to produce a steady jump under the worst flood conditions.
How is the jump length formula derived, and is it accurate?
The formula L = 220 y₁ tanh[(Fr₁ - 1)/22] is an empirical fit to flume data by Hager and Bremen (1989), and it is accurate to about 15% for Froude numbers between 1.5 and 13 in rectangular channels. Because jump length is notoriously hard to measure (the roller blends gradually into the main flow), all jump-length formulas carry significant uncertainty. Use the result as a conservative estimate and allow extra apron length as a factor of safety.
Does this calculator work for non-rectangular channels (trapezoidal, circular)?
No. The Belanger equation and the head-loss formula used here are valid only for horizontal, prismatic rectangular channels. Trapezoidal, triangular and circular channels require iterative solutions to the momentum equation that account for the varying width with depth. For those geometries, consult a specialist open-channel hydraulics solver or the HEC-RAS software.
What is jump efficiency and why does it matter?
Jump efficiency (η) is the ratio of downstream specific energy (E₂) to upstream specific energy (E₁), expressed as a percentage. An efficiency of 55% means the jump has dissipated 45% of the incoming kinetic energy as turbulence and heat. Higher efficiency values mean more energy is retained and therefore more remains to potentially cause erosion downstream. Knowing efficiency helps engineers decide whether additional apron protection or a secondary energy dissipator is needed.