Hydraulic Jump Calculator

Background.

Water rushing out from under a sluice gate or down a spillway chute is shallow and fast. Downstream, where the channel is deeper and the flow is slow, it cannot stay that way. The change happens abruptly, in a turbulent standing wave called a hydraulic jump: the surface leaps upward, the water churns and fills with air, and a large share of the flow's energy is turned into heat and noise. The same thing happens on a small scale in a kitchen sink, where the thin fast sheet spreading from the tap stream suddenly thickens into a ring.

Engineers build jumps on purpose. A stilling basin at the foot of a dam spillway is designed to hold a jump in place so that the destructive energy of the falling water is spent on a concrete apron rather than on the riverbed. Getting it right starts with one number: the depth the water will reach after the jump, called the conjugate or sequent depth. If the tailwater downstream is lower than that, the jump is swept downstream; if it is much higher, the jump drowns.

The conjugate depth comes from a momentum balance across the jump, first worked out by Bélanger in the nineteenth century. It needs only the incoming depth and the upstream Froude number, the ratio of the flow speed to the speed of a shallow-water wave. A jump can only form when that number is above 1, meaning the flow is supercritical. The relation is the ideal one for a horizontal rectangular channel. It assumes friction across the short jump is negligible and says nothing about where the jump will sit, how long it is, or how a sloping or widening channel changes it.

What is hydraulic jump calculator?

A hydraulic jump is the sudden transition from supercritical flow, shallow and fast with a Froude number above 1, to subcritical flow, deep and slow with a Froude number below 1. The conjugate depth y₂ is the downstream depth that balances the momentum of the incoming flow. For a horizontal rectangular channel the ratio y₂/y₁ depends only on the upstream Froude number, so the same Froude number always gives the same depth ratio whatever the size of the channel.

How to use this calculator.

1. 01Enter the upstream depth y1 in metres: the depth of the fast, shallow flow just before the jump, for example at the vena contracta below a sluice gate.
2. 02Enter the upstream Froude number. If you know the velocity V instead, compute it as V ÷ √(g × y1) with g = 9.81 m/s²; a value of 1 or less is refused because no jump can form.
3. 03Read the conjugate downstream depth in metres and compare it with the tailwater depth your channel actually provides.
4. 04Check the model-scope note: the result assumes a horizontal rectangular channel and does not size a stilling basin or locate the jump.

The formula.

y₂ = y₁ × [√(1 + 8Fr₁²) − 1] ÷ 2. Here y₁ is the upstream depth, Fr₁ the upstream Froude number V₁ ÷ √(g y₁), and y₂ the conjugate depth. The equation comes from equating the momentum flux plus hydrostatic pressure force on each side of the jump, per unit width of a rectangular channel, and solving the resulting quadratic for the positive root. Energy is not conserved across a jump, so energy cannot be used in its place; it is momentum that carries through. The constant 8 arises from that algebra, not from measurement. The Froude number dominates the answer: for large Fr₁ the ratio y₂/y₁ grows almost in proportion to it, approaching √2 × Fr₁, while y₁ simply scales the result, so doubling the incoming depth at the same Froude number doubles y₂.

A worked example.

Frequently asked questions.

References& sources.

1. [1]USBR, Hydraulic Design of Stilling Basins and Energy Dissipators, Engineering Monograph 25.
2. [2]USACE EM 1110-2-1603, Hydraulic Design of Spillways.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.