Hydraulic Jump Calculator
If you're interested in hydraulic engineering, this hydraulic jump calculator will be right up your alley. With its help, you can determine all the characteristics of a hydraulic jump, including the flow velocity up- and downstream, the height and length of the jump, and the total head loss. We will also provide you with the Froude number equation and teach you how to use it to identify the type of hydraulic jump.
What is a hydraulic jump?
By definition, a hydraulic jump happens when the flow of a fluid changes from supercritical to subcritical. The abrupt change in flow characteristics is accompanied by substantial energy losses, as well as turbulence in the flow. You can observe one as a sudden rise in water level with a turbulent transition zone — for example, just downstream of a spillway or a sluice gate.
To understand these terms, you first need to know what the critical depth is — the depth of flow at which the energy is at a minimum for a given discharge Q. Once you know the critical depth, you can instantly distinguish between:
- Supercritical flow (also called rapid flow) — the flow depth is lower than the critical depth; and
- Subcritical flow (also called slow flow) — the flow depth is higher than the critical depth.
The Froude number equation
Instead of analyzing the energy of the flow, hydraulic engineers use the Froude number to determine whether a flow is supercritical or subcritical. For any open channel flow, it can be calculated with the following formula:
Fr = v / √(g · D)
where:
- Fr — Froude number of the flow;
- v — Flow velocity;
- g — Gravitational acceleration; and
- D — Flow depth.
If the value of the Froude number is greater than one, the flow is supercritical. If it is smaller than one, the flow is subcritical. The value in the denominator, √(g·D), is the velocity of wave propagation, also called the wave celerity. For supercritical flow (Fr > 1), the velocity of the flow is higher than the celerity, so disturbances are transmitted downstream. For subcritical flow (Fr < 1), waves and disturbances will propagate upstream. When Fr = 1 (critical flow), any disturbance stays in one place.
Calculating the hydraulic jump properties
Our hydraulic jump calculator analyzes the phenomenon in detail. These formulas are only valid for the following assumptions: we are considering an open channel flow; the channel is rectangular and horizontal (without a slope); and the jump occurs from a supercritical to a subcritical flow.
1. Flow rate. For both upstream and downstream flows, the discharge per unit width q is equal to the flow velocity v multiplied by the flow depth:
q = v₁ · y₁ = v₂ · y₂
2. Sequent (downstream) depth. The ratio of the two conjugate depths is given by the Bélanger equation, derived from the momentum balance:
y₂ / y₁ = ½ · (√(1 + 8·Fr₁²) − 1)
3. Downstream velocity. Using continuity, once you know y₂ you get v₂ = v₁·y₁ / y₂.
4. Head loss. The energy dissipated in the jump is:
ΔE = (y₂ − y₁)³ / (4 · y₁ · y₂)
5. Jump height and length. The height of the jump is simply hj = y₂ − y₁, while the length is estimated with the empirical relation L ≈ 220 · y₁ · tanh[(Fr₁ − 1) / 22].
Types of hydraulic jumps
Based on the upstream Froude number Fr₁, hydraulic jumps are classified as follows:
- Undular jump (1.0 < Fr₁ < 1.7) — the water surface shows slight undulations; energy loss is minimal.
- Weak jump (1.7 ≤ Fr₁ < 2.5) — small rollers form on the surface; the downstream water surface remains smooth.
- Oscillating jump (2.5 ≤ Fr₁ < 4.5) — an oscillating jet enters the jump, generating large waves that can travel far downstream.
- Steady jump (4.5 ≤ Fr₁ < 9.0) — well-balanced and stable; this is the best range for energy-dissipating structures (45%–70% energy loss).
- Strong jump (Fr₁ ≥ 9.0) — very rough and intense, with up to 85% energy dissipation.
Units
This calculator supports both the metric system (depths in meters, velocities in m/s) and the American / imperial system (depths in feet, velocities in ft/s). Whatever unit system you choose, every result is converted so you can read the jump properties at a glance. If you also enter the channel width, the calculator reports the total volumetric discharge Q.
If you're interested in related fluid-mechanics tools, make sure to take a look at the Reynolds number calculator and the hydraulic radius calculator too!