What this calculator does
The Froude number compares how fast water is flowing against how fast a wave can travel on its surface. Below one, waves can travel upstream and the flow is called subcritical; above one they cannot, and the flow is supercritical.
That distinction governs how a channel behaves. Subcritical flow is deep, slow and tranquil; supercritical flow is shallow and fast, and the transition between them happens abruptly in a hydraulic jump.
The formula
Divide the flow velocity by the square root of gravity times the hydraulic mean depth, which is the cross-sectional area divided by the surface width.
| Term | Meaning |
|---|---|
| Froude number (Fr) | Flow velocity divided by wave speed. One is the critical value. |
| Subcritical flow | Fr below 1: tranquil flow where disturbances can propagate upstream. |
| Supercritical flow | Fr above 1: rapid, shallow flow where nothing can travel back upstream. |
| Hydraulic jump | The abrupt transition from supercritical to subcritical flow, which dissipates a great deal of energy. |
The inputs explained
| Field | What to enter |
|---|---|
| Flow velocity (u) (m/s) | The average flow velocity in metres per second. |
| Cross-section area (A) (m²) | The cross-sectional area of the flow, in square metres. |
| Channel width (W) (m) | A number, measured in m. Starts at 4. |
When to use it
Designing a spillway
Flow leaving a spillway is supercritical, and a stilling basin is designed to force a hydraulic jump that dissipates its energy safely.
Assessing a channel
Whether a channel runs subcritical or supercritical determines how it responds to obstructions and changes in slope.
Modelling ship resistance
The same dimensionless number governs wave-making resistance for vessels, which is why hull models are tested at matched Froude numbers.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
At what speed does the flow become supercritical?
The same channel at a range of flow velocities.
| Flow velocity | Froude number (Fr) | Flow regime |
|---|---|---|
| 1 m/s | 0.2607 | Subcritical (tranquil flow) |
| 2.5 m/s | 0.6518 | Subcritical (tranquil flow) |
| 5 m/s | 1.304 | Supercritical (rapid flow) |
Questions
Why does Fr = 1 matter so much?
Because at that point the flow moves exactly as fast as a surface wave. Below it, disturbances can travel upstream and influence conditions there; above it they cannot, so the upstream flow is unaware of what lies ahead.
What is a hydraulic jump?
The sudden transition from supercritical to subcritical flow, where the water depth rises abruptly and a great deal of energy is dissipated in turbulence. Spillway stilling basins are designed to produce one deliberately.
Why is it used for ship models?
Because wave-making resistance depends on the Froude number. Testing a scale model at the same Froude number as the full-size ship makes the wave patterns geometrically similar and the results transferable.
How does it differ from the Reynolds number?
Froude compares inertia against gravity and governs free-surface wave behaviour. Reynolds compares inertia against viscosity and governs whether flow is laminar or turbulent. Both matter for ships, which is why model testing is genuinely difficult.
For flow rate in an open channel, see the open channel flow calculator. For the laminar or turbulent question, see the Reynolds number calculator.