Froude Number Calculator
Compute Froude number from velocity and characteristic length.
Classifies flow as subcritical, critical, or supercritical for hydraulics and naval design.
The Froude number is one of the most useful dimensionless numbers in fluid mechanics. It compares the speed of a flow to the speed at which gravity waves can travel on its surface, and it tells you immediately whether gravity or inertia is in charge.
The formula:
Fr = v / √(g × L)
Where v is the flow velocity, g is gravitational acceleration (9.81 m/s² on Earth), and L is a characteristic length: depth for open channels, waterline length for ships, the radius of curvature for a bend, and so on. Fr is dimensionless.
What the value tells you:
| Froude number | Flow regime | What it means |
|---|---|---|
| Fr < 1 | Subcritical (tranquil) | Flow slower than wave propagation. Disturbances can travel upstream. Deep, slow rivers. |
| Fr = 1 | Critical | Flow matches wave speed. Hydraulic jumps form near this transition. |
| Fr > 1 | Supercritical (rapid) | Flow faster than waves. Shallow, fast water. Disturbances cannot move upstream. |
| Fr » 1 | Hyper-supercritical | Dam spillways, mountain streams, hydraulic jumps. |
William Froude, an English engineer working in the 1860s and 1870s, was the first to use scale models in towing tanks to test ship hull designs. He realized that for the wave patterns on a model to match the full-size ship, the dimensionless ratio v/√(gL) had to match. That ratio is now called the Froude number, and Froude similarity is still the backbone of naval architecture.
Ship hull design (why most ships sit around Fr ≈ 0.2-0.4): For a displacement hull (not planing), the bow wave it makes is governed by Froude similarity. A cargo ship 200 m long traveling 8 m/s has Fr = 8/√(9.81 × 200) = 0.181 (comfortable subcritical operation). As Fr approaches about 0.4 the wave-making resistance climbs sharply, and at Fr ≈ 0.5 it becomes prohibitive without huge power. That is why displacement hulls have a “hull speed” they cannot easily exceed without enormous power, and why fast boats become planing hulls (which break the Froude-similarity rule by lifting clear of their own bow wave).
Open channel flow (supercritical means hydraulic jumps): A drainage channel running 5 m/s with 0.8 m water depth has Fr = 5/√(9.81 × 0.8) = 1.78 (strongly supercritical). When such flow hits an obstacle (a slower-moving section, a step, a bridge pier) it cannot smoothly slow down because waves cannot propagate upstream. Instead it undergoes a hydraulic jump: a sudden, turbulent transition to subcritical flow with significant energy dissipation. Engineers deliberately design stilling basins below dam spillways to force hydraulic jumps and kill the kinetic energy before water re-enters the river.
Why naval engineers use Froude scaling but not Reynolds scaling for hull tests: A 1:25 scale model of a 200 m ship is only 8 m long. To get correct Reynolds (viscous) similarity at the same fluid, the model would need to move 25× faster than the prototype, which is physically impractical. Froude similarity is achievable because the model moves at √(1/25) = 1/5 the speed and the wave patterns scale correctly. The remaining mismatch (viscous drag) is corrected analytically using empirical friction-line formulas. This is the trick that lets every major ship be tested at affordable scale before construction.
Critical depth and minimum specific energy: In open channel hydraulics, the flow depth at Fr = 1 is called the critical depth. It is the depth at which a given discharge passes the channel with minimum specific energy. Anywhere along a channel where the flow transitions through critical depth (a free overfall, a sluice gate exit, a hydraulic jump) is a control point engineers use to set channel design parameters.
Where else the Froude number shows up:
- Granular flow and avalanche dynamics (Fr based on grain depth and flow velocity)
- Atmospheric flow over mountains and through valleys (with buoyancy frequency replacing √g)
- Biological locomotion: human walk-to-run transition occurs near Fr ≈ 0.5 with leg length as L. Long-legged animals walk faster than short-legged ones for this reason.
- Coastal engineering and tsunami modeling: shallow-water wave speed is √(gL), so wave behavior is fully characterized by Fr.
- Hydraulic structure design (spillways, weirs, drop structures, energy dissipators).
Worked example, hydraulic jump in a stilling basin: Water enters a stilling basin at v = 12 m/s and depth y = 0.3 m. Find the Froude number and determine if a hydraulic jump will form.
Fr = 12 / √(9.81 × 0.3) = 12 / 1.716 = 6.99
That is strongly supercritical (Fr » 1), well above the Fr = 1 transition. A strong hydraulic jump will form when this flow meets the slower downstream tailwater. The jump itself is highly turbulent and dissipates most of the kinetic energy. Exactly what the stilling basin is designed to do.
Earth vs other gravities: The √g in the denominator means Froude number scales the same way regardless of planet, but at lower gravity (Moon, Mars) the same flow velocity produces a higher Fr because waves travel more slowly. Fluid behavior on low-gravity worlds is qualitatively different: hydraulic jumps form at lower velocities, and the boundary between subcritical and supercritical shifts.