What Is the Froude Number?
The Froude number (Fr) is a dimensionless quantity that compares a flow's inertial forces to gravitational forces. Named after engineer William Froude, it is widely used in open-channel hydraulics, ship hydrodynamics, and the design of spillways and weirs. A Froude number tells you whether a flow is dominated by gravity or by momentum.
How to Use This Calculator
Enter the flow velocity v in metres per second, the characteristic length L in metres (for open channels this is usually the hydraulic depth), and the gravitational acceleration g (default 9.81 m/s²). The calculator returns the Froude number and classifies the flow regime.
The Formula Explained
The Froude number is defined as $$\text{Fr} = \frac{\text{Velocity (m/s)}}{\sqrt{\text{Gravity (m/s}^2\text{)} \times \text{Length (m)}}}$$ The denominator \(\sqrt{gL}\) represents the speed of a shallow-water gravity wave. When \(\text{Fr} < 1\) the flow is subcritical (tranquil, gravity-dominated, disturbances travel upstream). When \(\text{Fr} = 1\) the flow is critical. When \(\text{Fr} > 1\) the flow is supercritical (rapid, inertia-dominated, disturbances cannot travel upstream).
Worked Example
A channel carries water at \(v = 3 \text{ m/s}\) with a characteristic depth of \(L = 0.5 \text{ m}\), with \(g = 9.81 \text{ m/s}^2\). Then $$\sqrt{gL} = \sqrt{9.81 \times 0.5} = \sqrt{4.905} \approx 2.2147,$$ so $$\text{Fr} = \frac{3}{2.2147} \approx 1.355.$$ Since \(\text{Fr} > 1\), the flow is supercritical.
FAQ
Is the Froude number dimensionless? Yes. Because velocity divided by \(\sqrt{g \cdot L}\) cancels all units, Fr has no units, which lets it be applied across scales (model and full-scale).
What length should I use? For open-channel flow use the hydraulic depth (flow area ÷ top width); for ship resistance use the waterline length.
What does critical flow mean? At \(\text{Fr} = 1\) specific energy is minimised for a given discharge; flow transitions between subcritical and supercritical, often via a hydraulic jump.
