Froude Number Calculator
Enter the flow speed, flow depth and gravity to compute the Froude number and find out whether your open-channel flow is subcritical, critical or supercritical, with critical-speed details.
Check Whether Open-Channel Flow Is Subcritical, Critical or Supercritical
Froude Number Formula
Where:
- \(Fr\) — Froude number (dimensionless)
- \(v\) — mean flow velocity (m/s)
- \(g\) — gravitational acceleration (m/s²)
- \(L\) — characteristic length: for open-channel flow, the mean flow depth or hydraulic depth (m)
The formula compares the speed of the flow with the speed of a small gravity (surface) wave travelling on the water surface, \(v_c = \sqrt{gL}\). This wave speed is the natural information speed of a free-surface flow, so the ratio tells you whether disturbances can propagate upstream (\(Fr < 1\)) or are swept downstream (\(Fr > 1\)).
Froude Number – Calculation Example
Example: water flows at an average speed of 2 m/s in a rectangular channel with a water depth of 1.2 m. Take \(g = 9.81\ \text{m/s}^2\).
\[ Fr = \frac{v}{\sqrt{g \, L}} = \frac{2}{\sqrt{9.81 \times 1.2}} = \frac{2}{\sqrt{11.772}} = \frac{2}{3.431} \approx 0.583 \]
Result: \(Fr \approx 0.583 < 1\), so the flow is subcritical (tranquil). The critical speed at this depth is \(v_c = \sqrt{9.81 \times 1.2} \approx 3.431\ \text{m/s}\); the flow would have to accelerate to that speed to reach the critical state.
How to Use the Calculator
- Enter the Flow speed (v) and choose its unit (m/s, ft/s, km/h or mph). You can estimate it by timing a floating object over a known distance or measure it with a flow meter.
- Enter the Characteristic length / flow depth (L) and its unit. For a rectangular channel this is simply the water depth; for other cross-sections use the hydraulic depth (flow area divided by top width).
- Enter the Gravitational acceleration (g) — the default 9.81 m/s² (or 32.2 ft/s²) is standard on Earth and can be left as is.
- Read the results: the Froude number, the flow-regime classification (subcritical, near-critical or supercritical), and the critical speed and details, which include the critical speed at your depth, the specific energy head, and the critical depth for your speed.
What Is the Froude Number?
The Froude number is the standard dimensionless number for open-channel and free-surface flow. Just as the Mach number compares a gas flow speed to the speed of sound, the Froude number compares a water flow speed to the celerity of a small surface wave, \(\sqrt{gL}\). It measures the relative importance of inertia versus gravity: at low Fr, gravity dominates and the water surface adjusts gently; at high Fr, inertia dominates and the surface becomes steep and choppy.
The classification is straightforward. Subcritical flow (\(Fr < 1\)) is tranquil: waves can travel both upstream and downstream, so the flow depth is controlled by downstream conditions — this is why a weir or obstruction raises the water level far upstream (the backwater effect). Critical flow (\(Fr = 1\)) is an unstable transition where the depth equals the critical depth and the flow is extremely sensitive to small changes in slope or geometry. Supercritical flow (\(Fr > 1\)) is rapid: no disturbance can travel upstream, so the flow is controlled entirely by upstream conditions.
The transition from supercritical back to subcritical cannot happen smoothly — it occurs abruptly through a hydraulic jump, a standing wave that dissipates energy. Hydraulic jumps are seen downstream of spillways and sluice gates and are deliberately used in stilling basins to destroy the destructive energy of fast flow. The calculator's near-critical band (0.95–1.05) reflects the practical difficulty of holding flow exactly at \(Fr = 1\).
The result table also shows two useful derived quantities: the specific energy head \(E = L + v^2/(2g)\), the sum of depth and velocity head relative to the channel bottom, and the critical depth \(L_c = v^2/g\), the depth at which your current speed would be exactly critical. Comparing your actual depth with the critical depth is often the quickest way to judge the regime.
Assumptions, Limitations and Tips
- Use the right length. This calculator implements the depth-based Froude number for open-channel flow. For a rectangular channel, \(L\) is the water depth; for other shapes use the hydraulic depth \(A/T\) (flow area divided by top width). Ship and hull hydrodynamics uses a different, length-based Froude number — the two are not interchangeable.
- Use consistent reality. The calculator converts units internally, so you only need to enter values in whatever units you measured — just pick the matching unit in each dropdown.
- Uniform, gradually varied flow assumed. The classification assumes the velocity is reasonably representative of the cross-section. In highly non-uniform flow (near drops, junctions or jumps), Fr varies from point to point.
- Keep g realistic. Changing g only makes sense for other planets or teaching examples; on Earth use 9.81 m/s² (32.2 ft/s²).
- Depth and velocity interact with channel resistance: to explore related flow physics, see the Drag Force Calculator or the Speed of Sound in Air Calculator.
Frequently Asked Questions
How do I calculate the Froude number by hand?
Divide the mean flow velocity by the square root of gravity times depth: \( Fr = v/\sqrt{gL} \). For example, with \( v = 1.5\ \text{m/s} \) and \( L = 0.4\ \text{m} \): \( \sqrt{9.81 \times 0.4} \approx 1.98 \), so \( Fr \approx 0.76 \) — subcritical. You can do it on any calculator with a square-root key; only three inputs are needed.
What is a good or normal Froude number?
There is no single "good" value — it depends on the engineering goal. Rivers and canals are usually designed subcritical (\( Fr \) between about 0.2 and 0.8) for a stable water surface and easy control. Supercritical flow (\( Fr > 1 \)) is normal on steep chutes and spillways but requires careful design. Values near 1 are avoided because the flow becomes unstable and wavy.
Can I use feet instead of metres?
Yes, the Froude number is dimensionless, so any consistent unit system gives the same result. Use \( g = 32.2\ \text{ft/s}^2 \) with velocity in ft/s and depth in ft — or simply select your units in the calculator and it converts internally. The only mistake to avoid is mixing units, e.g. metres with ft/s².
What depth should I enter for a non-rectangular channel?
Use the hydraulic depth: flow area divided by top width, \( L = A/T \). For a rectangular channel this equals the water depth, since \( A = bL \) and \( T = b \). For a trapezoidal or circular channel, compute \( A \) and the top width \( T \) of the water surface at the actual depth, then divide. Using the maximum depth instead of \( A/T \) is a common error that misclassifies the regime.
Is this the same Froude number used for boats and ships?
No. Ship hydrodynamics uses a length-based Froude number, \( Fr = v/\sqrt{gL} \), where \( L \) is the hull's waterline length — the same formula but a completely different meaning of the length scale, used to predict wave-making resistance. The open-channel version uses flow depth and classifies free-surface flow. The two numbers are not interchangeable.
What happens when flow changes from supercritical to subcritical?
The transition cannot occur smoothly; it happens abruptly as a hydraulic jump — a stationary, foamy wave that dissipates energy and raises the water depth. The Froude number drops from above 1 to below 1 across the jump. Engineers deliberately create jumps in stilling basins below spillways and sluice gates to destroy the destructive energy of fast water.
How is the critical depth \( L_c = v^2/g \) different from my actual depth?
The critical depth is the depth at which your current speed would be exactly critical (\( Fr = 1 \)). If your actual depth is greater than \( L_c \), the flow is subcritical; if it is smaller, the flow is supercritical. Comparing the two depths is often the quickest sanity check on the regime — it works in any consistent unit system since both are lengths.