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What speed is √(gL)?
In shallow water, √(gh) is the speed scale at which small surface waves travel (h is the depth). The Froude number compares the flow speed with this wave speed. Below 1, surface disturbances can travel upstream against the flow; above 1, they are swept downstream. Density and viscosity do not appear, so the calculator asks for no fluid properties.
Variables and SI units
| Symbol | Meaning | SI unit | Example |
|---|---|---|---|
| v | Mean flow speed or ship speed | m/s | 3 |
| L | Flow depth (hydraulic depth) for channels, waterline length for ships | m | 5 |
| g | Gravitational acceleration | m/s² | 9.81 |
Choosing L
- For a wide rectangular channel, use the flow depth.
- For irregular channels, the hydraulic depth (flow area divided by surface width) is the usual choice.
- For ships, the length Froude number uses the waterline length. Subcritical and supercritical labels do not apply in that case.
- Use g = 9.81 m/s² unless you have a reason not to. Because L sits under a square root, entering it in mm by mistake (1000 times too large) makes Fr about 31.6 times too small.
Reading the result around 1
When L is a channel depth scale, Fr < 1 is subcritical, Fr = 1 critical and Fr > 1 supercritical. Where fast supercritical flow turns into slow subcritical flow, the water surface jumps up in a hydraulic jump. A USGS-listed study of lava channels relies on the fact that in water, undular hydraulic jumps are evidence of critical flow with Fr ≈ 1. If your result is very close to 1, the surface may be unstable, so consider the uncertainty in the measured depth as well. The calculator shows "critical" only for exactly 1, so values such as 0.98 or 1.02 are still labelled subcritical or supercritical.
Example and scale-model speeds
The example uses v = 3 m/s, L = 5 m and g = 9.81 m/s², giving Fr ≈ 0.43. To match Fr in a scale model, the model speed is the full-scale speed times √(scale). For a 1:10 model, 5 m/s becomes about 1.58 m/s. Matching Fr this way with the same water leaves the model Reynolds number at about 0.0316 times full scale. The scale-model note works through why the two conditions conflict.
Inputs that account for channel shape and measurement error
Example 1. Hydraulic depth of a trapezoidal channel
Given (assumed): a trapezoidal channel with a 2 m bottom width and 1:1 side slopes (1 horizontal per 1 vertical) carries 6 m³/s at 1 m depth.
Section geometry: flow area A = (2 + 1 × 1) × 1 = 3 m² and top width T = 2 + 2 × 1 = 4 m, so the hydraulic depth is D = A/T = 0.75 m and the mean speed is U = 6 ÷ 3 = 2 m/s.
Calculator entry: v = 2, L = 0.75 and g = 9.81 give Fr = 0.737, labelled subcritical.
Interpretation: entering the 1 m depth as L gives Fr = 0.639, about 13% lower. Both are subcritical here, but near critical flow the choice of L alone can flip the result, so use the hydraulic depth for channels with sloping sides.
Example 2. A 14.4 km/h flood flow and depth uncertainty
Given (assumed): in a wide river reach, tracking surface debris suggests a mean speed of 14.4 km/h, and the measured depth is 1.5 m with an assumed uncertainty of ±10%.
Unit conversion: 14.4 km/h ÷ 3.6 = 4 m/s.
Calculator entry: v = 4 and L = 1.5 give Fr = 1.04, labelled supercritical. With a depth 10% greater, 1.65 m, Fr = 0.994 (subcritical); with 10% less, 1.35 m, Fr = 1.10.
Interpretation: within the measurement uncertainty the classification switches between subcritical and supercritical. As the USGS-described study notes, undular jumps can form at Fr ∼ 1, so checking for a standing wave train on site is more reliable than a single computed value.
Using the Froude number calculator
When should g be something other than 9.81?
For channels and ships on Earth, 9.81 m/s² is enough. Enter a different value only when the effective acceleration differs, as on another planet or in a rotating device with strong centrifugal acceleration. The model similitude in MIT 2.154 also assumes model and ship share the same gravity field.
The calculator does not show Fr². What if I need the squared definition?
Square the result yourself. Fr = 1.04 in Example 2 is about 1.09 in the U²/(gL) definition. Before comparing with limits from another source, confirm which definition it uses.
I entered a ship length and got "subcritical". Is that wrong?
The number is correct, but the label only means something when L is a depth. A ship Fr relates the hull length to the waves the ship makes, so ignore the subcritical or supercritical wording and use the number alone.
Related guides and calculators
Guide
Calculation notes
- Can a scale model match both Re and Fr?: After setting a model speed here, see how far Re drifts under the same conditions.
- Surface tension and unsteady flow: connect force ratios to time ratios: Reuse the v and L from this page to compare Fr with We and St.
Related calculators
- Reynolds number (Re): Use the same v and L with the water viscosity to see whether viscous effects are negligible.
- Weber number (We): Check whether surface tension distorts results in a model only a few centimetres deep.
- Strouhal number (St): Non-dimensionalise the frequency when the surface oscillates periodically.
References
- U.S. Geological Survey: Can lava flow like water? Assessing applications of critical flow theory to channelized basaltic lava flows. Undular hydraulic jumps and critical flow at Fr ∼ 1
- MIT OpenCourseWare (2.154): Similitude lecture notes. Waterline-length Fr and Froude similitude in a shared gravity field
- University of Texas at Austin: Some Notes on Similitude. Why Re and Fr are hard to match together