Why compare flow speed with gravity-wave speed
Gravity restores a displaced free surface and so makes surface waves. Fr compares flow speed U with the gravity wave-speed scale √(gL). In an open channel, below 1 disturbances can travel upstream; above 1 they are swept downstream, which is why Fr decides whether a water-surface calculation takes its boundary condition upstream or downstream.
Its two main uses are open-channel flow and ship waves. MIT's manoeuvring notes (2.154) define Fr with vessel speed and waterline length, for problems with pressure boundary conditions such as ocean waves. Same formula, different L, different reading.
Fr = U/√(gL) and choosing L
U is mean flow or vessel speed (m/s), g gravity (usually 9.81 m/s²), L a length (m). In Fr², ρU² is the dynamic-pressure scale and ρgL the hydrostatic pressure of height L: inertia over gravity. Density cancels, so water, lava and mud can look alike at the same Fr.
L is depth for a wide rectangular channel, hydraulic depth (area over top width) for irregular channels, waterline length for a ship. Definitions vary too: this site uses U/√(gL), some references use U²/(gL), and NASA's 1968 dimensional analysis handbook at one point writes Lg/v². Check the formula before borrowing a number.
Open this calculatorShallow-water wave speed by depth
Small shallow-water waves travel at √(gh) (h the depth; g = 9.81 m/s²). A faster mean speed means Fr > 1.
| Depth h | √(gh) | Typical setting |
|---|---|---|
| 0.1 m | 0.990 m/s | Laboratory flume, roadside gutter |
| 0.5 m | 2.21 m/s | Small stream, irrigation canal |
| 1 m | 3.13 m/s | Medium river |
| 2 m | 4.43 m/s | River in flood |
| 5 m | 7.00 m/s | Large river, navigation canal |
So 3 m/s in a 5 m deep channel gives Fr = 3 ÷ 7.00 ≈ 0.43, subcritical. Thin fast flow on steep gutters or spillway chutes often exceeds 1; a deep river rarely approaches it.
Fr for a channel, a ship and a spillway model
Example 1. Supercritical flow below a sluice gate and the hydraulic jump
Given (all assumed): 6 m³/s flows in a 3 m wide rectangular channel, and the depth just below a sluice gate is 0.4 m. g = 9.81 m/s².
Unit conversion: discharge per unit width q = 6 ÷ 3 = 2 m²/s, so the mean speed is U₁ = q ÷ y₁ = 2 ÷ 0.4 = 5 m/s.
Calculation: √(9.81 × 0.4) = 1.98 m/s, so Fr₁ = 5 ÷ 1.98 = 2.52, or 6.37 in the U²/(gL) definition. Critical depth is yc = (q²/g)1/3 = 0.742 m. The textbook momentum relation for a jump in a rectangular channel, y₂/y₁ = ½(√(1 + 8Fr₁²) − 1), gives a ratio of 3.10, a depth after the jump of y₂ = 1.24 m, a speed of 1.61 m/s and Fr₂ = 0.461.
Interpretation: Supercritical flow below the gate jumps to subcritical if the tailwater holds about 1.24 m. The USGS study notes that undular jumps in water mark this transition and are evidence of critical flow (Fr ∼ 1). At Fr₁ near 2.5 the jump is stronger than undular, so check bed protection and stilling-basin length.
Example 2. A 120 m ship and a 1:25 towing-tank model
Given (assumed): a ship with a 120 m waterline length sails at 16 knots. The model scale is 1:25, and model and ship are in the same gravity field.
Unit conversion: 1 knot = 1852 m/h = 0.5144 m/s, so 16 knots is 8.23 m/s. The model length is 120 ÷ 25 = 4.8 m.
Calculation: Fr = 8.23 ÷ √(9.81 × 120) = 0.240. The Froude similitude condition in MIT 2.154, Umodel = Uship/√(scale ratio), gives a model speed of 8.23 ÷ 5 = 1.65 m/s. With the same water, model Re is 25−3/2 = 0.008 times the ship's. Approximating with ν = 1.0020 × 10⁻⁶ m²/s for water at 20 °C (IAEA), model Re is 7.89 × 10⁶ and ship Re is 9.86 × 10⁸.
Interpretation: Matching Re too would need a fluid with kinematic viscosity smaller by the scale ratio to the 3/2 power, which MIT notes is rarely workable in a large tank; turbulence stimulators are used instead and Fr is matched. Cf is insensitive to Re above about 500,000 per the same notes. Equating the wave phase speed √(gλ/2π) to ship speed gives λ = 2πFr²L = 0.362L, about 43.4 m: ship Fr relates hull length to its own waves, not to sub- or supercritical flow.
Example 3. A 1:36 spillway hydraulic model
Given (assumed): at the toe of a 1:36 spillway model, the measured speed is 1.2 m/s and the depth 0.05 m; the model discharge is 10 L/s.
Unit conversion: 10 L/s = 0.010 m³/s.
Calculation: model Fr = 1.2 ÷ √(9.81 × 0.05) = 1.71. The UT Austin similitude notes explain that free-surface problems such as flow over a spillway are scaled by matching Fr. Equal Fr means speed scales by √36 = 6 and depth by 36, so the prototype has 7.2 m/s, 1.8 m depth and the same Fr of 1.71. Discharge is speed times area, so it scales by 365/2 = 7776: 0.010 × 7776 = 77.76 m³/s.
Surface shape and jump position therefore scale geometrically, while Re with the same water differs by 363/2 = 216.
Interpretation: The prototype chute ends supercritical, so a jump must be accommodated downstream. At 5 cm depth surface tension and viscosity matter more than at full scale; do not scale up air entrainment or spray directly.
Reading results near Fr = 1
With a depth-based L, Fr < 1 is subcritical, 1 critical, above 1 supercritical. In a real river, Fr of about 0.9 to 1.1 gives an undulating surface, where the USGS undular jumps occur. Fr ∝ depth−1/2, so a 10% depth error shifts Fr about 5%, enough to flip the class; compute Fr across the depth uncertainty.
The USGS study reports that undular jumps in the 2018 Kīlauea near-vent lava channel also reflected critical flow, with wavelengths scaling with depth and speed as theory predicts. Property-free Fr allows such comparisons, but check Re for very viscous fluids.
Common mistakes when interpreting Fr
- Calling a ship Fr subcritical: a waterline-length Fr of 0.4 or 0.5 is not "subcritical".
- Mixing definitions: 1 in U²/(gL) is 1 here, but 0.25 there is 0.5 here.
- Units of L: a millimetre value entered as metres makes Fr about 31.6 times too small.
- Matching Re at the same time: UT Austin notes it is not feasible with one fluid; pick the dominant force and correct for the other.
- Limits of the mean speed: compute Fr separately around piers or in bends where local speeds differ.
Froude number points that cause confusion
How do I compare with sources that define Fr as U²/(gL)?
Take the square root of their value. Fr₁ = 2.52 in Example 1 is 6.37 squared; only the limit at 1 is shared.
What does a ship Fr of 0.24 mean?
A wave travelling at ship speed is 2πFr², about 0.36, hull lengths long. Higher Fr lengthens the waves toward hull length and raises wave-making resistance; it is not read against 1.
Does the Fr ≈ 1 criterion apply to lava or mud flows?
The calculation is the same, and USGS found Kīlauea lava jumps reflected critical flow, but the study assesses when this works; check Re and viscous losses too.
If a model matches Fr, how is the Re mismatch handled?
MIT 2.154 describes turbulence stimulators giving a ship-like boundary layer, with Cf insensitive to Re above about 500,000, then matching Fr. Friction scale effects are corrected separately.
Related dimensionless numbers
Numbers paired with Fr in free-surface problems to check scale effects and surface tension.
- Reynolds number (Re): A Froude-scaled model has Re smaller by the scale ratio to the 3/2 power, so friction resistance needs correcting.
- Weber number (We): We/Fr² = ρgL²/σ has no speed in it, showing how much surface tension gains on gravity in a small model.
- Strouhal number (St): For periodic surface motion such as sloshing, match the speed ratio Fr and the time ratio St together.
References
- U.S. Geological Survey: Can lava flow like water? Assessing applications of critical flow theory to channelized basaltic lava flows. Undular jumps as evidence of critical flow (Fr ∼ 1) and of the supercritical-to-subcritical transition; the Kīlauea lava channel case
- MIT OpenCourseWare (2.154): Similitude lecture notes. Waterline-length Fr, surface-wave phase speed, Froude-similitude model speed and viscosity condition, turbulence stimulators and Re of about 500,000
- University of Texas at Austin: Some Notes on Similitude. Re and Fr cannot be matched together; Fr is matched for bodies at or under a free surface and spillway flow
- NASA: Dimensional Analysis Handbook. Gravity g leading to the Froude number in dimensional analysis; the Lg/v² form