Library / Dimensional analysis and similarity / DIM-05
Dimensional analysis and similarity
Froude number
Ratio of the flow speed to the speed of a gravity wave.
- Ref
- DIM-05
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Fr
Fr = U/√(gL) compares the flow speed with √(gL), the speed at which a long gravity wave travels in shallow water. In shallow open-channel flow take L as the hydraulic depth, the flow area divided by the free-surface width. Fr = 1 is critical flow; below 1 the flow is subcritical (slow and deep, and disturbances can travel upstream), and above 1 it is supercritical (fast and shallow, and a hydraulic jump can form). A model of a ship, spillway or river reproduces the gravity effects of the prototype when it has the same Froude number.
Ratio of the flow speed to the speed of a gravity wave. It governs flows with a free surface, such as open channels, ship hulls and spillways, and Fr = 1 marks critical flow.
Calculator
- Gravity-wave speed, √(gL)3.132 m/s
- Fr²0.9174
- Speed at which Fr = 1 (critical)3.132 m/s
- Model speed at the same Fr for a 1:25 scale model0.6 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A flow of 3 m/s over a length of 1 m gives Fr = 0.958, just below the critical value of 1, so the flow is subcritical.
- Equal Froude numbers need the model speed to scale with the square root of the length: a 1:25 model is tested at one fifth of the prototype speed (here 0.6 m/s). Matching the Reynolds number as well would need a different fluid, so gravity-driven tests usually match Fr only.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Characteristic flow speed | 0.01 to 50 m/s | 0.003193 | 15.96 | 15.96 | +10% | +10 % |
| L Characteristic length (hydraulic depth for a shallow channel) | 0.01 to 100 m | 0.09578 | 9.578 | 9.482 | +10% | −4.65 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.5477 | 3 | 2.452 | +10% | −4.65 % |
Sweep of Characteristic flow speed
| Characteristic flow speed (m/s) | Froude number |
|---|---|
| 0.01 | 0.003193 |
| 0.029 | 0.009258 |
| 0.08409 | 0.02685 |
| 0.2438 | 0.07785 |
| 0.7071 | 0.2258 |
| 2.05 | 0.6547 |
| 5.946 | 1.898 |
| 17.24 | 5.505 |
| 50 | 15.96 |
Sweep one variable
Variables and how each one contributes
Froude number
What the equation gives you. Everything else feeds it.
Characteristic flow speed
The characteristic flow speed, in m/s: the mean speed in a channel, or the ship’s speed through the water. Raises the Froude number in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 0.9578 to 1.054.
Gravitational acceleration
The gravitational acceleration, in m/s² (9.81 on Earth). It matters only for a different planet or a quick model comparison. Lowers the Froude number with the square root of g: stronger gravity makes waves faster, so the same flow is relatively slower. Raising it by 10 % (to 10.79) moves the result by −4.65 %, from 0.9578 to 0.9133.
Characteristic length (hydraulic depth for a shallow channel)
The characteristic length, in m: the hydraulic depth for a shallow channel, or the waterline length for a ship. Lowers the Froude number with the square root of L: a longer length means faster waves. Raising it by 10 % (to 1.1) moves the result by −4.65 %, from 0.9578 to 0.9133.
Worked example with the values above
- Wave speed: √(gL) = √(9.81 × 1) = 3.132 m/s.
- Divide the flow speed by it: 3 / 3.132.
- Fr = 0.9578, so the flow is subcritical.
Assumptions and limits
- Gravity is the force that matters, and the free surface is the interface of interest.
- For a channel, L is the hydraulic depth (flow area over free-surface width), and the pressure is hydrostatic.
Common mistakes
- Using the channel width or the wetted perimeter in place of the hydraulic depth.
- Comparing Froude numbers that use different reference lengths.
- Matching only the Froude number and expecting the viscous forces to scale as well: the Reynolds number will differ in a model.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (U × 1, g × ½, L × ½). Speed dominates: a 2 % speed error gives a 2 % Froude error, while the same error in g or L gives only 1 %. In an open channel the hydraulic depth is the hardest input to measure well.
Where it is used
- Classifying open-channel flow as subcritical, critical or supercritical before analysing a hydraulic jump or a weir.
- Setting the speed of a towed ship model or a hydraulic model of a spillway or river.
Maths you need
- Free-surface flow: flow with a free liquid surface acted on by gravity
- Dimensional analysis: dimensionless groups and similarity
Related formulas
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