Learn: Froude number
The Froude number compares the mean speed of open-channel flow with the speed scale of shallow gravity waves. It helps distinguish subcritical flow, which can respond upstream, from supercritical flow.
What the formula is saying
Compute the gravity-wave speed √(gD), then divide flow speed v by it. For a general section D is hydraulic depth A/top width, not hydraulic radius A/wetted perimeter.
Read the symbols in plain language
- v
- Mean velocity
Mean velocity. A ratio above one means the section-average flow is faster than this gravity-wave scale.
m/sUse m/s as the base unit shown here. Use v in m/s, g in m/s² and hydraulic depth D in m. Both numerator and denominator are speeds, so Fr is dimensionless.
- g
- Gravity
Gravity. Hydraulic depth times gravity has units of speed squared before taking the square root.
m/s²Use m/s² as the base unit shown here. Use v in m/s, g in m/s² and hydraulic depth D in m. Both numerator and denominator are speeds, so Fr is dimensionless.
- D
- Hydraulic depth A/T
Flow area divided by free-surface top width; this differs from hydraulic radius and from pipe diameter.
mMetres measure length; 1 m = 1000 mm.
- Fr
- Result to find
Froude number. A ratio above one means the section-average flow is faster than this gravity-wave scale.
ratio / no unit
Sort out the units first
Use v in m/s, g in m/s² and hydraulic depth D in m. Both numerator and denominator are speeds, so Fr is dimensionless.
Assumptions before calculating
Assume an open-channel shallow-water description with a hydrostatic pressure distribution and a suitable section-average velocity. Use positive hydraulic depth and gravity.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Fr and explain the result in the stated output unit.
- v · Mean velocity
- 2 m/s
- g · Gravity
- 9.81 m/s²
- D · Hydraulic depth A/T
- 1 m
Find the gravity-wave speed scale
Hydraulic depth times gravity has units of speed squared before taking the square root.
√((9.81) × (1)) ≈ 3.132091953 m/sCompare flow speed with wave speed
A ratio above one means the section-average flow is faster than this gravity-wave scale.
(2) ÷ (3.132091953) ≈ 0.6385508568
Does this worked answer make sense?
At fixed depth, twice the speed gives twice Fr. At fixed speed, four times the hydraulic depth halves Fr.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- v · Mean velocity
- 4 m/s
- g · Gravity
- 9.81 m/s²
- D · Hydraulic depth A/T
- 0.5 m
Find the gravity-wave speed scale
Hydraulic depth times gravity has units of speed squared before taking the square root.
√((9.81) × (0.5)) ≈ 2.214723459 m/sCompare flow speed with wave speed
A ratio above one means the section-average flow is faster than this gravity-wave scale.
(4) ÷ (2.214723459) ≈ 1.806094564
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- v · Mean velocity
- 3 m/s
- g · Gravity
- 9.81 m/s²
- D · Hydraulic depth A/T
- 0.75 m
Find: Learn: Froude number
A hint, not the answer
Compute the gravity-wave speed √(gD), then divide flow speed v by it. For a general section D is hydraulic depth A/top width, not hydraulic radius A/wetted perimeter.
Use v in m/s, g in m/s² and hydraulic depth D in m. Both numerator and denominator are speeds, so Fr is dimensionless.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find the gravity-wave speed scale
Hydraulic depth times gravity has units of speed squared before taking the square root.
√((9.81) × (0.75)) ≈ 2.712471198 m/sCompare flow speed with wave speed
A ratio above one means the section-average flow is faster than this gravity-wave scale.
(3) ÷ (2.712471198) ≈ 1.106002527
Avoid the common trap
Do not insert pipe diameter by habit or use hydraulic radius instead of hydraulic depth. Do not square the whole denominator twice.
When this method applies — and when it does not
Fr < 1 is subcritical, Fr = 1 is critical and Fr > 1 is supercritical within this model. This is not Reynolds number and does not classify laminar versus turbulent flow. Complicated sections and non-hydrostatic flow need more care.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Froude number. Water-measurement principles; for discharge devices, read the orifice/weir chapters and the installation and head-measurement conditions, not only the coefficient formula.
- U.S. Bureau of Reclamation — Water Measurement Manual, 3rd edition (1997; revised reprint 2001)
- Dawei Han, University of Bristol — Concise Hydraulics (2008, Ventus Publishing)
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.