Skip to main content

Froude Number

 

Froude number formula

\( Fr \;=\;   \dfrac{  v }{   \sqrt{ g \cdot h_m } } \)    (Froude Number)

\( v \;=\;   Fr  \cdot \sqrt{ g \cdot h_m  }  \) 

\( g \;=\;  \dfrac{ v^2 }{ h_m  \cdot Fr^2 }\) 

\( h_m \;=\;   \dfrac{ v^2 }{ g  \cdot Fr^2} \) 
Symbol English Metric 
\( Fr  \) = Froude Number \(dimensionless\) \( dimensionless \)
\( v  \) = Flow Velocity \(ft\;/\;sec\) \(m\;/\;s\)
\( g  \) = Gravitational Acceleration \(ft\;/\;sec^2\) \(m\;/\;s^2\)
\( h_m \) = Mean Depth \(ft\) \(m\)

Froude number, abbreviated as \(Fr\), a dimensionless number, is use to understand how water (or any fluid with a free surface) moves in open channels, like rivers, canals, spillways, or around ships.  In simple terms, it compares the inertial forces (the momentum or push of the flowing water) to the gravitational forces (the pull of gravity that tries to keep the surface level and influences waves).  This matters most in situations with a free surface, where the top of the water isn't confined by a pipe or closed conduit, because gravity shapes how surface waves and disturbances behave.  The Froude number is calculated as the flow velocity divided by the square root of gravity times a characteristic length (usually the water depth in open channels).

Froude Number Regime
Froude Number Less than 1 (Fr < 1)  -  Subcritical Flow, (also called tranquil or slow)  Gravity dominates here, so the water moves relatively slowly and deeply.  Small disturbances, like ripples or waves, can travel upstream against the current, and the flow is controlled from downstream (like backwater effects in a mild-sloped river). the flow is slow, and gravitational forces dominate (calm river flow).
Froude Number Exactly 1 (Fr = 1)  -  Critical Flow  This is the balanced point where inertial and gravitational forces are equal.  It often marks a transition zone, such as the minimum energy state for a given discharge, and is key for features like weirs or the narrowest point in a channel constriction.
Froude Number Greater than 1 (Fr > 1)  -  Supercritical Flow, (also called rapid or shooting)  Inertia takes over, so the water rushes fast and shallow.  Waves and disturbances can't move upstream—they're swept along with the flow—and control comes from upstream. This creates fast, energetic conditions, like in steep mountain streams or spillways.

This classification helps engineers design safe and efficient systems.  For example, knowing the regime predicts where a hydraulic jump might occur (when fast supercritical flow suddenly slows to subcritical, creating a turbulent rise in water level).  It also ensures that small-scale physical models in the lab match real-world behavior through Froude scaling, so tests on rivers, dams, or ship hulls are reliable.  Overall, the Froude number is one of the most practical tools for making sense of free-surface flows in everyday hydraulic engineering.

Piping Designer Logo 1