Froude Number Formula
Calculate the Froude number using Fr = v / sqrt(gL).
Dimensionless ratio of flow inertia to gravity, used in ship design and open-channel hydraulics.
The Formula
The Froude number is a dimensionless number that compares the speed of a flow to the speed of gravity waves on its surface. It was named after the English engineer William Froude, who pioneered the use of scale models for ship hull testing in the 1860s and 1870s. The Froude number tells you whether gravity or inertia dominates the behavior of a fluid flow. When the Froude number is less than 1, the flow is called subcritical, meaning gravity waves can travel upstream and the flow is relatively calm and deep. When the Froude number is greater than 1, the flow is supercritical, meaning the flow moves faster than gravity waves can propagate, resulting in shallow, fast-moving water. A Froude number of exactly 1 is the critical condition, where the flow velocity matches the wave speed.
This concept is absolutely essential in naval architecture. When designing a ship hull, engineers build scale models and test them in towing tanks. The Froude number ensures that the wave patterns produced by the model accurately represent what happens at full scale. Two vessels with the same Froude number produce geometrically similar wave patterns, regardless of their actual size. This principle, called Froude similarity, saves enormous time and money during hull design. The Froude number is also critical in civil engineering for designing spillways, weirs, and drainage channels. Hydraulic jumps, the turbulent transition from supercritical to subcritical flow that you can see at the base of a dam spillway, occur when the Froude number drops below 1. River engineers use the Froude number to predict flood behavior and design bridges that can withstand high-flow conditions.
Variables
| Symbol | Meaning |
|---|---|
| Fr | Froude number (dimensionless) |
| v | Flow velocity (m/s or ft/s) |
| g | Gravitational acceleration (9.81 m/s² or 32.17 ft/s²) |
| L | Characteristic length, such as water depth or ship waterline length (m or ft) |
Example 1: Ship Hull Design
A cargo ship has a waterline length of 200 m and travels at 8 m/s. What is its Froude number?
Fr = v / √(g · L)
Fr = 8 / √(9.81 × 200)
Fr = 8 / √1962
Fr = 8 / 44.29
Fr ≈ 0.181 (subcritical, typical for large cargo vessels)
Example 2: Open Channel Flow
Water flows through a drainage channel at 5 m/s with a depth of 0.8 m. Is the flow subcritical or supercritical?
Fr = v / √(g · L)
Fr = 5 / √(9.81 × 0.8)
Fr = 5 / √7.848
Fr = 5 / 2.801
Fr ≈ 1.785 (supercritical flow, fast and shallow)
When to Use It
The Froude number appears whenever gravity-driven waves interact with fluid flow.
- Ship and boat hull design using scale model testing
- Open-channel hydraulics for rivers, canals, and spillways
- Predicting hydraulic jumps in dam spillways and weirs
- Flood modeling and bridge scour analysis
- Designing harbors and coastal structures
- Scaling wave tank experiments for offshore platforms