Froude Number — Physica

Equivalent forms

Fr^2 = \frac{v^2}{gL}

Fr = \frac{\text{inertial force}}{\text{gravitational force}}

Cross Fr = 1 and the whole character of the flow flips — tranquil to rapid.

Symbol	Name	SI unit	Dimension	Range
$v$	Flow Velocity Characteristic flow or body speed	m/s	L·T⁻¹	0 – 50
$g$	Gravitational Acceleration Acceleration due to gravity	m/s²	L·T⁻²	1 – 25
$L$	Characteristic Length Waterline length or flow depth	m	L	0.01 – 300
$Fr$	Froude Numberoutput Dimensionless ratio of inertial to gravitational effects	dimensionless	1	0 – 5

Unit systems

Symbol	SI	CGS	Imperial
$v$	m/s	cm/s	ft/s
$g$	m/s²	cm/s²	ft/s²
$L$	m	cm	ft
$Fr$	dimensionless	dimensionless	dimensionless

Where it holds

Applies to free-surface and gravity-driven flows (open channels, ship hulls, spillways). Not relevant to fully enclosed pressurized pipe flow where gravity waves play no role.

Dimensional analysis

$[v/\sqrt{gL}] = (L\cdot T^{-1})/\sqrt{L\cdot T^{-2}\cdot L} = (L\cdot T^{-1})/(L\cdot T^{-1}) = 1 \checkmark$ (dimensionless)

Discovery

William Froude · 1868

Froude, a British naval engineer, towed scale ship models to study wave-making resistance and established the similarity law that lets tank tests predict full-size hull drag. The dimensionless group was later named in his honor.

Try this

Why can a duck never outrun its own bow wave — and why do ship models in tanks predict real hull drag?

A ship moves at 10 m/s and has a waterline length of 100 m. What is its Froude number, and is the flow sub- or super-critical?

Research status: stable

Real-world applications

Ship and submarine hull design via towing-tank models
Open-channel and spillway hydraulics (hydraulic jumps)
Animal locomotion (the walk-run transition occurs near $Fr \approx 0.5)$
Tsunami and shallow-water wave speed estimates

Common misconceptions

Froude scaling matches gravity-wave effects, but viscous (Reynolds) effects cannot be matched simultaneously at model scale — hence Froude's ingenious split of resistance into wave and friction parts
$Fr = 1$ is not a fluid breaking the 'sound barrier'; that is the Mach number — Fr concerns gravity waves
A higher Froude number does not always mean faster in absolute terms; it depends on length too

Experimental verification

Froude's own towing-tank experiments, and a century of model-basin testing, confirm that hulls at equal Froude number produce geometrically similar wave patterns and scalable wave drag. Open-channel flumes reproduce predicted critical depth at

Fr = 1

.

Derivation

Form the ratio of inertial force per unit volume

(\rho v^{2}/L)

to gravitational force per unit volume

(\rho g)

.

The square root of this ratio,

v/\sqrt{gL}

, is the Froude number.

Equivalently it compares flow speed v to the shallow-water gravity-wave speed

\sqrt{gL}

.

Limiting cases

Fr < 1⟶ Subcritical (tranquil) flowGravity waves outrun the flow, so disturbances travel upstream — deep, slow river flow.

Fr = 1

⟶ Critical flowFlow speed equals wave speed; minimum specific energy, and where a hydraulic jump can form.

Fr > 1⟶ Supercritical (rapid) flowFlow outruns its own waves; disturbances cannot move upstream — shooting flow below a spillway.

What if…

What if a small boat tries to exceed

Fr \approx 0.4

?

Wave-making resistance spikes as the hull tries to climb its own bow wave (hull speed). Planing hulls break free of this limit by lifting out of the water.

What if gravity were weaker (on the Moon)?

$\sqrt{gL}$ drops, so the same speed gives a higher Froude number; waves are slower and hydraulic transitions occur at lower speeds.

1

Froude number of a ship

Given ·

v:: 10
g:: 9.81
L:: 100

Find · Fr

Steps

Compute the wave speed $\sqrt{gL} = \sqrt{981} \approx 31.3\,\mathrm{m/s}$
$Fr = 10 / 31.3$
$Fr \approx 0.32$ — subcritical, well below the wave-making hump near $Fr \approx 0.4$

Result ·

Fr = v/\sqrt{gL} = 10/\sqrt{9.81\times 100} = 10/31.3 \approx 0.32

Reynolds Number→Mach Number→Weber Number→