Definitions

Froude number

The Froude number is a dimensionless number comparing inertial and gravitational forces. It may be used to quantify the resistance of an object moving through water, and compare objects of different sizes. Named after William Froude, the Froude number is based on his speed/length ratio.

Dimensionless form

The dimensionless Froude number is defined as:

mathrm{Fr} = frac{V}{c} where $V$ is a characteristic velocity , and $c$ is a characteristic water wave propagation velocity. The Froude number is thus the hydrodynamic equivalent to the Mach number.

Origins

Quantifying resistance of floating objects is generally credited to William Froude, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Wave-Line Theory which first described the resistance of a shape as being a function of the waves caused by varying pressures around the hull as it moves through the water. The Naval Constructor Ferdinand Reech had put forward the concept in 1832 but had not demonstrated how it could be applied to practical problems in ship resistance. Speed/length ratio was originally defined by Froude in his Law of Comparison in 1868 in dimensional terms as:

$mathrm\left\{Speed Length Ratio\right\} =frac \left\{V\right\}\left\{sqrt mathrm\left\{LWL\right\} \right\}$

where:

v = speed in knots
LWL = length of waterline in feet

The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. It is sometimes called Reech-Froude number after Ferdinand Reech.

Definitions of the Froude number in different applications

Ship hydrodynamics

For a ship, the Froude number is defined as:
$mathrm\left\{Fr\right\} = frac\left\{V\right\}\left\{sqrt\left\{gL\right\}\right\},$
where V is the velocity of the ship, g is the acceleration due to gravity, and L is the length of the ship.

The ship generates waves, the longest of which have a length of about the ship's length L, and an associated propagation velocity of about

$c_text\left\{wave\right\} = sqrt\left\{frac\left\{gL\right\}\left\{2pi\right\}\right\}$.
For a velocity V above cwave, the boat will start planing: the boat overtakes its own generated surface wave, similar to supersonic flight where a plane travels faster than the speed of sound waves. The associated critical Froude number is:
$mathrm\left\{Fr\right\}_text\left\{critical\right\} = frac\left\{1\right\}\left\{sqrt\left\{2pi\right\}\right\} approx 0.4,$
above which there is a strong decrease in wave making resistance.

Shallow water waves

For shallow water waves, like for instance tidal waves and the hydraulic jump, the characteristic velocity V is the average flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, c, is equal to the square root of gravitational acceleration g, times cross-sectional area A, divided by free-surface width B:

c = sqrt{g frac{A}{B}}, so the Froude number in shallow water is:

mathrm{Fr} = frac{V}{sqrt{displaystyle g frac{A}{B}}}. For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:

mathrm{Fr} = frac{V}{sqrt{gd}}. For Fr < 1 the flow is called a subcritical flow, while a supercritical flow has Fr > 1.

An alternate definition used in fluid mechanics is

$widehat\left\{mathrm\left\{Fr\right\}\right\}=frac\left\{V^2\right\}\left\{gd\right\},$
where each of the terms on the right have been squared. This form is the reciprocal of the Richardson number.

Stirred tanks

In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is proportional to $Nd$, where $N$ is the impeller speed (rev/s) and $d$ is the impeller diameter, the Froude number then takes the following form:
$mathrm\left\{Fr\right\}=frac\left\{N^2d\right\}\left\{g\right\}.$

Densimetric Froude Number

When used in the context of the Boussinesq approximation the densimetric Froude number is defined as

$mathrm\left\{Fr\right\}=frac\left\{u\right\}\left\{sqrt\left\{g\text{'} h\right\}\right\}$
where $g\text{'}$ is the reduced gravity:
$g\text{'} = g\left\{rho_1-rho_2over \left\{rho\right\}\right\}$

The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the Richardson number which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a gravity current moves with a front Froude number of about unity.

Uses

The Froude number is used to compare the wave making resistance between bodies of various sizes and shapes.

In free-surface flow, the nature of the flow (supercritical or subcritical) depends upon whether the Froude number is greater than or less than unity.