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.

Origins

File:DSCN0415-swan-raven crop 600x350.JPG

The hulls of swan (above) and raven (below). A sequence of 3, 6 and 12 (shown in the picture) foot scale models were constructed by Froude and used in towing trials to establish resistance and scaling laws.

The quantification of the resistance of floating objects is generally credited to 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:

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

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.

Dimensionless form

The dimensionless Froude number is defined as

$\mathrm {Fr} ={\frac {V}{c}}$

where $V$ is an average velocity , and $c$ is the propagation velocity of a shallow water wave. The Froude number is thus the hydrodynamic equivalent to the Mach number.

The wave velocity, $c$, is equal to the square root of gravitational acceleration times cross-sectional area divided by free-surface width:

$c={\sqrt {g{\frac {A}{B}}}}$

and so the Froude number can often be simplified to

$\mathrm {Fr} ={\frac {V}{\sqrt {gd}}}$

where $d$ is a depth or length scale.

It is also common in fluid mechanics to use the form

$\mathrm {Fr} ^{2}={\frac {V^{2}}{gd}}$

which is simply the square of the previous version. This form is proportional to inertia rather than velocity, and is the reciprocal of the Richardson number.

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:

$\mathrm {Fr} ={\frac {N^{2}d}{g}}$

Densimetric Froude Number

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

$\mathrm {Fr} ={\frac {u}{\sqrt {g'h}}}$

where $g'$ is the reduced gravity:

$g'=g{\rho _{1}-\rho _{2} \over {\rho }}$

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.

External links

• More than 300, freely available, published research articles on open channel flow, hydraulic engineering and related topics by Professor Hubert Chanson, Department of Civil Engineering, The University of Queensland
• http://www.qub.ac.uk/waves/fastferry/reference/MCA457.pdf