Froude number

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The Froude number is a dimensionless number 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 the speed/length ratio discovered by Froude, and on which the Froude number is based.

[edit] Origins

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:

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

where:

v = speed in knots

LWL is 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.

[edit] Dimensionless forms

The dimensionless Froude number is defined as

$Fn = \frac{v}{\sqrt{g LWL}}$

where $v$ is the speed in m/s, g is the acceleration due to gravity, and $L W L$ is the Waterline length.

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

In fluid dynamics, the Froude number is the reciprocal of the square root of the Richardson number. When used in the context of the Boussinesq approximation it is defined as

${u\over \sqrt{g' h}}$

where g' the reduced gravity and $h$ a representative vertical lengthscale. Strictly, this is known as the densimetric Froude number.

  Thus the Froude number is given by the ratio of inertial to gravitational forces in flow.

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.