Froude number
In continuum mechanics, the Froude number (Fr, after William Froude, /ˈfruːd/[1]) is a dimensionless number defined as the ratio of the flow inertia to the external force field (the latter in many applications simply due to gravity). The Froude number is based on the speed–length ratio which he defined as:[2][3] where u is the local flow velocity (in m/s), g is the local gravity field (in m/s2), and L is a characteristic length (in m).
The Froude number has some analogy with the Mach number. In theoretical fluid dynamics the Froude number is not frequently considered since usually the equations are considered in the high Froude limit of negligible external field, leading to homogeneous equations that preserve the mathematical aspects. For example, homogeneous Euler equations are conservation equations. However, in naval architecture the Froude number is a significant figure used to determine the resistance of a partially submerged object moving through water.
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In open channel flows, Belanger 1828 introduced first the ratio of the flow velocity to the square root of the gravity acceleration times the flow depth. When the ratio was less than unity, the flow behaved like a fluvial motion (i.e., subcritical flow), and like a torrential flow motion when the ratio was greater than unity.[4]
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. The naval constructor Frederic Reech had put forward the concept much earlier in 1852 for testing ships and propellers but Froude was unaware of it.[5] Speed–length ratio was originally defined by Froude in his Law of Comparison in 1868 in dimensional terms as:
where:
The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. In France, it is sometimes called Reech–Froude number after Frederic Reech.[6]
Usage[edit]
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.
One can easily see the line of "critical" flow in a kitchen or bathroom sink. Leave it unplugged and let the faucet run. Near the place where the stream of water hits the sink, the flow is supercritical. It 'hugs' the surface and moves quickly. On the outer edge of the flow pattern the flow is subcritical. This flow is thicker and moves more slowly. The boundary between the two areas is called a "hydraulic jump". The jump starts where the flow is just critical and Froude number is equal to 1.0.
The Froude number has been used to study trends in animal locomotion in order to better understand why animals use different gait patterns[15] as well as to form hypotheses about the gaits of extinct species.[16]
In addition particle bed behavior can be quantified by Froude number (Fr) in order to establish the optimum operating window.[18]