The Froude number is a dimensionless number comparing inertia 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 the speed/length ratio as defined by him.
Dimensionless form
The dimensionless Froude number is defined as:
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
In open channel flows, Bélanger (1828 [1]) 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 (Chanson 2009 [2]).
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.
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:
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. In France, it is sometimes called Reech–Froude number after Ferdinand Reech.[3]
Definitions of the Froude number in different applications
Ship hydrodynamics
For a ship, the Froude number is defined as:[4]
where V is the velocity of the ship, g is the acceleration due to gravity, and L is the length of the ship at the water line level, or Lwl in some notations. It is an important parameter with respect to the ship's drag, or resistance, including the 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:
so the Froude number in shallow water is:
For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:
For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterised as supercritical flow. When Fr ≈ 1 the flow is denoted as critical flow.
An alternate definition used in fluid mechanics is
where each of the terms on the right have been squared.[5] 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:
Densimetric Froude number
When used in the context of the Boussinesq approximation the densimetric Froude number is defined as
where g' is the reduced gravity:
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.
Walking Froude number
In studying the dynamics of bipedal walking, walking is modeled as an inverted pendulum, where the center of mass goes through a circular arc centered at the foot.[6] The Froude number is the ratio of the centrifugal force around the center of motion, the foot, and the weight of the person walking.
where m is the mass, l is the leg length, g is the acceleration due to gravity and V is the velocity. The theoretical maximum speed of walking is with Fr=1 since any higher value would result in 'take-off' and the foot missing the ground. The typical transition speed from running to walking occurs with 
.
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.
Notes
- ^ Bélanger, Jean-Baptiste (1828). Paris, France, 38 pages & 5 tables: Carilian-Goeury.
- ^ Chanson, Hubert (2009). Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828). Vol. 135, No. 3, pp. 159-163 (DOI: 10.1061/(ASCE)0733-9429(2009)135:3(159)): Jl of Hyd. Engrg, ASCE.
- ^ Chanson (2004), p. xxvii.
- ^ Newman, John Nicholas (1977). Marine hydrodynamics. Cambridge, Massachussets: MIT Press. ISBN 0-262-14026-8. , p. 28.
- ^ Frank M. White, Fluid Mechanics, 4th edition, McGraw-Hill (1999), 294.
- ^ Vaughan, C. L.; OʼMalley, M. J. (2005), "Froude and the contribution of naval architecture to our understanding of bipedal locomotion", Gait & Posture 21 (3): 350–362, doi:10.1016/j.gaitpost.2004.01.011
References
See also
External links
