Froude number

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The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of a partially submerged object moving through water, and permits the comparison of objects of different sizes. Named after William Froude, the Froude number is based on the speed/length ratio as defined by him.

The 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 analogous to the Mach number. The greater the Froude number, the greater the resistance.

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]).

File:Boat models by William Froude.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.
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\[\mathrm{Speed\ Length\ Ratio} =\frac {V}{\sqrt \mathrm{LWL} }\]

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. 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]

\[\mathrm{Fr} = \frac{V}{\sqrt{gL}},\]

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. Note that the Froude number used for ships, by convention, is the square root of the Froude number as defined above.

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, 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 \[\widehat{\mathrm{Fr}}=\frac{V^2}{gd},\] where each of the terms on the right have been squared.[5] This form is the reciprocal of the Richardson number.

Extended Froude number

Geophysical mass flows such as avalanches and debris flows take place on inclined slopes which then merges into a gentle and flat run-out zones.[6] [7] So, these flows are associated with the elevation of the topographic slopes that induce the gravity potential energy together with the pressure potential energy during the flow. Therefore, the classical Froude number should include this additional effect. For such a situation, Froude number needs to be re-defined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:[8] \[\mathrm{Fr}=\frac{u}{\sqrt{\beta h + s_g (x_d - x)}},\] where \(u\) is the mean flow velocity, \(\beta = g K\cos\zeta\), (\(K\) is the earth pressure coefficient, \(\zeta\) is the slope), \(s_g = g\sin\zeta\), \(x\) is the channel downslope position and \(x_d\) is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; \( E_{pot}^p = \beta h\) and \(E_{pot}^g = s_g(x_d - x)\) are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallow-water or granular flow Froude number, the potential energy associated with the surface elevation, \(E_{pot}^g =s_g(x_d - x)\), is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term \(\beta h\) emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows \(\beta h\) is of order \(\ll 1 \), while \(u\) and \(s_g(x_d - x)\) are both of order unity. If the mass is shallow with a virtually bed-parallel free-surface, then \(\beta h\) can be disregarded. In this situation, if the gravity potential is not taken into account, then Fr is unbounded even though the kinetic energy is bounded. So, formally considering the additional contribution due to the gravitational potential energy, the singularity in Fr is removed.

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: \[\mathrm{Fr}=\frac{N^2d}{g}.\]

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.

Walking Froude number

The Froude number may be used to study trends in animal gait patterns. In analyses of the dynamics of legged locomotion, a walking limb is often modeled as an inverted pendulum, where the center of mass goes through a circular arc centered at the foot.[9] The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:

\[\mathrm{Fr}=\frac{\text{centripetal force}}{\text{gravitational force}}=\frac{mV^2/l}{mg}=\frac{V^2}{gl}\]

where \(m\) is the mass, \(l\) is the characteristic length, \(g\) is the acceleration due to gravity and \(V\) is the velocity. The characteristic length, \(l\), may be chosen to suit the study at hand. For instance, some studies have used the vertical distance of the hip joint from the ground,[10] while others have used total leg length.[9][11]

The Froude number may also be calculated from the stride frequency \(f\) as follows[10]:

\[\mathrm{Fr}=\frac{V^2}{gl}=\frac{(lf)^2}{gl}=\frac{lf^2}{g}.\]

If total leg length is used as the characteristic length, then the theoretical maximum speed of walking has a Froude number of \(1.0\) since any higher value would result in 'take-off' and the foot missing the ground. The typical transition speed from bipedal running to walking occurs with \(\mathrm{Fr} \approx 0.5\).[12] R. McN. Alexander found that animals of different sizes and masses travelling at different speeds, but with the same Froude number, consistently exhibit similar gaits. This study found that animals typically switch from an amble to a symmetric running gait (e.g. a trot or pace) around a Froude number of \(1.0\). A preference for asymmetric gaits (e.g. a canter, transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between \(2.0\) and \(3.0\).[10]

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.

The Froude number has been used to study trends in animal locomotion in order to better understand why animals use different gait patterns [10] as well as to form hypotheses about the gaits of extinct species.[11]

Notes

  1. ↑ Script error
  2. ↑ Script error
  3. ↑ Chanson (2004), p. xxvii.
  4. ↑ Script error, p. 28.
  5. ↑ Frank M. White, Fluid Mechanics, 4th edition, McGraw-Hill (1999), 294.
  6. ↑ Script error
  7. ↑ Script error
  8. ↑ Script error
  9. ↑ 9.0 9.1 Script error
  10. ↑ 10.0 10.1 10.2 10.3 Script error
  11. ↑ 11.0 11.1 Script error
  12. ↑ Script error

References

  • Script error, 650 pp.

External links

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