Froude number
In continuum mechanics, the Froude number (Fr) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). Named after William Froude, the Froude number is based on the speed–length ratio as defined by him, and is defined as:^{[1]}^{[2]}
 <math>
\mathrm{Fr} = \frac{u_0}{\sqrt{g_0 l_0}} </math>
where u_{0} is a characteristic flow velocity, g_{0} is in general a characteristic external field, and l_{0} is a characteristic length. 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 very significant figure used to determine the resistance of a partially submerged object moving through water, and permits the comparison of similar objects of different sizes, because the wave pattern generated is similar at the same Froude number only.
The Denny Ship Model Experiment tank in Dumbarton, Scotland, has a bust of Froude near the front door.
Contents
Origins[edit]
In open channel flows, Bélanger (1828^{[3]}) 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. Froude's observations led him to derive the WaveLine 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 1852 for testing ships and propellers. Speed/length ratio was originally defined by Froude in his Law of Comparison in 1868 in dimensional terms as:
<math>\text{Speed Length Ratio} =\frac{u}{\sqrt {\text{LWL}} }</math>
where:
 u = flow speed
 LWL = length of waterline
The term was converted into nondimensional 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.^{[5]}
Definition and main application[edit]
To show how the Froude number is linked to general continuum mechanics and not only to hydrodynamics we start from the nondimesionalisation of Cauchy momentum equation.
Cauchy momentum equation[edit]
In order to make the equations dimensionless, a characteristic length r_{0}, and a characteristic velocity u_{0}, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
<math> \rho^*\equiv \frac \rho {\rho_0}, </math> <math> u^*\equiv \frac u {u_0}, </math> <math> r^*\equiv \frac r {r_0}, </math> <math>t^*\equiv \frac {u_0}{r_0} t, </math> <math>\nabla^*\equiv r_0 \nabla , </math> <math>\mathbf g^* \equiv \frac {\mathbf g} {g_0}, </math> <math>\boldsymbol \sigma^* \equiv \frac {\boldsymbol \sigma} {p_0}, </math>
Substitution of these inversed relations in the Euler momentum equations, and definition of the Froude number:
 <math>\mathrm{Fr}=\frac{u_0}{\sqrt{g_0 r_0}},</math>
and the Euler number:
 <math>\mathrm{Eu}=\frac{p_0}{\rho_0 u_0^2},</math>
the equations are finally expressed (with the material derivative and now omitting the indexes):

Cauchy momentum equation (nondimensional convective form)  <math>
{D \bold u \over D t} + \mathrm{Eu} \, \frac 1 \rho \nabla \cdot \boldsymbol \sigma = \frac 1 {\mathrm{Fr}^2} \mathbf g </math>
Cauchytype equations in the high Froude limit Fr → ∞ (corresponding to negligible external field) are named free equations. On the other hand in the low Euler limit Eu → 0 (corresponding to negligible stress) general Cauchy momentum equation becomes an inhomogeneous Burgers equation (here we explicit the material derivative):

Burgers equation (nondimensional conservation form)  <math>
{\partial \bold u \over \partial t} + \nabla \cdot \left(\frac 1 2 \mathbf u \otimes \mathbf u \right) = \frac 1 {\mathrm{Fr}^2} \mathbf g </math>
This is an inhomogeneous pure advection equation, as much as the Stokes equation is a pure diffusion equation.
Euler momentum equation[edit]
Euler momentum equation is a Cauchy momentum equation with the Pascal law being the stress constitutive relation:
<math>\boldsymbol \sigma = p \boldsymbol I </math>
in nondimensional Lagrangian form is:
 <math>{D \bold u \over D t} + \mathrm{Eu} \, \frac {\nabla p}{\rho}= \frac 1 {\mathrm{Fr}^2} \hat g </math>
Free Euler equations are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.
[edit]
Incompressible NavierStokes momentum equation is a Cauchy momentum equation with the Pascal law and Stokes's law being the stress constitutive relations:
<math>\boldsymbol \sigma = p \mathbf I + \mu (\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\text{T}) </math>
in nondimensional convective form it is:^{[6]}
 <math>{D \bold u \over D t} + \mathrm{Eu} \, \frac {\nabla p}{\rho} =  \frac 1 {\mathrm{Re}} \, \nabla^2 u + \frac 1 {\mathrm{Fr}^2} \hat g </math>
where Re is the Reynolds number. Free NavierStokes equations are dissipative (non conservative).
Other applications[edit]
Ship hydrodynamics[edit]
In marine hydrodynamic applications, the Froude number usually referenced with the notation Fn and is defined as:^{[7]}
 <math>\mathrm{Fn_L} = \frac{u}{\sqrt{gL}},</math>
where u is the relative flow velocity between the sea and ship, g is in particular the acceleration due to gravity, and L is the length of the ship at the water line level, or L_{wl} in some notations. It is an important parameter with respect to the ship's drag, or resistance, especially in terms of wave making resistance.
In the case of planing crafts, where the waterline length is too speeddependent to be meaningful, the Froude number is best defined as displacement Froude number and the reference length is taken as the cubic root of the volumetric displacement of the hull:
 <math>\mathrm{Fn_V} = \frac{u}{\sqrt{gV^{1/3}}}.</math>
Shallow water waves[edit]
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 crosssection perpendicular to the flow direction. The wave velocity, c, is equal to the square root of gravitational acceleration g, times crosssectional area A, divided by freesurface width B:
 <math>
c = \sqrt{g \frac{A}{B}}, </math> so the Froude number in shallow water is:
 <math>
\mathrm{Fr} = \frac{U}{\sqrt{\displaystyle g \frac{A}{B}}}. </math> For rectangular crosssections with uniform depth d, the Froude number can be simplified to:
 <math>
\mathrm{Fr} = \frac{U}{\sqrt{gd}}. </math> 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.
Extended Froude number[edit]
Geophysical mass flows such as avalanches and debris flows take place on inclined slopes which then merge into gentle and flat runout zones.^{[8]} 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 redefined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:
 <math>\mathrm{Fr}=\frac{u}{\sqrt{\beta h + s_g (x_d  x)}},</math>
where <math>u</math> is the mean flow velocity, <math>\beta = g K\cos\zeta</math>, (<math>K</math> is the earth pressure coefficient, <math>\zeta</math> is the slope), <math>s_g = g\sin\zeta</math>, <math>x</math> is the channel downslope position and <math>x_d</math> is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; <math> E_{pot}^p = \beta h</math> and <math>E_{pot}^g = s_g(x_d  x)</math> are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallowwater or granular flow Froude number, the potential energy associated with the surface elevation, <math>E_{pot}^g =s_g(x_d  x)</math>, is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term <math>\beta h</math> emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows <math>\beta h</math> is of order <math>\ll 1 </math>, while <math>u</math> and <math>s_g(x_d  x)</math> are both of order unity. If the mass is shallow with a virtually bedparallel freesurface, then <math>\beta h</math> 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[edit]
In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is ωr (circular motion), where ω is the impeller frequency (usually in rpm) and r is the impeller radius (in engineering the diameter is much more frequently employed) , the Froude number then takes the following form:
 <math>
\mathrm{Fr}=\omega \sqrt \frac{r}{g}. </math>
Densimetric Froude number[edit]
When used in the context of the Boussinesq approximation the densimetric Froude number is defined as
 <math>
\mathrm{Fr}=\frac{u}{\sqrt{g' h}} </math> where <math>g'</math> is the reduced gravity:
 <math>
g' = g{\rho_1\rho_2\over {\rho_1}} </math>
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[edit]
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:
 <math>
\mathrm{Fr}=\frac{\text{centripetal force}}{\text{gravitational force}}=\frac{mv^2/l}{mg}=\frac{v^2}{gl} </math>
where <math>m</math> is the mass, <math>l</math> is the characteristic length, <math>g</math> is the acceleration due to gravity and <math>v</math> is the velocity. The characteristic length, <math>l</math>, 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]}
 <math>
\mathrm{Fr}=\frac{v^2}{gl}=\frac{(lf)^2}{gl}=\frac{lf^2}{g}. </math>
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 'takeoff' and the foot missing the ground. The typical transition speed from bipedal walking to running occurs with <math>\mathrm{Fr} \approx 0.5</math>.^{[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[edit]
The Froude number is used to compare the wave making resistance between bodies of various sizes and shapes.
In freesurface flow, the nature of the flow (supercritical or subcritical) depends upon whether the Froude number is greater than or less than unity. You can easily see the line of "critical" flow in you 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 fast. 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". That's 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 ^{[10]} as well as to form hypotheses about the gaits of extinct species.^{[11]}
Notes[edit]
 ↑ Shih, 6.Incompressible fluid flow, p.7
 ↑ Frank M. White, Fluid Mechanics, 4th edition, McGrawHill (1999), 294.
 ↑ Bélanger, JeanBaptiste (1828). Essai sur la Solution Numérique de quelques Problèmes Relatifs au Mouvement Permanent des Eaux Courantes. Paris, France, 38 pages & 5 tables: CarilianGoeury.
 ↑ Chanson, Hubert (2009). "Development of the Bélanger Equation and Backwater Equation by JeanBaptiste Bélanger (1828)". Journal of Hydraulic Engineering. 135 (3): 159–63. doi:10.1061/(ASCE)07339429(2009)135:3(159).
 ↑ Chanson (2004), p. xxvii.
 ↑ Shih, 6.Incompressible fluid flow
 ↑ Newman, John Nicholas (1977). Marine hydrodynamics. Cambridge, Massachusetts: MIT Press. ISBN 0262140268., p. 28.
 ↑ Takahashi, T. (2007). Debris Flow: Mechanics, Prediction and Countermeasures. Taylor and Francis, Leiden.^{[page needed]}
 ↑ ^{9.0} ^{9.1} Vaughan, Christopher L.; O’Malley, Mark J. (2005). "Froude and the contribution of naval architecture to our understanding of bipedal locomotion". Gait & Posture. 21 (3): 350–62. PMID 15760752. doi:10.1016/j.gaitpost.2004.01.011.
 ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} Alexander, R. McN. (1984). "The Gaits of Bipedal and Quadrupedal Animals". The International Journal of Robotics Research. 3 (2): 49–59. doi:10.1177/027836498400300205.
 ↑ ^{11.0} ^{11.1} Sellers, William Irvin; Manning, Phillip Lars (2007). "Estimating dinosaur maximum running speeds using evolutionary robotics". Proceedings of the Royal Society B: Biological Sciences. 274 (1626): 2711–6. JSTOR 25249388. PMC 2279215 Freely accessible. PMID 17711833. doi:10.1098/rspb.2007.0846.
 ↑ Alexander, RM (1989). "Optimization and gaits in the locomotion of vertebrates". Physiological reviews. 69 (4): 1199–227. PMID 2678167.
See also[edit]
 Flow velocity
 External field
 Cauchy momentum equation
 Burgers equation
 Euler equations (fluid dynamics)
References[edit]
 Chanson, Hubert (2004). Hydraulics of Open Channel Flow: An Introduction (2nd ed.). Butterworth–Heinemann. ISBN 0750659785., 650 pp