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).
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.
However, in naval architecture the Froude number is a significant figure used to determine the resistance of a partially submerged object moving through water.
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:
[6] To show how the Froude number is linked to general continuum mechanics and not only to hydrodynamics we start from the Cauchy momentum equation in its dimensionless (nondimensional) form.
Substitution of these inverse relations in the Euler momentum equations, and definition of the Froude number:
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 make explicit the material derivative):
The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.
In marine hydrodynamic applications, the Froude number is usually referenced with the notation Fn and is defined as:[8]
In the case of planing craft, where the waterline length is too speed-dependent 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:
The wave velocity, termed celerity c, is equal to the square root of gravitational acceleration g, times cross-sectional area A, divided by free-surface width B:
When considering wind effects on dynamically sensitive structures such as suspension bridges it is sometimes necessary to simulate the combined effect of the vibrating mass of the structure with the fluctuating force of the wind.
The Froude number has also been applied in allometry to studying the locomotion of terrestrial animals,[9] including antelope[10] and dinosaurs.
The extended Froude number is defined as the ratio between the kinetic and the potential energy:
is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; Eppot = βh and Egpot = sg(xd − 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, Egpot, is not considered.
The term βh emerges from the change of the geometry of the moving mass along the slope.
Dimensional analysis suggests that for shallow flows βh ≪ 1, while u and sg(xd − x) are both of order unity.
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.
In the study of stirred tanks, the Froude number governs the formation of surface vortices.
If Fr<1, the particles are just stirred, but if Fr>1, centrifugal forces applied to the powder overcome gravity and the bed of particles becomes fluidized, at least in some part of the blender, promoting mixing[13] When used in the context of the Boussinesq approximation the densimetric Froude number is defined as
For example, the leading edge of a gravity current moves with a front Froude number of about unity.
[14] The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:
For instance, some studies have used the vertical distance of the hip joint from the ground,[15] while others have used total leg length.
The typical transition speed from bipedal walking to running occurs with Fr ≈ 0.5.
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.
[15] The Froude number is used to compare the wave making resistance between bodies of various sizes and shapes.
[16] In addition particle bed behavior can be quantified by Froude number (Fr) in order to establish the optimum operating window.
