Cauchy momentum equation

In order to make the equations dimensionless, a characteristic length r₀, and a characteristic velocity u₀, 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:

$${\displaystyle \rho ^{*}\equiv {\frac {\rho }{\rho _{0}}},\quad u^{*}\equiv {\frac {u}{u_{0}}},\quad r^{*}\equiv {\frac {r}{r_{0}}},\quad t^{*}\equiv {\frac {u_{0}}{r_{0}}}t,\quad \nabla ^{*}\equiv r_{0}\nabla ,\quad \mathbf {g} ^{*}\equiv {\frac {\mathbf {g} }{g_{0}}},\quad {\boldsymbol {\sigma }}^{*}\equiv {\frac {\boldsymbol {\sigma }}{p_{0}}},}$$

Substitution of these inverse relations in the Euler momentum equations, and definition of the Froude number:

$${\displaystyle \mathrm {Fr} ={\frac {u_{0}}{\sqrt {g_{0}r_{0}}}},}$$

and the Euler number:

$${\displaystyle \mathrm {Eu} ={\frac {p_{0}}{\rho _{0}u_{0}^{2}}},}$$

the equations are finally expressed (with the material derivative and now omitting the indexes):

Cauchy momentum equation (nondimensional convective form)

$${\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}={\frac {1}{\mathrm {Fr} ^{2}}}\mathbf {g} }$$

Cauchy-type 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 make explicit the material derivative):

Burgers equation (nondimensional conservation form)

$${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot \left({\frac {1}{2}}\mathbf {u} \otimes \mathbf {u} \right)={\frac {1}{\mathrm {Fr} ^{2}}}\mathbf {g} }$$

This is an inhomogeneous pure advection equation, as much as the Stokes equation is a pure diffusion equation.

Euler momentum equation

Euler momentum equation is a Cauchy momentum equation with the Pascal law being the stress constitutive relation:

$${\displaystyle {\boldsymbol {\sigma }}=p\mathbf {I} }$$

in nondimensional Lagrangian form is:

$${\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {\nabla p}{\rho }}={\frac {1}{\mathrm {Fr} ^{2}}}{\hat {g}}}$$

Free Euler equations are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.

Incompressible Navier–Stokes momentum equation

Incompressible Navier–Stokes momentum equation is a Cauchy momentum equation with the Pascal law and Stokes's law being the stress constitutive relations:

$${\displaystyle {\boldsymbol {\sigma }}=p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathsf {T}}\right)}$$

in nondimensional convective form it is:^[7]

$${\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {\nabla p}{\rho }}={\frac {1}{\mathrm {Re} }}\nabla ^{2}u+{\frac {1}{\mathrm {Fr} ^{2}}}{\hat {g}}}$$

where Re is the Reynolds number. Free Navier–Stokes equations are dissipative (non conservative).

Ship hydrodynamics

Wave pattern versus speed, illustrating various Froude numbers.

In marine hydrodynamic applications, the Froude number is usually referenced with the notation Fn and is defined as:^[8]

$${\displaystyle \mathrm {Fn} _{L}={\frac {u}{\sqrt {gL}}},}$$

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 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:

$${\displaystyle \mathrm {Fn} _{V}={\frac {u}{\sqrt {g{\sqrt[{3}]{V}}}}}.}$$

Shallow water waves

For shallow water waves, such as tsunamis and hydraulic jumps, the characteristic velocity U is the average flow velocity, averaged over the cross-section perpendicular to the flow direction. 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:

$${\displaystyle c={\sqrt {g{\frac {A}{B}}}},}$$

so the Froude number in shallow water is:

$${\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g{\dfrac {A}{B}}}}}.}$$

For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:

$${\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {gd}}}.}$$

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.

Wind engineering

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. In such cases, the Froude number should be respected. Similarly, when simulating hot smoke plumes combined with natural wind, Froude number scaling is necessary to maintain the correct balance between buoyancy forces and the momentum of the wind.

Allometry

The Froude number has also been applied in allometry to studying the locomotion of terrestrial animals,^[9] including antelope^[10] and dinosaurs.^[11]

Stirred tanks

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:

$${\displaystyle \mathrm {Fr} =\omega {\sqrt {\frac {r}{g}}}.}$$

The Froude number finds also a similar application in powder mixers. It will indeed be used to determine in which mixing regime the blender is working. 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]

Densimetric Froude number

When used in the context of the Boussinesq approximation the densimetric Froude number is defined as

$${\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g'h}}}}$$

where g′ is the reduced gravity:

$${\displaystyle g'=g{\frac {\rho _{1}-\rho _{2}}{\rho _{1}}}}$$

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.^[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:

$${\displaystyle \mathrm {Fr} ={\frac {\text{centripetal force}}{\text{gravitational force}}}={\frac {\;{\frac {mv^{2}}{l}}\;}{mg}}={\frac {v^{2}}{gl}}}$$

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,^[15] while others have used total leg length.^[14][16]

The Froude number may also be calculated from the stride frequency f as follows:^[15]

$${\displaystyle \mathrm {Fr} ={\frac {v^{2}}{gl}}={\frac {(lf)^{2}}{gl}}={\frac {lf^{2}}{g}}.}$$

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 takeoff and the foot missing the ground. The typical transition speed from bipedal walking to running occurs with Fr ≈ 0.5.^[17] R. M. 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.^[15]