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nless number ratio of a fluid s flow inertia to the external field in continuum mechanics the froude number fr after william froude ˈ f r uː d 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 the froude number is based on the speed length ratio which he defined as 2 3 f r u g l displaystyle mathrm fr frac u sqrt gl where u is the local flow velocity in m s g is the local gravity field in m s 2 and l is a characteristic length in m 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 significant figure used to determine the resistance of a partially submerged object moving through water origins edit in open channel flows belanger 1828 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 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 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 speed length ratio u lwl displaystyle text speed length ratio frac u sqrt text lwl where u flow speed lwl length of waterline 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 frederic reech 6 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 cauchy momentum equation in its dimensionless nondimensional form cauchy momentum equation edit see also cauchy momentum equation 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 ρ ρ ρ 0 u u u 0 r r r 0 t u 0 r 0 t r 0 g g g 0 σ σ p 0 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 f r u 0 g 0 r 0 displaystyle mathrm fr frac u_ 0 sqrt g_ 0 r_ 0 and the euler number e u p 0 ρ 0 u 0 2 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 d u d t e u 1 ρ σ 1 f r 2 g 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 u t 1 2 u u 1 f r 2 g 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 edit see also euler equations fluid dynamics euler momentum equation is a cauchy momentum equation with the pascal law being the stress constitutive relation σ p i displaystyle boldsymbol sigma p mathbf i in nondimensional lagrangian form is d u d t e u p ρ 1 f r 2 g 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 edit see also navier stokes equations incompressible flow incompressible navier stokes momentum equation is a cauchy momentum equation with the pascal law and stokes s law being the stress constitutive relations σ p i μ u u t displaystyle boldsymbol sigma p mathbf i mu left nabla mathbf u nabla mathbf u mathsf t right in nondimensional convective form it is 7 d u d t e u p ρ 1 r e 2 u 1 f r 2 g 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 other applications edit ship hydrodynamics edit 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 f n l u g l 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 f n v u g v 3 displaystyle mathrm fn _ v frac u sqrt g sqrt 3 v shallow water waves edit 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 c g a b displaystyle c sqrt g frac a b so the froude number in shallow water is f r u g a b 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 f r u g d 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 edit 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 edit the froude number has also been applied in allometry to studying the locomotion of terrestrial animals 9 including antelope 10 and dinosaurs 11 rotary kilns edit the froude number in rotary kiln engineering is a parameter that characterizes the balance between centrifugal forces caused by kiln rotation and gravitational forces acting on the bed material it is defined as f r ω 2 r i g displaystyle mathrm fr frac omega 2 r _ i g where ω is the angular velocity of the kiln in rad s r i is the kiln s internal radius and g is the gravitational acceleration it is used to describe the motion regime of the granular bed so whether the solids slip roll cascade cataract or are centrifuged to the shell low froude numbers typically fr 10 4 correspond to a slipping or slumping regime with very little mixing of the solid bed because particle motion governs mixing heat and mass transfer as well as the residence time in the kiln the froude number serves as a key scaling parameter for rotary kilns a similar froude numbers is a good initial indication for a similar flow behavior and bed profile across laboratory pilot and industrial rotary kilns 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 run out zones 12 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 f r u β h s g x d x displaystyle mathrm fr frac u sqrt beta h s_ g left x_ d x right where u is the mean flow velocity β gk cos ζ k is the earth pressure coefficient ζ is the slope s g g sin ζ x is the channel downslope position and x d displaystyle 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 p pot βh and e g pot 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 g pot is not considered the extended froude number differs substantially from the classical froude number for higher surface elevations 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 s g x d x are both of order unity if the mass is shallow with a virtually bed parallel free surface then βh 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 f r ω r g 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 edit when used in the context of the boussinesq approximation the densimetric froude number is defined as f r u g h displaystyle mathrm fr frac u sqrt g h where g is the reduced gravity g g ρ 1 ρ 2 ρ 1 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 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 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 f r centripetal force gravitational force m v 2 l m g v 2 g l 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 f r v 2 g l l f 2 g l l f 2 g 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 usage edit 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 one can easily see the line of critical flow in a 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 quickly 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 the jump starts 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 15 as well as to form hypotheses about the gaits of extinct species 16 in addition particle bed behavior can be quantified by fr...