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e 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 froude number fr in order to establish the optimum operating window 18 see also edit flow velocity vector field which is used to mathematically describe the motion of a continuum body force force which acts throughout the volume of a body cauchy momentum equation burgers equation partial differential equation euler equations fluid dynamics set of quasilinear hyperbolic equations governing adiabatic and inviscid flow reynolds number ratio of inertial to viscous forces acting on a liquid notes edit merriam webster online for brother james anthony froude shih 2009 p 7 white 1999 p 294 chanson 2009 pp 159 163 normand 1888 pp 257 261 chanson 2004 p xxvii shih 2009 newman 1977 p 28 alexander r mcneill 2013 10 01 chapter 2 body support scaling and allometry functional vertebrate morphology harvard university press pp 26 37 doi 10 4159 harvard 9780674184404 c2 isbn 978 0 674 18440 4 alexander r mcn 1977 allometry of the limbs of antelopes bovidae journal of zoology 183 1 125 146 doi 10 1111 j 1469 7998 1977 tb04177 x issn 0952 8369 alexander r mcneill 1991 how dinosaurs ran scientific american 264 4 130 137 bibcode 1991sciam 264d 130a doi 10 1038 scientificamerican0491 130 issn 0036 8733 jstor 24936872 takahashi 2007 p 6 powder mixing powder mixers design ribbon blender paddle mixer drum blender froude number powderprocess net n d retrieved 31 may 2019 1 2 vaughan o malley 2005 pp 350 362 1 2 3 4 alexander 1984 1 2 sellers manning 2007 alexander 1989 jikar dhokey shinde 2021 references edit 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 s2cid 120138903 alexander rm 1989 optimization and gaits in the locomotion of vertebrates physiological reviews 69 4 1199 227 doi 10 1152 physrev 1989 69 4 1199 pmid 2678167 belanger jean baptiste 1828 essai sur la solution numerique de quelques problemes relatifs au mouvement permanent des eaux courantes an essay on the numerical solution to some problems relative to the steady movement of running water in french paris carilian goeury chanson hubert 2004 hydraulics of open channel flow an introduction 2nd ed butterworth heinemann p 650 isbn 978 0 7506 5978 9 chanson hubert 2009 development of the bélanger equation and backwater equation by jean baptiste bélanger 1828 pdf journal of hydraulic engineering 135 3 159 63 doi 10 1061 asce 0733 9429 2009 135 3 159 jikar p c dhokey n b shinde s s 2021 numerical modeling simulation and experimental study of dynamic particle bed counter current reactor and its effect on solid gas reduction reaction mining metallurgy exploration 39 springer 139 152 doi 10 1007 s42461 021 00516 6 issn 2524 3462 s2cid 244507908 newman john nicholas 1977 marine hydrodynamics cambridge massachusetts mit press isbn 978 0 262 14026 3 normand j a 1888 on the fineness of vessels in relation to size and speed transactions of the institution of naval architects 29 257 261 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 doi 10 1098 rspb 2007 0846 jstor 25249388 pmc 2279215 pmid 17711833 shih y c spring 2009 chapter 6 incompressible inviscid flow pdf fluid mechanics takahashi tamotsu 2007 debris flow mechanics prediction and countermeasures crc press isbn 978 0 203 94628 2 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 doi 10 1016 j gaitpost 2004 01 011 pmid 15760752 white frank m 1999 fluid mechanics 4th ed wcb mcgraw hill isbn 978 0 07 116848 9 external links edit https web archive org web 20070927085042 http www qub ac uk waves fastferry reference mca457 pdf v t e fluid mechanics fluid statics hydraulics archimedes principle fluid dynamics computational fluid dynamics aerodynamics navier stokes equations boundary layer entrance length dimensionless numbers alfvén mach archimedes atwood bagnold bejan biot bodenstein bond brinkman capillary cauchy chandrasekhar damköhler darcy dean deborah dukhin eckert ekman eötvös euler froude galilei graetz grashof görtler hagen iribarren kapitza keulegan carpenter knudsen laplace lewis mach marangoni morton nusselt ohnesorge péclet prandtl magnetic turbulent rayleigh reynolds magnetic richardson roshko rossby rouse schmidt scruton sherwood shields stanton stokes strouhal stuart suratman taylor ursell weber weissenberg womersley retrieved from https en wikipedia org w index php title froude_number oldid 1347609668 categories dimensionless numbers of fluid mechanics fluid dynamics naval architecture hidden categories articles with short description short description is different from wikidata cs1 french language sources fr this page was last edited on 7 april 2026 at 17 56 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy 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