How to compute the Froude number: introduced by William Froude (Great Britain 1810-1879) Plymouth naval engineer founder of modern naval hydrodynamics, it is a fundamental dimensionless parameter measuring the ratio between inertial force and gravitational force in a fluid moving with free surface. Fundamental formula: Fr = v / sqrt(g * L), where v characteristic flow velocity (m/s), g gravitational acceleration (9.80665 m/s^2 SI standard), L characteristic length (m) - hydraulic depth for open channels, waterline length Lwl for ships. Essential tool for: civil hydraulic engineering (irrigation canals, dam spillways Hoover Dam Glen Canyon Bureau Reclamation, energy-dissipating hydraulic jumps Belanger 1828), naval hydrodynamics (ship resistance Maersk MSC Carnival Cruise Lines, yachts Bertram Hatteras Sea Ray, US Navy DDG Arleigh Burke-class destroyers), fluvial morphology (mountain torrents vs lowland rivers Mississippi Ohio Colorado Hudson USA), aerospace (space launchers atmospheric phase hydrodynamic tracking SpaceX NASA), geophysics (granular avalanches, mud flows, volcanic lava flows). Flow regime classification: Fr<1 subcritical (tranquil fluvial flow, perturbations propagate upstream), Fr=1 critical (critical condition discovered Jean-Baptiste Belanger France 1828, regime transition), Fr>1 supercritical (rapid torrential flow, perturbations only downstream, hydraulic jumps possible).

Froude number formulas

Basic definition: Fr = v/sqrt(g*L). Dimensionless, independent of unit system choice (SI, US, CGS). Fr^2 = v^2/(g*L) = kinetic energy / surface potential energy ratio. Equivalently: Fr = v/c, where c=sqrt(g*L) celerity gravitational surface waves (long wave speed shallow water). For Fr1: waves only downstream (supercritical, analogy aerodynamic shock wave sonic).

Open channel - hydraulic depth D: D = A/T, wetted cross-section area A divided by free surface width T. For rectangular channel width b, depth y: D=y. For trapezoidal channel base b, slopes m:1, depth y: A=y(b+my), T=b+2my, D=A/T. For partially filled circular channel: D depends on filling angle. Manual calculation or software HEC-RAS (USACE U.S. Army Corps Engineers).

Naval Froude number Fn: Fn = v/sqrt(g*Lwl), where Lwl waterline length of ship (m), v velocity knots converted m/s (1 knot=0.5144 m/s). Savitsky 1964 hull classification: Fn<0.4 displacement (cargo ships), 0.4<Fn1.0 planing (sport boats, F1 powerboats). Volumetric Froude number FnV=v/sqrt(g*displacement^(1/3)).

Belanger 1828 hydraulic jump: for Fr1>1 (upstream supercritical), Fr2<1 (downstream subcritical): depth ratio y2/y1 = 0.5*(sqrt(1+8*Fr1^2)-1) (Belanger conjugate depth equation). Energy dissipated: dE/E1 = (y2-y1)^3 / (4*y1*y2*(y1+y2)). Jump types: Fr19 strong (70-85% violent rough). USBR US Bureau Reclamation Denver Colorado standard.

Critical velocity: vc = sqrt(g*L). At Fr=1 v=vc, flow is minimum specific energy E_min=1.5*Dc for given discharge Q. Critical discharge for rectangular channel: qc = sqrt(g*Dc^3) = sqrt(g)*Dc^1.5 (per unit width). Critical depth yc=(q^2/g)^(1/3). Critical condition important hydraulic node fluvial (Manning Chezy formulas), throat dam spillways.

History and context

William Froude (Great Britain 1810-1879) Devon naval engineer Plymouth worked with Isambard Kingdom Brunel railways and then Royal Navy admiralty from 1859 dedicating to ship hydrodynamics. Published "On the Rolling of Ships" 1861 demonstrating similarity law. His main contribution 1868 "Law of Comparison" defined dimensionless number Fn=v/sqrt(g*L) for scaling reduced model tests Royal Navy admiralty. Built first experimental towing tank world 1872 Torquay Devon, base 100:1 scale ship model testing. His son Robert Edmund Froude (1846-1924) continued work Admiralty Experiment Works AEW Haslar Portsmouth (closed 1989). Jean-Baptiste Belanger (France 1790-1874) Ecole Polytechnique Paris published 1828 "Essai sur la solution numerique de quelques problemes relatifs au mouvement permanent des eaux courantes" introducing critical flow Fr=1 concept and conjugate equation hydraulic jump y2/y1=0.5*(sqrt(1+8Fr1^2)-1). Eduard Mach (Austria 1838-1916) introduced 1887 Mach number analogy (velocity/sound speed ratio) Froude inspired. Lord Kelvin Thomson (Scotland 1824-1907) 1887 derived Kelvin wave theory angle 19.47° produced ships. Henri Poincare (France 1854-1912) floating wave stability theory. John Henry Michell (Australia 1863-1940) 1898 fundamental paper "Wave Resistance of a Ship" Phil Mag 45:106 first analysis viscous resistance vs wave-making. Thomas Havelock (Great Britain 1877-1968) further wave-making development 1909-1934.

Daniel Savitsky (USA 1916-2014) Stevens Institute Technology Hoboken NJ 1964 fundamental "Hydrodynamic Design of Planing Hulls" modern semi-displacement planing classification. USA naval hydrodynamics: David Taylor Model Basin Carderock MD US Navy Naval Surface Warfare Center towing tank facility, Massachusetts Institute Technology MIT Naval Architecture Marine Engineering Department Cambridge MA, Webb Institute Glen Cove NY private naval architecture, US Naval Academy Annapolis MD. Industry: General Dynamics NASSCO San Diego CA shipbuilding, Huntington Ingalls Newport News VA Pascagoula MS major US shipyards Navy contractors, Bath Iron Works ME, Maersk Line Norfolk VA US arm Danish container leader. US naval ships: Arleigh Burke-class destroyers DDG-51 (89 in fleet 2025), Nimitz-class CVN aircraft carriers (10 active), Ohio-class SSBN ballistic missile submarines (14 active), Virginia-class SSN attack submarines (23 active 2025). USBR US Bureau Reclamation Denver Colorado Hoover Dam Glen Canyon Grand Coulee Lake Powell hydraulic dams. NOAA National Oceanic Atmospheric Administration Boulder CO marine forecast models. NSF National Science Foundation funding hydrodynamics research. American Society Civil Engineers ASCE Reston VA, American Society Naval Engineers ASNE Alexandria VA professional associations. Software hydraulics HEC-RAS USACE U.S. Army Corps Engineers Hydrologic Engineering Center Davis California free worldwide standard.

Universities: MIT 2.20 Marine Hydrodynamics, Stanford CEE263 Open Channel Flow, UC Berkeley CE154, University of Michigan NA&ME naval architecture, Scripps Institution Oceanography La Jolla CA marine geology, Woods Hole Oceanographic Institution Falmouth MA.

Typical Froude regime and parameters table

• Subcritical Fr<0.5: tranquil lowland river, irrigation canal
• Subcritical 0.5<Fr<1: fast lowland river, gravity conduits
• Critical Fr=1: minimum energy condition, regime transition
• Supercritical 1<Fr<3: mountain torrent, dam spillways
• Supercritical 3<Fr<10: violent rapids, granular avalanches
• Ship displacement Fn<0.4: cargo VLCC supertanker, container Maersk
• Semi-displacement 0.4<Fn<1.0: yachts Bertram Hatteras Sea Ray
• Planing Fn>1.0: sport boats Donzi Apache, F1 powerboats
• Undular jump Fr1<1.7: no dissipation, smooth surface
• Weak jump 1.7<Fr1<2.5: surface rollers, 5-15% energy
• Oscillating jump 2.5<Fr1<4.5: unstable, 15-45% dissipation
• Stable jump 4.5<Fr1<9: optimum dissipation 45-70%
• Strong jump Fr1>9: violent rough 70-85% dissipation
• Shallow water wave celerity: c=sqrt(g*y), c=3.13 m/s for y=1 m
• Kelvin wave half-angle: 19.47° regardless ship velocity
• SI standard gravity: g=9.80665 m/s^2 (1901 CGPM)
• 1 knot: 0.5144 m/s = 1.852 km/h (nautical mile/hour)
• Critical velocity y=2 m: vc=sqrt(9.81*2)=4.43 m/s=8.6 knots
• Critical velocity Lwl=100 m: vc=sqrt(9.81*100)=31.3 m/s=60.9 knots
• Mach number analog: Ma=v/c_sound, Fr=v/c_grav

Detailed numerical examples

Example 1 - Lowland river Mississippi depth 3 m velocity 2 m/s. Mississippi USA lowland river. v=2 m/s, hydraulic depth D=3 m (assuming wide channel). Fr=2/sqrt(9.81*3)=2/5.42=0.369. SUBCRITICAL Fr<1, tranquil fluvial regime, perturbations propagate upstream. Critical velocity vc=sqrt(9.81*3)=5.42 m/s. Mississippi at Memphis TN typically Fr=0.2-0.4 low-high water. Applications: river navigation Memphis TN-New Orleans LA barge transport, agriculture irrigation USACE flood control levees.

Example 2 - Cargo ship Maersk container Lwl=300 m velocity 20 knots. Container ship Maersk Triple-E class Lwl=300 m, v=20 knots=10.29 m/s. Fn=10.29/sqrt(9.81*300)=10.29/54.25=0.190. DISPLACEMENT Fn<0.4, optimal cargo ship hull regime. Resistance dominated by viscous friction (high Reynolds), wave-making minor. Suez Canal max v=8 knots (Fn=0.076 ultra-displacement). Maersk Line Copenhagen Denmark global container leader 700 ships 4 million TEU Twenty-foot Equivalent Unit capacity worldwide.

Example 3 - Mountain torrent Colorado depth 0.5 m velocity 5 m/s. Colorado River tributary Rocky Mountains, depth 0.5 m, v=5 m/s (fast mountain course). Fr=5/sqrt(9.81*0.5)=5/2.21=2.26. SUPERCRITICAL 1<Fr<5, rapid torrential flow. Hydraulic jump possible at waterfall base: y2/y1=0.5*(sqrt(1+8*2.26^2)-1)=0.5*(sqrt(41.9)-1)=0.5*(6.47-1)=2.74. Jump depth y2=0.5*2.74=1.37 m. Energy dissipation 28% (oscillating). Application Colorado USACE Denver flood control hydraulics, weir gates regulation.

Example 4 - California irrigation canal trapezoidal. Central Valley Project California: A=10 m^2, B=5 m free surface width, v=10 km/h=2.78 m/s. D=A/B=10/5=2 m. Fr=2.78/sqrt(9.81*2)=2.78/4.43=0.628. SUBCRITICAL 0.5<Fr<1, fast lowland flow. Critical velocity vc=4.43 m/s. California State Water Project 1100 km canal system 1960s aqueduct, Bureau Reclamation Sacramento San Joaquin delta supply. Manning Chezy formulas complementary to Froude for friction resistance.

Example 5 - Hoover Dam spillway high head historical. Hoover Dam Bureau Reclamation 221 m altezza Black Canyon Colorado-Arizona-Nevada border, spillway test: D=0.3 m spillway, v=15 m/s high head. Fr=15/sqrt(9.81*0.3)=15/1.716=8.74. HYPER-SUPERCRITICAL Fr>5, violent flow. Strong hydraulic jump: y2/y1=0.5*(sqrt(1+8*76.4)-1)=11.87. Jump depth y2=0.3*11.87=3.56 m. Energy dissipation 76% (strong). Application USBR Type I-IV stilling basins design Hoover Dam Glen Canyon Grand Coulee dams. Hoover Dam Boulder City NV Bureau Reclamation engineering icon 1936 inauguration.

Limits and considerations

1. Froude number valid only for free-surface fluids (open channels, ships). For closed conduits use Reynolds Re.
2. Characteristic length L: hydraulic depth D=A/T channels, Lwl waterline ships. Inappropriate choice alters Fr.
3. Belanger 1828 hydraulic jump valid only for horizontal rectangular channels. Trapezoidal corrections Chow.
4. Naval Fn classification Savitsky 1964 modern hulls. Sailboat Fn depends also on v_wind.
5. Viscosity (Re), surface tension (We) effects neglected: valid for L>1 m, v>0.5 m/s typically.
6. Hydraulic depth D=A/T overestimates Fr for narrow free surface width channels.
7. Compressibility neglected: Fr<<sound_speed/sqrt(g*L) always true water (Mach<<1 water).
8. Tide influence not considered: estuary river Fr requires tidal cycle correction.
9. Sediment transport fluvial hydraulics requires Shields parameter Fr*shear not just macroscopic Fr.
10. Kelvin wave angle 19.47° valid only for deep water h>>L/2 (deep water condition).

Common errors

1. Confusing hydraulic depth D=A/T with water height h: rectangular channels equal, trapezoidal different.
2. Calculating ship Fr with Lwl height: correct waterline length, never draft.
3. Neglecting knot-m/s conversion: 1 knot=0.5144 m/s, 14% error if 0.5 used.
4. Confusing Froude Fr with Reynolds Re: Fr inertia/gravity, Re inertia/viscosity.
5. Assuming ship Fr=open channel Fr: different applications, different scaling laws.
6. Neglecting Belanger 1828 when Fr1>1 jump upstream: y2/y1 critical for design dissipation.
7. Calculating wrong critical velocity: vc=sqrt(g*L), NOT v_c=sqrt(g*L^2) or other.
8. Assuming hydraulic jump for Frsubcritical transition.
9. Ignoring scale Froude similarity effects: ship model test scaling requires Fr_model=Fr_full.
10. Confusing naval Fn with volumetric FnV: Fn=v/sqrt(gLwl), FnV=v/sqrt(g*displacement^(1/3)).

Numbered applications

1. Naval hydrodynamics: David Taylor Model Basin Carderock MD US Navy, MIT Naval Architecture, Webb Institute Glen Cove NY.
2. US Navy ships: Arleigh Burke-class DDG-51 (89 in fleet), Nimitz-class CVN aircraft carriers (10), Ohio Virginia submarines.
3. US shipbuilding: General Dynamics NASSCO San Diego, Huntington Ingalls Newport News Pascagoula, Bath Iron Works ME.
4. Maritime commerce: Maersk Line Norfolk VA US arm, MSC container leader, Carnival Cruise Lines Miami FL.
5. Hydraulic engineering USA: USACE Davis CA HEC-RAS standard, Bureau Reclamation Denver Hoover Dam Glen Canyon.
6. Irrigation systems: California State Water Project 1100 km, Central Valley Project Bureau Reclamation Sacramento.
7. River navigation: Mississippi River barge transport Memphis TN-New Orleans LA, Ohio Cincinnati Pittsburgh, Hudson NY Albany.
8. Geophysics avalanches: USGS US Geological Survey Menlo Park CA, NOAA Boulder CO marine forecast models.
9. Yachts recreation: Bertram Yachts FL, Hatteras Yachts NC, Sea Ray Knoxville TN, Donzi Apache F1 powerboats.
10. Educational teaching: MIT 2.20 Marine Hydrodynamics, Stanford CEE263 Open Channel Flow, UC Berkeley CE154.

Support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary hydraulics and hydrodynamics. For professional engineering Froude W 1868 'On the Influence of the Length on the Resistance of Ships' Trans Inst Naval Arch, Belanger JB 1828 'Essai sur la solution numerique des problemes relatifs au mouvement permanent des eaux courantes' Paris, Chow VT 'Open-Channel Hydraulics' 1959 McGraw-Hill, Savitsky D 1964 'Hydrodynamic Design of Planing Hulls' Marine Tech, USBR US Bureau Reclamation 'Design of Small Dams' 1987 Denver Colorado.

Updated July 2026