Breakthroughs in Fluid Engineering#
“When you can measure what you are speaking about, and express it in numbers, you know something about it.” – William Thomson (Lord Kelvin), lecture to the Institution of Civil Engineers, 1883
Engineering hydraulics is older than the calculus. Archimedes and Stevin
settled fluids at rest, Torricelli timed the jet from an orifice, and the
eighteenth century learned to measure moving water with Pitot’s and
Venturi’s instruments while Euler wrote the equation of every turbine.
Viscous flow in pipes then took a century to understand, from Hagen and Poiseuille’s capillaries
through Reynolds’s dye streak to the friction charts of Nikuradse,
Colebrook and Moody. Alongside them came the scaling laws of Froude and
Buckingham, Stokes’s drag and Couette’s viscometer, the weirs, jumps and
specific energy of open channels, Pelton’s wheel, the pressure waves of
water hammer, cavitation in turbines and the hand methods for city water
networks. The entries trace the ideas behind
engineeringkit.fluid_engineering. Each links to the implementing
function and to a gallery example devoted to it.
c. 250 BC – Archimedes’s Principle#
In the first book of On Floating Bodies Archimedes proved, from the postulate that fluid parts pressed less are pushed by those pressed more, that a solid lighter than a fluid sinks until it displaces its own weight, and that a heavier solid is lightened by the weight of the fluid it displaces. The buoyant force \(F_B = \rho g V\) underlies every float, hull and hydrometer, and the weighing of bodies in water to find their density.
Implementation: engineeringkit.fluid_engineering.buoyancy_force().
The example weighs King Hiero’s crown in water, reads its silver content
back from the apparent weight and floats an iceberg.
References: Archimedes, On Floating Bodies, Book I, Props. 5-7, in T. L. Heath (ed.), The Works of Archimedes (Cambridge: Cambridge University Press, 1897).
1586 – Stevin’s Hydrostatic Paradox#
Simon Stevin, the Flemish engineer of the Dutch dikes and sluices, showed that the force of water on the bottom of a vessel equals the weight of a column standing on the bottom, whatever the shape of the vessel and however little water it holds. He went on to find the force on a vertical or sloping wall by slicing it into strips, the first calculation of engineering hydrostatics. In modern form, \(p = \rho g h\), and the force on a plane is the pressure at its centroid times its area.
Implementation:
engineeringkit.fluid_engineering.hydrostatic_pressure(),
engineeringkit.fluid_engineering.pressure_head(),
engineeringkit.fluid_engineering.hydrostatic_force_plane() and
engineeringkit.fluid_engineering.center_of_pressure_depth(). The
example fills three vessels of different shape to one depth and finds the
thrust on a dam face and where it acts.
References: S. Stevin, De Beghinselen des Waterwichts (Leiden: Plantijn, 1586).
1644 – Torricelli’s Law of Efflux#
Evangelista Torricelli, who succeeded Galileo at Florence, stated at the end of his treatise on the motion of heavy bodies that water spouting from a small hole in a vessel has the speed a body would gain falling from the free surface, so that a jet pointed upward would rise back almost to the surface. In modern form \(v = \sqrt{2gh}\). The jet contracts just past a sharp-edged hole, the vena contracta, and the discharge coefficient of about 0.61 that measures this contraction became a staple of hydraulic tables. Daniel Bernoulli later derived Torricelli’s law from the conservation of energy.
Implementation: engineeringkit.fluid_engineering.orifice_flow()
and engineeringkit.fluid_engineering.tank_drain_time(). The example
drains a tank through an orifice and checks the square-root law of its
level against a step-by-step integration.
References: E. Torricelli, “De motu aquarum,” in De motu gravium naturaliter descendentium et proiectorum, Opera geometrica (Florence: Masse and de Landis, 1644) 191-203. The date and page range were checked against secondary sources (Whewell; S. R. Bistafa’s translation), not the 1644 printing.
1732 – Pitot’s Tube#
Henri Pitot faced the open end of a bent glass tube into the current of the Seine and saw the water inside rise above the river surface by an amount that grew with the speed. His readings showed that the river flows fastest near the surface, against the belief that speed grows with depth. The rise is the velocity head \(v^2/(2g)\), a relation Bernoulli’s Hydrodynamica explained six years later, and Pitot’s tube, combined with a static port, became the standard probe for water and air speed.
Implementation: engineeringkit.fluid_engineering.pitot_velocity()
and engineeringkit.fluid_engineering.dynamic_pressure(). The example
reads river speeds from the rise in the tube and finds the Mach number at
which the incompressible formula overestimates airspeed by 1%.
References: H. Pitot, “Description d’une machine pour mesurer la vitesse des eaux courantes et le sillage des vaisseaux,” Hist. Acad. R. Sci. (1732) 363-376.
1754 – Euler’s Turbine Equation#
Leonhard Euler, at the Berlin academy, analysed the reaction water wheel that Segner had built, and in a long memoir written in 1753 he worked out the theory of turbines in general. Its central result is that the torque on a rotor equals the change in the angular momentum of the water passing through it, so the head a rotor gives or takes is \(H = (u_2V_{t2} - u_1V_{t1})/g\). The equation still opens the design of every pump, fan, compressor and turbine; Jakob Ackeret built and tested a turbine to Euler’s proportions in 1944.
Implementation:
engineeringkit.fluid_engineering.euler_turbomachine_head(), with
engineeringkit.fluid_engineering.affinity_laws(). The example draws
the ideal head of an impeller with backward, radial and forward blades and
recovers the head affinity law from the velocity triangle.
References: L. Euler, “Théorie plus complette des machines qui sont mises en mouvement par la réaction de l’eau,” Mém. Acad. Sci. Berlin 10 (for 1754, publ. 1756) 227-295 (Eneström E222); J. Ackeret, “Untersuchung einer nach den Euler’schen Vorschlägen (1754) gebauten Wasserturbine” (1944), English translation by S. R. Bistafa, arXiv:2108.12048 (2021). The volume and pages are from the Euler Archive; the memoir itself was not consulted.
1797 – Venturi’s Tube#
Giovanni Battista Venturi, professor of physics at Modena, studied water flowing through conical contractions and expansions. He found that the pressure falls in the throat and is mostly recovered when the cone widens gently. Ninety years later Clemens Herschel built the Venturi water meter on the effect: with continuity and Bernoulli’s equation, the pressure drop from inlet to throat gives the volume flow, and the recovery makes the meter cheap to run.
Implementation: engineeringkit.fluid_engineering.venturi_flow(),
engineeringkit.fluid_engineering.continuity_velocity() and
engineeringkit.fluid_engineering.bernoulli_pressure(). The example
draws the pressure along a meter and reads the flow back from it.
References: G. B. Venturi, Recherches expérimentales sur le principe de la communication latérale du mouvement dans les fluides (Paris, 1797); C. Herschel, “The Venturi water meter,” Trans. ASCE 17 (1887) 228-258.
1828 – Bélanger’s Hydraulic Jump#
Jean-Baptiste Bélanger, an engineer of the Ponts et Chaussées, published in 1828 an essay on the numerical solution of steady flow in open channels. It gave the backwater equation for gradually varied flow with a step method to integrate it, identified the critical flow condition where that equation is singular, and treated the abrupt rise of a fast, shallow stream to a deep, slow one: the hydraulic jump. In 1828 he applied Bernoulli’s energy balance across the jump, which is wrong, since the roller dissipates energy. His later lectures (1838, printed 1841) used the momentum principle and gave the relation still known as the Bélanger equation, \(y_2/y_1 = (\sqrt{1 + 8\mathrm{Fr}_1^2} - 1)/2\).
Implementation: engineeringkit.fluid_engineering.sequent_depth()
and engineeringkit.fluid_engineering.hydraulic_jump_head_loss().
The example finds the jump below a sluice gate, checks that momentum is
conserved and energy is not, and draws the depth ratio and loss against
the upstream Froude number.
References: J.-B. Bélanger, Essai sur la solution numérique de quelques problèmes relatifs au mouvement permanent des eaux courantes (Paris: Carilian-Goeury, 1828); H. Chanson, “Development of the Bélanger equation and backwater equation by Jean-Baptiste Bélanger (1828),” J. Hydraul. Eng. 135 (2009) 159-163, doi:10.1061/(ASCE)0733-9429(2009)135:3(159). The account of the 1828 and 1838 derivations follows Chanson; the essay itself was not consulted.
1839-1840 – Hagen and Poiseuille: Laminar Pipe Flow#
Gotthilf Hagen, a Prussian hydraulic engineer, timed water through brass tubes, and Jean Léonard Marie Poiseuille, a physician studying the circulation of the blood, timed liquids through glass capillaries. Both found that slow flow through a tube is proportional to the pressure drop and to the fourth power of the radius. Stokes and others later derived the parabolic velocity profile behind it. In Darcy’s form the same law is the laminar friction factor \(f = 64/Re\).
Implementation: engineeringkit.fluid_engineering.poiseuille_flow(),
engineeringkit.fluid_engineering.poiseuille_velocity() and
engineeringkit.fluid_engineering.laminar_friction_factor(). The
example integrates the parabolic profile, checks the fourth-power law and
recovers the pressure drop from \(64/Re\).
References: G. Hagen, “Über die Bewegung des Wassers in engen cylindrischen Röhren,” Ann. Phys. Chem. 46 (1839) 423-442; J. L. M. Poiseuille, “Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres,” C. R. Acad. Sci. 11 (1840) 961-967, 1041-1048.
Hagen and Poiseuille: laminar pipe flow (1839-1840)
1845-1857 – Weisbach and Darcy: Head Loss in Pipes#
Julius Weisbach, at the Freiberg mining academy, wrote the friction loss in a pipe as a coefficient times \((L/D)\,v^2/(2g)\) and tabulated loss coefficients for bends, valves and contractions. Henry Darcy, engineer of the Dijon water supply, measured losses in new and incrusted pipes up to 0.5 m in diameter and showed that the coefficient depends on the diameter and the state of the wall. Their equation, with Darcy’s name on the friction factor, is how every pipeline is sized.
Implementation: engineeringkit.fluid_engineering.darcy_head_loss(),
engineeringkit.fluid_engineering.minor_head_loss() and
engineeringkit.fluid_engineering.velocity_head(), with the
plot_grade_lines()
visualizer. The example follows the energy and hydraulic grade lines from a
reservoir through two pipes to a free jet.
References: J. Weisbach, Lehrbuch der Ingenieur- und Maschinen-Mechanik, vol. 1 (Braunschweig: Vieweg, 1845); H. Darcy, Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux (Paris: Mallet-Bachelier, 1857).
Weisbach and Darcy: head loss in pipes (1845-1857)
1851 – Stokes’s Drag on a Sphere#
Pendulum clocks and the pendulums used to measure gravity were corrected for the air they carried with them, and the corrections disagreed with experiment. George Gabriel Stokes, Lucasian professor at Cambridge, solved the slow viscous flow around a sphere and a cylinder in a paper read in December 1850. For the sphere he found the drag \(3\pi\mu dV\), valid while inertia is negligible, and from it the speed at which a fine particle settles, which he used to explain why clouds float. At larger Reynolds numbers the measured drag curve takes over, levelling off near \(C_D = 0.4\).
Implementation: engineeringkit.fluid_engineering.stokes_drag(),
engineeringkit.fluid_engineering.stokes_settling_velocity(),
engineeringkit.fluid_engineering.sphere_drag_coefficient() and
engineeringkit.fluid_engineering.terminal_velocity(). The example
draws the sphere drag curve and finds how fast sand grains of four sizes
fall through water, with and without Stokes’s law.
References: G. G. Stokes, “On the effect of the internal friction of fluids on the motion of pendulums,” Trans. Camb. Phil. Soc. 9, Part II (1851) 8-106 (read 9 December 1850); F. M. White, Viscous Fluid Flow, 3rd ed. (New York: McGraw-Hill, 2006), for the drag-curve fit.
1858 – James Thomson’s Triangular Notch#
James Thomson, professor of engineering at Belfast and elder brother of William Thomson (Lord Kelvin), proposed to the British Association in 1858 that water be measured over a right-angled V-notch cut in a weir board. Because the wetted section stays the same shape as the head changes, the discharge coefficient is nearly constant, and because the flow grows as \(H^{5/2}\), small flows still give a readable head. He reported experiments on 90° and wider notches in 1861. Rectangular sharp-crested weirs, with coefficients refined by Francis (1855) and Rehbock (1929), remain the companion device for larger flows.
Implementation:
engineeringkit.fluid_engineering.v_notch_weir_flow(),
engineeringkit.fluid_engineering.rectangular_weir_flow() and
engineeringkit.fluid_engineering.rehbock_discharge_coefficient().
The example compares the rating curves of a V-notch and a rectangular weir
and the precision with which each must be read.
References: J. Thomson, “On experiments on the measurement of water by triangular notches in weir boards,” Rep. Br. Assoc. Adv. Sci. (1858), Trans. Sect., 181; (1861), Trans. Sect., 151. Later sources disagree on the 1858 page (133 or 181); the report itself was not consulted.
1868 – Froude’s Ship Models#
William Froude proposed to the Admiralty that a ship’s resistance could be found by towing a model at the “corresponding speed”, the ship’s speed times the square root of the length ratio. At that speed model and ship make geometrically similar waves: their Froude numbers \(v/\sqrt{gL}\) agree. Because skin friction scales differently, Froude subtracted it, measured on towed planks, and scaled only the rest. The Admiralty paid for his tank at Torquay, and full-scale towing of HMS Greyhound in 1871 confirmed the method.
Implementation: engineeringkit.fluid_engineering.froude_number()
and engineeringkit.fluid_engineering.reynolds_number(). The example
scales a model test to a ship with Froude’s split, using the ITTC 1957
friction line.
References: W. Froude, “Observations and suggestions on the subject of determining by experiment the resistance of ships,” memorandum to the Admiralty (1868), in The Papers of William Froude (London: Institution of Naval Architects, 1955).
1880 – Pelton’s Water Wheel#
The hydraulic mines of the California gold country drove their stamps and pumps with hurdy-gurdy wheels: jets of water from high heads striking flat paddles. Lester Allan Pelton, a millwright at Camptonville, built a wheel whose buckets split the jet with a central ridge and turned each half almost back on itself, and patented it in October 1880. The momentum equation explains the gain: a jet turned through \(\theta\) delivers a fraction \((1 - \cos\theta)/2\) of its power at the best bucket speed, half the jet speed, which is one half for flat paddles and nearly one for Pelton’s buckets. Pelton wheels still run the highest-head hydroelectric plants.
Implementation: engineeringkit.fluid_engineering.jet_force() and
engineeringkit.fluid_engineering.pelton_wheel_power(), with
engineeringkit.fluid_engineering.bend_force() for the same momentum
balance on a pipe bend. The example compares the efficiency of flat
paddles and Pelton buckets against bucket speed.
References: L. A. Pelton, “Water-wheel,” US Patent 233,692 (granted 26 October 1880; filed 3 July 1880).
1883 – Reynolds’s Dye Experiment#
Osborne Reynolds let water flow from a tank into a glass tube with a bell mouth and fed a thin streak of dye into the entrance. Below a critical speed the streak stayed a straight thread; above it the streak broke up into eddies that filled the tube. The critical speed changed with the diameter and the temperature, but always at the same value of \(\rho v D/\mu\). Reynolds also showed that the resistance changes from proportional to the speed to nearly its square at the same point.
Implementation: engineeringkit.fluid_engineering.reynolds_number().
The example computes the critical speed against tube diameter and water
temperature and shows the jump in friction factor across the transition.
References: O. Reynolds, “An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels,” Phil. Trans. R. Soc. 174 (1883) 935-982.
1889 – Manning’s Open-Channel Formula#
Robert Manning, chief engineer of the Irish Board of Works, tested the many open-channel formulas of his day against the best measurements of canals and rivers. He found that Chézy’s coefficient grows roughly as the sixth root of the hydraulic radius and proposed \(V = R^{2/3}S^{1/2}/n\) in metric units, with a roughness coefficient \(n\). He read the paper to the Institution of Civil Engineers of Ireland in December 1889; it appeared in print in 1891. Tabulated values of \(n\) made it the standard for channels, culverts and sewers.
Implementation: engineeringkit.fluid_engineering.manning_velocity(),
engineeringkit.fluid_engineering.manning_flow() and
engineeringkit.fluid_engineering.hydraulic_radius(). The example
draws the rating curve of a concrete channel, finds the normal depth for a
design flow and checks that the flow is subcritical.
References: R. Manning, “On the flow of water in open channels and pipes,” Trans. Inst. Civil Eng. Ireland 20 (1891) 161-207.
1890 – Couette’s Rotating-Cylinder Viscometer#
Maurice Couette, in his doctoral work under Gabriel Lippmann at Paris, measured the viscosity of water between two coaxial cylinders, the outer one rotating and the inner one hung from a torsion wire. In a narrow gap the flow is plain shear between a fixed and a sliding wall, with a straight velocity profile and a uniform stress \(\mu U/h\), so the torque on the inner cylinder gives the viscosity directly. He found the torque proportional to the speed until the flow became unstable, and his instrument is the ancestor of the rotational viscometers and rheometers used today. Arnulph Mallock built a similar apparatus at about the same time.
Implementation:
engineeringkit.fluid_engineering.couette_viscometer_torque(),
engineeringkit.fluid_engineering.couette_velocity(),
engineeringkit.fluid_engineering.couette_flow_rate() and
engineeringkit.fluid_engineering.couette_shear_stress(). The example
reads a viscosity back from a measured torque and adds a pressure
gradient to plane Couette flow.
References: M. Couette, “Études sur le frottement des liquides,” Ann. Chim. Phys., 6th ser., 21 (1890) 433-510. The volume and pages are from secondary sources; the paper itself was not consulted.
1898 – Joukowsky’s Water Hammer#
Bursts in the new Moscow water mains led Nikolai Joukowsky to experiment on pipes up to 760 m long. He showed that a sudden valve closure raises the pressure by \(\rho a\,\Delta v\), where \(a\) is the speed of the pressure wave, slowed below the speed of sound in water by the elasticity of the wall as Korteweg had calculated in 1878. The wave travels to the reservoir and back in \(2L/a\), and any closure faster than that produces the full rise. He reported the work in 1898, and the memoir appeared in 1900.
Implementation:
engineeringkit.fluid_engineering.water_hammer_pressure(),
engineeringkit.fluid_engineering.pipe_wave_speed() and
engineeringkit.fluid_engineering.critical_closure_time(). The
example compares the surge in steel, iron and PVC pipe and draws the
pressure at the valve after a sudden closure.
References: N. Joukowsky, “Über den hydraulischen Stoss in Wasserleitungsröhren,” Mém. Acad. Imp. Sci. St.-Pétersbourg, 8th ser., 9(5) (1900) 1-71; English translation by O. Simin, Proc. AWWA 24 (1904) 341-424; D. J. Korteweg, “Ueber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Röhren,” Ann. Phys. 241 (1878) 525-542.
1912 – Bakhmeteff’s Specific Energy#
Boris Bakhmeteff, then teaching hydraulics at St Petersburg, measured the energy of an open-channel stream from its own bed rather than from a fixed datum. Plotted against depth, this specific energy \(E = y + q^2/(2gy^2)\) has a least value at the critical depth, where the Froude number is 1, and any larger energy can be carried at two alternate depths, one tranquil and one rapid. Bélanger had found the critical condition in 1828 as a singularity of the backwater equation; Bakhmeteff tied it to minimum energy, and the specific-energy diagram of his 1932 Hydraulics of Open Channels became the standard way to reason about gates, steps and contractions. Salas Edwards and Domínguez reached the same idea independently at about the same time.
Implementation: engineeringkit.fluid_engineering.specific_energy(),
engineeringkit.fluid_engineering.critical_depth() and
engineeringkit.fluid_engineering.alternate_depths(). The example
draws the specific-energy diagram and finds the water surface over a
raised step in the bed.
References: B. A. Bakhmeteff, O neravnomernom dvizhenii zhidkosti v otkrytom rusle [Varied flow of liquids in open channels] (St Petersburg, 1912); B. A. Bakhmeteff, Hydraulics of Open Channels (New York: McGraw-Hill, 1932); O. Castro-Orgaz, H. Chanson et al., “Boris A. Bakhmeteff and the development of specific energy and momentum concepts,” J. Hydraul. Eng. (2018). The 1912 book was not consulted; its priority follows Castro-Orgaz and Chanson.
1914 – Buckingham’s Pi Theorem#
Edgar Buckingham, a physicist at the US Bureau of Standards, proved that any physically meaningful relation among \(n\) quantities whose dimensions involve \(r\) independent base units can be written as a relation among \(n - r\) dimensionless products. Rayleigh and Reynolds had used dimensional reasoning case by case; Buckingham made it a method, and his paper named the groups \(\Pi\). The exponents of the groups form the null space of the dimension matrix.
Implementation: engineeringkit.fluid_engineering.buckingham_pi()
(exact rational row reduction in
engineeringkit.fluid_engineering.utils.integer_null_space()),
engineeringkit.fluid_engineering.euler_number() and
engineeringkit.fluid_engineering.weber_number(). The example derives
the four groups of pipe friction and the groups of droplet break-up, and
shows three dissimilar pipes with equal groups giving one Euler number.
References: E. Buckingham, “On physically similar systems; illustrations of the use of dimensional equations,” Phys. Rev. 4 (1914) 345-376.
1924 – Thoma’s Cavitation Parameter#
Faster, lower-head water turbines of the early twentieth century had their runners pitted by cavitation: vapour bubbles that form where the pressure drops to the vapour pressure and collapse against the blades. Dieter Thoma, at the Technische Hochschule München, showed that turbines of one design begin to cavitate at the same ratio \(\sigma\) of the suction head margin to the machine head. The Thoma number, and the net positive suction head behind it, set the setting level of every turbine and the suction lift of every pump.
Implementation: engineeringkit.fluid_engineering.thoma_sigma(),
engineeringkit.fluid_engineering.npsh_available() and
engineeringkit.fluid_engineering.cavitation_number(). The example
finds the largest suction lift of a pump at three water temperatures.
References: D. Thoma, “Die Kavitation bei Wasserturbinen,” Wasserkraft-Jahrbuch (Munich, 1924).
1933 – Nikuradse’s Sand-Roughened Pipes#
Johann Nikuradse, in Ludwig Prandtl’s institute at Göttingen, coated brass pipes with sieved sand of six grain sizes and measured their friction over a wide range of Reynolds numbers. After his smooth-pipe measurements of 1932, which fixed Prandtl and von Kármán’s logarithmic law, he showed that rough pipes follow the smooth law until the grains reach through the viscous sublayer and then settle on a friction factor that depends on the relative roughness alone.
Implementation:
engineeringkit.fluid_engineering.nikuradse_friction_factor() and
engineeringkit.fluid_engineering.smooth_pipe_friction_factor(). The
example draws the smooth and fully rough limits for Nikuradse’s six
roughnesses and finds where each becomes fully rough.
References: J. Nikuradse, “Strömungsgesetze in rauhen Rohren,” VDI-Forschungsheft 361 (1933); J. Nikuradse, “Gesetzmäßigkeiten der turbulenten Strömung in glatten Rohren,” VDI-Forschungsheft 356 (1932).
1936 – Hardy Cross’s Pipe-Network Method#
Hardy Cross, professor of structural engineering at the University of Illinois, had already relaxed the moments in frames one joint at a time. He applied the same idea to water distribution systems. Starting from flows that satisfy continuity at every junction, he corrected one loop at a time by a Newton step that makes the loop’s head losses sum to zero. The corrections preserve continuity, so the method converges on both sets of conditions with only hand arithmetic. Network analysis in water utilities relied on it for thirty years.
Implementation: engineeringkit.fluid_engineering.hardy_cross()
(returning a NetworkResult) and
engineeringkit.fluid_engineering.pipe_resistance(). The example
balances a two-loop network and checks it against a direct Newton solve.
References: H. Cross, “Analysis of flow in networks of conduits or conductors,” Univ. Illinois Eng. Exp. Stn. Bull. 286 (1936).
1939 – Colebrook’s Turbulent Friction Factor#
Commercial pipes do not have Nikuradse’s uniform sand, and their friction leaves the smooth law gradually instead of with his dip. Cyril Colebrook, after experiments with Cedric White on pipes coated with mixed grains, added the smooth and rough terms inside one logarithm. The result fits commercial pipes from smooth to fully rough with an “equivalent sand roughness” for each material. It is implicit in the friction factor, so it is solved by iteration.
Implementation:
engineeringkit.fluid_engineering.colebrook_friction_factor(). The
example solves the equation by fixed-point substitution and shows how it
joins the smooth and fully rough limits.
References: C. F. Colebrook, “Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws,” J. Inst. Civil Eng. 11 (1939) 133-156; C. F. Colebrook and C. M. White, “Experiments with fluid friction in roughened pipes,” Proc. R. Soc. A 161 (1937) 367-381.
1944 – Moody’s Diagram#
Lewis Moody, professor of hydraulics at Princeton, plotted Colebrook’s equation for a range of relative roughnesses on one log-log chart with the laminar line, and tabulated equivalent roughnesses for commercial materials. He showed how to use the chart for the three classic pipe problems: finding the loss, the flow or the diameter. Hunter Rouse had drawn a similar chart in 1943, but Moody’s became the one in every textbook.
Implementation:
plot_moody_diagram(),
with engineeringkit.fluid_engineering.colebrook_friction_factor()
and engineeringkit.fluid_engineering.laminar_friction_factor(). The
example draws the chart, reads one point from it, and solves Moody’s
diameter problem.
References: L. F. Moody, “Friction factors for pipe flow,” Trans. ASME 66 (1944) 671-684.
1976 – Swamee and Jain’s Explicit Friction Factor#
Iterating Colebrook’s equation was tedious on the programmable calculators of the 1970s. Prabhata Swamee and Akalank Jain fitted explicit formulas for the friction factor, the flow and the diameter of a pipe, each within a few percent of the implicit solution over the range of engineering pipes. Their friction factor became the standard explicit substitute for Colebrook in hand calculations and spreadsheets.
Implementation:
engineeringkit.fluid_engineering.swamee_jain_friction_factor(). The
example maps its error against Colebrook and checks their explicit
diameter formula against an iterative solution.
References: P. K. Swamee and A. K. Jain, “Explicit equations for pipe-flow problems,” J. Hydraul. Div. ASCE 102 (1976) 657-664.
