Chanson H. Jean-Baptiste Charles Joseph Belanger (1790-1874), the Backwater Equation and the Belanger Equation (796976), страница 2
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CORIOLIS who introduced firstthe velocity correction coefficient.DARCY: Henri Philibert Gaspard DARCY (1805-1858) was a French civil engineer. He studied at EcolePolytechnique between 1821 and 1823, and later at the Ecole Nationale Supérieure des Ponts etChaussées (BROWN 2002). He performed numerous experiments of flow resistance in pipes (DARCY1858) and in open channels (DARCY and BAZIN 1865), and of seepage flow in porous media (DARCY1856a,b). He gave his name to the Darcy-Weisbach friction factor and to the Darcy law in porous media.DUPUIT: Arsène Jules Etienne Juvénal DUPUIT (1804-1866) was a French engineer and economist. Hisexpertise included road construction, economics, statics and hydraulics.Ecole Nationale Supérieure des Ponts et Chaussées, Paris: French civil engineering school founded in 1747.The direct translation is : 'National School of Bridge and Road Engineering'.
Among the directors therewere the famous hydraulicians A. CHEZY and G. de PRONY. Other famous professors included B.F. deBELIDOR, J.B.C.J. BELANGER, J.A.C. BRESSE, G.G. CORIOLIS and L.M.H. NAVIER.Ecole Polytechnique, Paris: Leading French engineering school founded in 1794 during the FrenchRévolution under the leadership of Lazare CARNOT and Gaspard MONGE. It absorbed the state artilleryschool in 1802 and was transformed into a military school by Napoléon BONAPARTE in 1804. Famousprofessors included Augustin Louis CAUCHY, Jean Baptiste Joseph FOURIER, Siméon-DenisPOISSON, Jacques Charles François STURM, among others.EYTELWEIN: Johann EYTELWEIN (1764-1848) was a German mathematician and engineer.Fawer jump: undular hydraulic jump.Hydraulic jump: stationary transition from a rapid, high-velocity flow to a slower fluvial flow motion.LAGRANGE: Joseph-Louis LAGRANGE (1736-1813) was a French mathematician (CHANSON 2007b).During the 1789 Revolution, he worked on the committee to reform the metric system.
He was Professorof mathematics at the École Polytechnique from the start.Left bank: looking downstream, towards the river mouth, the left bank is on the left.MONGE: Gaspard MONGE (1746-1818), Comte de Péluse, was a French mathematician who inventeddescriptive geometry and pioneered the development of analytical geometry. He was a prominent figureduring the French Revolution, helping to establish the Système métrique and the École Polytechnique,and being Minister for the Navy and colonies between 1792 and 1793.PITOT: Henri PITOT (1695-1771) was a French mathematician, astronomer and hydraulician. He was amember of the French Académie des Sciences from 1724. He invented the Pitot tube to measure flowvelocity in the Seine river (first presentation in 1732 at the Académie des Sciences de Paris).POISSON: Siméon Denis POISSON (1781-1840) was a French mathematician and scientist.
He developedthe theory of elasticity, a theory of electricity and a theory of magnetism.PRONY: Gaspard Clair François Marie Riche de PRONY (1755-1839) was a French mathematician andengineer. He succeeded A. CHEZY as director general of the Ecole Nationale Supérieure des Ponts etChaussées, Paris during the French Revolution.viiRapidly varied flow: open channel flow characterised by large changes over a short distance (e.g.
sharpcrested weir, sluice gate, hydraulic jump).REECH: Ferdinand REECH (1805-1880) was a French naval instructor who proposed first the ReechFroude number in 1852 for the testing of model ships and propellers.Right bank: looking downstream, towards the river mouth, the right bank is on the right.Roller: in hydraulic engineering, a series of large-scale turbulent eddies : e.g., the roller of a hydraulic jump.Shock waves: in high-velocity, supercritical flows, a flow disturbance (e.g.
change of direction, contraction)induces the development of shock waves propagating at the free-surface across the channel. Shock wavesare called also lateral shock waves, oblique hydraulic jumps, Mach waves, cross-waves, diagonal jumps.Stilling basin: hydraulic structure for dissipating the energy of the flow downstream of a spillway, outletwork, chute or canal structure. In many cases, a hydraulic jump is used as the energy dissipator within thestilling basin.Supercritical flow: open channel flow characterised by a Froude number greater than unity.Undular hydraulic jump: stationary hydraulic jump characterised by steady free-surface undulationsdownstream of the jump and by the absence of a formed roller. An undular jump flow is called a Fawerjump in homage to C.
FAWER's (1937) work.Weak jump: A weak hydraulic jump is characterised by a marked roller, no free-surface undulation and lowenergy loss. It is usually observed after the disappearance of undular hydraulic jump with increasingupstream Froude numbers.viii1. IntroductionThe hydraulic jump is the rapid and sudden transition from a high-velocity supercritical open channel flow toa subcritical flow (Fig. 1).
Hydraulic jumps are commonly experienced in rivers and canals, in industrialapplications and in manufacturing processes. A hydraulic jump is a flow singularity and discontinuity. For ahorizontal rectangular channel and neglecting boundary friction, the continuity and momentum principlesgive a series of dimensionless relationships between the upstream and downstream flow properties:d2 1 ⎛= ⎜ 1 + 8 Fr12 − 1⎞⎟d1 2 ⎝⎠(1)Fr223 / 2=3/ 2Fr1 ⎛⎞2⎜ 1 + 8 Fr1 − 1⎟⎝⎠(2)where the subscripts 1 and 2 refer to the upstream and downstream flow conditions respectively, Fr is theFroude number: Fr = V / g d , d and V are the flow depth and velocity respectively, and g is the gravityacceleration. A hydraulic jump is typically classified in terms of its inflow Froude number Fr1 = V1 / g d 1that is always greater than unity (BÉLANGER 1828, HENDERSON 1966, CHANSON 2004).
For a Froudenumber slightly above unity, the hydraulic jump is characterised by a smooth rise of the free-surfacefollowed by a train of stationary free-surface undulations (Fig. 1A). For larger Froude numbers, the jump hasa marked roller with large scale vortices, and the flow is characterised by significant kinetic energydissipation and air bubble entrainment (Fig. 1B).Historical contributions on the hydraulic jumps included the physical experiments of BIDONE (1819)performed in France in 1818, the theoretical analyses of BÉLANGER (1828,1841), the experiments ofDARCY and BAZIN (1865), the solutions of BOUSSINESQ (1877) and the work of BAKHMETEFF(1932).
Recent reviews encompassed HAGER (1992) and CHANSON (2007a,2009).Jean-Baptiste BÉLANGER (Fig. 2) is commonly linked to the application of the momentum principle to thehydraulic jump: i.e., the Bélanger equation. But few people appreciate that his original paper was focused onthe study of gradually varied open channel flows (BÉLANGER 1828), while his considerable influence onhis contemporaries is sometimes lost. For example, his name is written on the border of one of the fourfacades of the Eiffel Tower together with that of the famous hydraulic engineers Jean Charles BORDA,Gaspard de PRONY, Jean-Victor PONCELET, and Jacques Antoine Charles BRESSE (Fig.
3).The contribution of Jean-Baptiste BÉLANGER to open channel flows is re-considered herein. It ishighlighted that his development of the backwater equation was remarkable for a period when numericalintegration calculations were performed by hand (BÉLANGER 1828). Jean-Baptiste BÉLANGERintroduced the notion of critical flow conditions as a singularity of the backwater calculations, and showedthat the backwater equation cannot be solved across a hydraulic jump.
He understood the rapidly-variednature of the jump flow and the concept of supercritical inflow. Although his initial treatment of thehydraulic jump was erroneous, a later development gave the hydraulic jump equation (BÉLANGER 1841).1(A) Undular hydraulic jump: Fr1 = 1.1, d1 = 0.104 m, Re = 1.1 105, B = 0.5 m- Flow from right to left(B) Hydraulic jump with roller: Fr1 = 7.9, d1 = 0.018 m, Re = 5.9 104, B = 0.5 m- Flow from right to leftFig.
1 - Photographs of hydraulic jumps in the Gordon McKay Hydraulics Laboratory at the University ofQueenslandFig. 2 - Photograph of Jean-Baptiste BÉLANGER (Courtesy of the Bibliothèque de l'Ecole NationaleSupérieure des Ponts et Chaussées)2Fig. 3 - Inscription BÉLANGER on the Eiffel Tower (Tour Eiffel) between LAGRANGE and CUVIER,with BRESSE on the left - Photograph taken on 25 July 20082. Life of Jean-Baptiste BÉLANGER (1790-1874)Born in Valenciennes, in northern France, on 4 April 1790 (Ref.: Birth Certificate, Parish of St Vaast enVille, App. A), Jean-Baptiste Charles Joseph BÉLANGER was the son of Charles Antoine Aimé JosephBÉLANGER, master locksmith, and of Jeanne Françoise Joseph FAUCONNIER. He studied in Paris at theEcole Polytechnique (1), finishing second, and later at the Ecole des Ponts et Chaussées.As Ingénieur du Corps des Ponts et Chaussées (Bridges and Roads Corps of Engineers), he started hisengineering career in 1816 at La Réole.
From 1821, he moved to work on the Somme navigation canal andafter 1826 on the Ardennes navigation canal (La Houille Blanche 1960). It was during these two missionsthat he studied specifically the hydraulics of gradually-varied open channel flows. He later became a lecturerat the Ecole Centrale des Arts et Manufactures between 1838 and 1864 (Fig. 4), at the Ecole des Ponts et1in the 1808 cohort (promotion 1808) together with Gustave Gaspard CORIOLIS (1792-1843) (Journal de l'Ecole3Chaussées from 1841 to 1855, and at the Ecole Polytechnique from 1851 to 1860 (CHATZIS 1995). At theEcole Centrale, one of his students was Gustave EIFFEL (1832-1923) who built the Eiffel tower andengraved his name around the first floor together with the names of 71 other scientists (Fig.
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