Chanson H. Jean-Baptiste Charles Joseph Belanger (1790-1874), the Backwater Equation and the Belanger Equation (796976)
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THE UNIVERSITY OF QUEENSLANDDIVISION OFCIVIL ENGINEERINGREPORT CH69/08JEAN-BAPTISTE CHARLES JOSEPH BÉLANGER(1790-1874), THE BACKWATER EQUATION ANDTHE BÉLANGER EQUATIONAUTHOR: Hubert CHANSONHYDRAULIC MODEL REPORTSThis report is published by the Division of Civil Engineering at the University ofQueensland. Lists of recently-published titles of this series and of other publicationsare provided at the end of this report. Requests for copies of any of these documentsshould be addressed to the Civil Engineering Secretary.The interpretation and opinions expressed herein are solely those of the author(s).Considerable care has been taken to ensure accuracy of the material presented.Nevertheless, responsibility for the use of this material rests with the user.Division of Civil EngineeringThe University of QueenslandBrisbane QLD 4072AUSTRALIATelephone:Fax:(61 7) 3365 3619(61 7) 3365 4599URL: http://www.eng.uq.edu.au/civil/First published in 2008 byDivision of Civil EngineeringThe University of Queensland, Brisbane QLD 4072, Australia© ChansonThis book is copyrightISBN No.
9781864999211The University of Queensland, St Lucia QLDJEAN-BAPTISTE CHARLES JOSEPH BÉLANGER (1790-1874),THE BACKWATER EQUATION ANDTHE BÉLANGER EQUATIONbyHubert CHANSONProfessor, Division of Civil Engineering, School of Engineering,The University of Queensland, Brisbane QLD 4072, AustraliaPh.: (61 7) 3365 3619, Fax: (61 7) 3365 4599, Email: h.chanson@uq.edu.auUrl: http://www.uq.edu.au/~e2hchans/REPORT No. CH69/08ISBN 9781864999211Division of Civil Engineering, The University of QueenslandAugust 2008Jean-Baptiste BÉLANGER (1790-1874)(Courtesy of the Bibliothèque de l'Ecole Nationale Supérieure des Ponts et Chaussées)AbstractIn an open channel, the transition from a high-velocity open channel flow to a fluvial motion is a flowsingularity called a hydraulic jump.
The application of the momentum principle to the hydraulic jump iscommonly called the Bélanger equation, but few know that his original treatise was focused on the study ofgradually varied open channel flows (BÉLANGER 1828).. The originality of BÉLANGER's (1828) essaywas the successful development of the backwater equation for steady, one-dimensional gradually-variedflows in an open channel, together with the introduction of the step method, distance calculated from depth,and the concept of critical flow conditions. In 1828, Jean-Baptiste BÉLANGER understood the rapidlyvaried nature of the jump flow, but he applied incorrectly the Bernoulli principle to the hydraulic jump. Thecorrect application of momentum considerations to the hydraulic jump flow was derived 10 years later andfirst published by BÉLANGER (1841) Altogether Jean-Baptiste BÉLANGER's (1828,1841,1849)contributions to modern open channel hydraulics were remarkable and influenced the works by J.A.C.BRESSE, H.P.G.
DARCY, A.J.C. BARRÉ de SAINT VENANT, and J.V. BOUSSINESQ.Keywords: Jean-Baptiste BÉLANGER, Backwater equation, Gradually-varied flows, Critical flowconditions, Direct step method, Bélanger equation, Hydraulic jumps, Momentum equation, Energy equation,Open channel flows, Hydraulic engineering.RésuméEn hydraulique des écoulements à surface libre, le nom de Jean-Baptiste BÉLANGER est souvent associé auressaut hydraulique, et à l'application de l'équation de conservation de la quantité de mouvement, appeléel'équation de Bélanger. Une étude de son essai BÉLANGER (1828) montre que le thème principal était lecalcul des écoulements à surface libre graduellement variés, avec le développement de l'équation du remous.Dans ce travail, Jean-Baptiste BÉLANGER dériva l'équation du remous avec une lucidité remarquable, et sadérivation donne des résultats d'une précision étonnante.
En discutant les singularités de l'équation duremous, il introduisit aussi les conditions d'écoulement critique, et le concept de profondeur critique, bienavant ses contemporains. Par contre, en 1828, BÉLANGER appliqua incorrectement le principe deconservation d'énergie au ressaut hydraulique; le résultat pour le cas d'un canal rectangulaire et horizontalétait fondamentalement impropre, et il fut corrigé 10 ans plus tard (BÉLANGER 1841). Quoiqu'il en soit, lacontribution de Jean-Baptiste BÉLANGER à l'hydraulique des écoulements à surface libre étaitexceptionnelle, en avance sur temps, et elle précéda les travaux de J.A.C.
BRESSE, H.P.G. DARCY, A.J.C.BARRÉ de SAINT VENANT, and J.V. BOUSSINESQ.Mots-clef: Jean Baptiste BÉLANGER, Equation du remous, Ressaut hydraulique, Conditions d'écoulementcritique, Equation de conservation de la quantité de mouvement, Ecoulements à surface libre.iiTABLE OF CONTENTSPageAbstractiiKeywordsiiRésuméiiMots-clésiiTable of contentsiiiList of SymbolsivGlossaryvi1. Introduction12. Life of Jean-Baptiste BÉLANGER (1790-1874)33.
The analysis of the hydraulic jump: the "Bélanger equation"44. Gradually-varied flow calculations: the backwater equation95. Discussion136. Conclusion157. Acknowledgments16Appendix A - Birth certificate of Jean-Baptiste BÉLANGER17Appendix B - The names of the 72 scientists written around the Eiffel Tower, Paris19Appendix C - The Prony flow resistance formula22REFERENCES24Internet references26Bibliographic reference of the Report CH69/0827iiiList of symbolsThe following symbols are used in this report :Aflow cross-section area (m2);acoefficient of the Prony resistance formula;Bchannel width (m);bcoefficient of the Prony resistance formula;DHhydraulic diameter : DH = 4 A/Pw;dwater depth (m) measured normal to the invert;dccritical flow depth (m) : d c = q 2 / g in a rectangular, horizontal channel;Especific energy (m); for a horizontal channel with hydrostatic pressure distribution:E=d+V2;2gFrFroude number defined as : Fr = V / g d ;fDarcy-Weisbach friction factor;ggravity constant (m/s2);Htotal head (m);ksequivalent sand roughness height (m);Llength (m);Pwwetted perimeter (m);Qvolume flow rate (m3/s);ReReynolds number: Re = ρ V d/μ;qvolume flow rate per unit width (m2/s): q = Q/B;Sffriction slope;Sobed slope : So = sinθ;Vflow velocity (m/s) positive downstream;xlongitudinal flow direction (m);zobed elevation (m) positive upwards;Greek symbolsαkinetic energy correction coefficient, also called Coriolis coefficient;α'velocity correction coefficient;μdynamic viscosity of water (Pa s);θbed slope angle with the horizontal, positive downwards;ρwater density (kg/m3);∅diameter (m);Subscriptccritical flow conditions;1upstream flow conditions;iv2downstream flow conditions;AbbreviationsD/Sdownstream;U/Supstream;Notation∂∂xpartial differentiation with respect to x.vGlossaryBARRÉ de SAINT-VENANT: Adhémar Jean Claude BARRÉ de SAINT-VENANT (1797-1886), Frenchengineer of the 'Corps des Ponts-et-Chaussées', developed the equation of motion of a fluid particle interms of the shear and normal forces exerted on it (BARRÉ de SAINT-VENANT 1871).Bélanger equation: momentum equation applied across a hydraulic jump in a horizontal channel; theequation was first derived by BÉLANGER (1841) and named after him.BERNOULLI: Daniel BERNOULLI (1700-1782) was a Swiss mathematician, physicist and botanist whodeveloped the Bernoulli equation in his "Hydrodynamica, de viribus et motibus fluidorum" textbook (1stdraft in 1733, 1st publication in 1738, Strasbourg).BIDONE: Giorgio BIDONE (1781-1839) was an Italian hydraulician.
His experimental investigations on thehydraulic jump were published between 1819 and 1826.BORDA: Jean-Charles de BORDA (1733-1799) was a French mathematician and military engineer. Heachieved the rank of Capitaine de Vaisseau and participated to the U.S. War of Independence with theFrench Navy. He investigated the flow through orifices and developed the Borda mouthpiece.BOSSUT: Abbé Charles BOSSUT (1730-1804) was a French ecclesiastic and experimental hydraulician,author of a hydrodynamic treaty (BOSSUT 1772).BOUSSINESQ: Joseph Valentin BOUSSINESQ (1842-1929) was a French hydrodynamicist and Professorat the Sorbonne University (Paris).
His treatise "Essai sur la théorie des eaux courantes" (1877) remainsan outstanding contribution in hydraulic engineering literature.Boussinesq coefficient: momentum correction coefficient named after J.V. BOUSSINESQ who firstproposed it.BRESSE: Jacques Antoine Charles BRESSE (1822-1883) was a French applied mathematician andhydraulician.
He was Professor at the Ecole Nationale Supérieure des Ponts et Chaussées, Paris as thesuccessor of J.B. BELANGER. His contribution to gradually-varied flows in open channel hydraulics isconsiderable (BRESSE 1860).BUAT: Comte Pierre Louis George du BUAT (1734-1809) was a French military engineer and hydraulician.He was a friend of Abbé C. BOSSUT. Du BUAT is considered as the pioneer of experimental hydraulics.His textbook (BUAT 1779) was a major contribution to flow resistance in pipes, open channel hydraulicsand sediment transport.CARNOT: Lazare N.M.
CARNOT (1753-1823) was a French military engineer, mathematician, general andstatesman who played a key-role during the French Revolution.CAUCHY: Augustin Louis de CAUCHY (1789-1857) was a French engineer from the 'Corps des Ponts-etChaussées'. He devoted himself later to mathematics and he taught at Ecole Polytechnique, Paris, and atthe Collège de France. He worked with Pierre-Simon LAPLACE and J.
Louis LAGRANGE. In fluidmechanics, he contributed greatly to the analysis of wave motion.CHEZY: Antoine CHEZY (1717-1798) (or Antoine de CHEZY) was a French engineer and member of theFrench 'Corps des Ponts-et-Chaussées'. He designed canals for the water supply of the city of Paris. In1768 he proposed a resistance formula for open channel flows called the Chézy equation. In 1798, hebecame Director of the Ecole Nationale Supérieure des Ponts et Chaussées after teaching there for manyyears.viCORIOLIS: Gustave Gaspard CORIOLIS (1792-1843) was a French mathematician and engineer of the'Corps des Ponts-et-Chaussées' who first described the Coriolis force (i.e. effect of motion on a rotatingbody).Coriolis coefficient: kinetic energy correction coefficient named after G.G.
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