Jean-Baptiste Charles Joseph BÉLANGER (1790-1874), the Backwater Equation and the Bélanger Equation

In an open channel, the transition from a high-velocity open channel flow to a fluvial motion is a flow singularity called a hydraulic jump. The application of the momentum principle to the hydraulic jump is commonly called the Belanger equation, but few know that his original treatise was focused on the study of gradually varied open channel flows (BELANGER 1828).. The originality of BELANGER's (1828) essay was the successful development of the backwater equation for steady, one-dimensional gradually-varied flows 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 BELANGER understood the rapidly-varied nature of the jump flow, but he applied incorrectly the Bernoulli principle to the hydraulic jump. The correct application of momentum considerations to the hydraulic jump flow was derived 10 years later and first published by BELANGER (1841) Altogether Jean-Baptiste BELANGER'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. BARRE de SAINT VENANT, and J.V. BOUSSINESQ. // En hydraulique des ecoulements a surface libre, le nom de Jean-Baptiste BELANGER est souvent associe au ressaut hydraulique, et a l'application de l'equation de conservation de la quantite de mouvement, appelee l'equation de Belanger. Une etude de son essai BELANGER (1828) montre que le theme principal etait le calcul des ecoulements a surface libre graduellement varies, avec le developpement de l'equation du remous. Dans ce travail, Jean-Baptiste BELANGER deriva l'equation du remous avec une lucidite remarquable, et sa derivation donne des resultats d'une precision etonnante. En discutant les singularites de l'equation du remous, il introduisit aussi les conditions d'ecoulement critique, et le concept de profondeur critique, bien avant ses contemporains. Par contre, en 1828, BELANGER appliqua incorrectement le principe de conservation d'energie au ressaut hydraulique; le resultat pour le cas d'un canal rectangulaire et horizontal etait fondamentalement impropre, et il fut corrige 10 ans plus tard (BELANGER 1841). Quoiqu'il en soit, la contribution de Jean-Baptiste BELANGER a l'hydraulique des ecoulements a surface libre etait exceptionnelle, en avance sur temps, et elle preceda les travaux de J.A.C. BRESSE, H.P.G. DARCY, A.J.C. BARRE de SAINT VENANT, and J.V. BOUSSINESQ.

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