Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger „1828..

where subscripts 1 and 2 refer to the upstream and downstream flow conditions, respectively; F Froude number; F=V / gd; d and V flow depth and velocity, respectively; and g=gravity acceleration. The hydraulic jump is typically classified in terms of its inflow Froude number F1=V1 / gd1, which must be greater than unity Bélanger 1828; Henderson 1966 . For F1 slightly above unity, the hydraulic jump is characterized by a train of stationary free-surface undulations. For larger Froude numbers, the jump has a marked roller with large-scale vortices, and the flow is characterized by significant kinetic energy dissipation and air bubble entrainment. Historical contributions on the hydraulic jumps included the experiments of Bidone 1819 , the theoretical analysis of Bélanger 1828, 1841 , the experiments of Darcy and Bazin 1865 , the solutions of Boussinesq 1877 , and the work of Bakhmeteff 1932 . Recent technical reviews encompass Hager 1992 and Chanson 2007, 2009 . Bélanger is commonly linked to the application of the momentum principle to the hydraulic jump i.e., the Bélanger equation , but few people realize that his 1828 treaty was focused on the study of gradually varied open channel flows. The original work of Bélanger 1828 is reconsidered herein and it is highlighted that his development of the backwater equation was remarkable for a period when numerical integration calculations were performed by hand. Bélanger introduced the notion of critical flow conditions as a singularity of the backwater equation, and showed that the backwater equation cannot be solved across a hydraulic jump. Instead, he understood the rapidly varied nature of the jump flow, though his 1828 theoretical treatment was incorrect.