Development Of The Belanger Equation And Backwater Equation By Jean-baptiste Belanger (1828)

摘要

A hydraulic jump is the sudden and rapid transition from a sub-critical to a subcritical flow motion, and may be considered as a flow singularity. For a horizontal rectangular channel and neglecting boundary friction, the continuity and momentum principles yield a relationship between the upstream and downstream flow depthsrnd_2/D_1 = 1/2 × ((1+8f_1~2)~(1/2)-1)(1)rnwhere subscripts 1 and 2 refer to the upstream and downstream flow conditions, respectively; F=Froude number; F=V/(gd)~(1/2); 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 F_1 = V_1/(gd_1)~(1/2), which must be greater than unity (Belanger 1828; Henderson 1966). For F_1 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 Belanger (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).