Modeling overfalls using vertically averaged and moment equations

A new set of vertically averaged and moment equations, which assumes a linear longitudinal velocity distribution and quadratic vertical velocity and pressure distributions, is used for modeling how in the vicinity of horizontal rectangular free overfall with smooth and rough beds and sharp-crested weirs with sloping upstream faces. These equations are modeled using a hybrid Petrov-Galerkin and Bubnov-Galerkin finite-element scheme. For the rectangular free overfalls, the predicted water surface profiles upstream of the overfall and the free jet trajectory agree well with the measured data. The computed vertical velocity and pressure distributions at the brink and upstream of the overfall are found to be in good agreement with the measured data; while the computed longitudinal velocity distributions compare well with a two-dimensional potential flow model. The computed results for sharp-crested weirs with sloping upstream faces agree well with the measured data for an upstream weir slope of up to 27 degrees with the horizontal. For an upstream slope of 45 degrees and steeper and for a large weir height the predicted water surface upstream of a weir shows numerical instability.