Author(s): G.D. Ransford

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Abstract: An analytical study is made of pressures on a spillway apron downstream from a partly raised gate, and the results compared with experiment. A first step is to consider pressures produced by a thin nappe emerging on a parabolic (or “trajectory” type) apron from a gate without contraction, i.e. from a gate fitted with a hypothetical bellmouth type of inlet. This case can be analysed exactly, using a power series development for the pressure as a function of time, or distance, after emergence. The result obtained, eqn. (10) in the article, consists of the product of two factors. The first is an expression, Y(l-Z/H) for the pressure on the sill immediately after emergence which, as to be expected, reduces to atmospheric should the operating head Z equal the design head H of the parabola, i.e. the head for which the jet follows the parabola exactly. The second factor represents the reduction in pressure with distance travelled after emergence. In the treatment of a thick jet, the expression previously quoted, Y(l-Z/H), has to be integrated over the height Y of the gate opening, or more accurately (should contraction occur, with a contraction coefficient C), over height CY. In order to do so, a hypothesis has to be made concerning the variation of the radius of curvature of the streamlines with height. Thus, by assuming a fixed centre of curvature, one finds a formula for the pressure which is closely related to an earlier approximation suggested by Maitre. Comparison with experiment suggests that this approach is in fact inadequate, and has prompted the analysis of the change of curvature occurring at the location of the vena contracta, where surface curvature becomes nil momentarily. The formulae obtained on this basis, one with a linear reduction of curvature with height above the crest up to the surface at the vena contracta, and the other assuming a quadratic law of reduction, are shown to be satisfactory when compared with available data. A linear reduction of curvature with height appears to fit a parabolic spillway best, but, for a Creager or “trajectory” type, with sharper curvatures towards the summit, the second quadratic type formula appears more suitable (in order to treat a Creager apron, a procedure has been suggested for establishing an equivalence with the parabolic type of apron for which the theory was originally developed).

DOI: https://doi.org/10.1080/00221688309499436

Year: 1983