Solution of Highly Curvilinear Gravity Flows

Application of the proposed method to flow over a vertical sharp-crested weir yields profiles and discharge coefficients for head-to-weir-height ratios in the range of 0.1 < h/w < 3. Comparison with experimental data shows good agreement. The solution consists in mapping the complex-potential plane into an auxiliary plane in which the flow boundaries lie on the real axis. A postulated line distribution of vorticity, proportional to the unknown tangential derivative of elevation with respect to velocity potential, together with the Bernoulli equation on the free surfaces, leads to an integral equation descriptive of the flow. An iterative procedure analogous to the Neumann treatment of Fredholm integral equations is developed and is used to solve the equation by a digital computer to any desired accuracy. The analysis gives theoretical corroboration of the well-established but poorly explained observation that in the usual free case of weir flow, in which the depth passes through critical near the crest, the discharge cannot be varied independently of the flow geometry, h/w.