Standard horseshoe cross section geometry

The standard horseshoe cross section is found in many tunnels for open-channel and pressurized conduit water conveyance. The exact geometric relationships for this cross section are needed for hydraulic analysis and mathematical modeling purposes because look-up tables of numerically approximated values are typically as slow, or slower, when implemented in computer programs. Implicit solutions of the unsteady hydraulic equations require exact, analytical partial derivatives of some of the relationships, both for accuracy and numerical stability. The equations for cross-sectional area, wetted perimeter, top width, and depth to area centroid are presented as functions of the depth of water in standard horseshoe sections, in addition to the partial derivatives of wetted perimeter with respect to water depth, and depth to area centroid with respect to water depth. Sample unsteady-flow simulation results for a horseshoe cross section are compared to those of a comparably sized circular cross section.