Direct calculation formulas for normal depths of four kinds of parabola-shaped channels

Abstract

Channels with parabola-shaped cross-section are widely used in water conservancy and hydropower engineering, agricultural irrigation and drainage engineering, so it is necessary to determine the accurate normal depth values for design of open channels, operation management and analysis of gradually varied flow. However, the governing equations are nonintegrable for cubic and semi-quintic parabola-shaped channels in terms of the wetted perimeters. In this study, the Simpson's numerical integral method is introduced to approximate the two nonintegrable wetted perimeters for cubic and semi-quintic parabola-shaped channels which can meet the requirements of engineering design well in the commonly using range of engineering. Subsequently, the approximate wetted perimeters are substituted into the uniform flow equation and the uniform flow equation deforms optimal model identically. The optimal model parameters are determined by the 1stOpt software based on Marquardt algorithm and two explicit calculation formulas for the normal depths of cubic and semi-quintic channels are proposed. At the same time, two direct calculation formulas for normal depths of semi-cubic and quadratic parabola-shaped channels are presented as well. Four sets of formulas for normal depths of parabola-shaped channels are presented and their maximum relative errors are 0.52%, − 0.37%, − 0.22% and 0.57%, respectively, which have wide range of application, high precision, concise form and can provide some better guidance for engineering design and operation management.