悬链线形断面收缩水深的直接计算公式

Analytical Method for Calculating Water Depth at Contracted Section in Catenary Channel

【目的】致力于寻求悬链线形断面收缩水深的直接计算公式,以解决悬链线形断面收缩水深在理论上无法直接求解的问题。【方法】引入恰当的无量纲参数,对悬链线形断面收缩水深的基本方程进行数学变换,得到无量纲收缩水深的隐函数方程,根据特殊一元二次方程的定义,在工程适用范围内,对无量纲收缩水深a和收缩水深与悬链线形断面形状参数之比x的值进行回归分析,得到无量纲收缩水深隐函数方程的一元二次替代方程,解该一元二次方程,即可得到悬链线形断面收缩水深的直接计算公式。【结果】对公式进行误差分析及比较发现,在工程适用范围内,公式计算值的最大相对误差绝对值小于0.94%,高于现有计算公式精度。【结论】推求的悬链线形断面收缩水深的直接计算公式适用范围广、形式简捷、精度高,完全满足工程的实际需要。

【Objective】The purpose of this paper to present a method to calculate water depth at contracted section in catenary channel.【Method】Using dimensionless parameters, the equation for water depth at contracted section in catenary channel was mathematically transformed to derive an implicit function for the dimensionless water depth. Based on the definition of a special equation with one unknown variable, a regression model with α and x in the range of α∈[0.01,0.5] and x∈[0.5,7] was established to obtain a alternative to the special quadratic equation for the dimensionless contracted water depth. Solving this special equation using the root-finding formula can obtain the contracted water depth. We also analyzed the errors of this method【. Result】For engineering application, the absolute value of the maximum relative error of the formula was less than 0.94%, more accurate than the existing formulae.【Conclusion】The calculation formula for the contracted water depth in catenary channel is applicable to a wide range of problems. It is simple and accurate, meeting the requirement of most projects.