悬链线形断面正常水深与临界水深的直接计算

近年来,悬链线形断面渠道得到越来越广泛的应用。悬链线形断面的临界水深和正常水深的计算需求解超越方程,无解析解。首先,结合悬链线形断面的几何特征、水力要素、均匀流与临界流的基本方程,引入合适的无量纲参数,导出悬链线形断面的均匀流和临界流方程;经数学变换后得到悬链线形断面正常水深和临界水深的牛顿迭代式,并利用优化拟合原理求出二者的初值计算公式,一次迭代后得到正常水深和临界水深的直接计算公式。最后对公式进行误差分析及比较,表明在工程适用范围内,正常水深和临界水深直接计算公式的最大相对误差绝对值分别为5.33×10^-5%和5.05×10^-5%,二次迭代后精度可分别提升10^8和10^6倍。

Direct Calculation of Normal and Critical Water Depths for Catenary Linear Sections

In recent years,catenary channel has been widely used.There is no analytical solution to the transcendental equation for the calculation of critical depth and normal depth of catenary linear section.Firstly,the uniform flow and critical flow equations of the catenary linear section are derived by introducing appropriate dimensionless parameters in combination with the geometric characteristics of the catenary linear section,the hydraulic elements,and the basic equations of uniform flow and critical flow.After mathematical transformation,the Newtonian iterative formulae of normal and critical water depths of catenary linear section are obtained,and the initial value calculation formulas of them are obtained by using the optimization fitting principle,and the direct calculation formulas of normal and critical water depths are obtained after an iteration.Finally,the error analysis and comparison of the formula show that within the scope of engineering application,the maximum absolute relative error of the direct calculation formula of normal water depth and critical water depth is 5.33×10^-5%and 5.05×10^-5%,respectively.After the second iteration,the accuracy can be improved by 10^8 times and 10^6 times,respectively.