抛物线断面河渠收缩水深的直接计算公式

为便于工程设计应用,提出了一种抛物线形断面河渠收缩水深近似解直接计算公式.通过对收缩水深基本方程进行数学变换,将未知量与已知量分别用无量纲参数相对收缩水深λ和无量纲综合已知量参数β代替,用不动点迭代法建立迭代公式.在分析函数单调性和凹凸性以及方程的根的基础上,结合函数的几何图像,应用数值计算方法初步选取迭代初值,再以迭代次数最少且相对误差最小为目标对初值进行优化计算,最后得到了收缩水深直接计算公式,并进行相对误差分析和应用举例.结果表明直接计算公式形式较简单、适用范围广、结果精确,其最大误差小于0.43%.

Formula for direct calculation of contracted depth of parabola-shaped canal

The analytical solution formula of contracted depth in parabola-shaped canal is very complex.In order to facilitate engineering,a direct calculation formula for approximate solution of contracted depth is proposed.After mathematical transformation of the basic equation for contracted depth,the amount of unknown and known are replaced by relative contraction water depth λ and integrated known quantity β,which are dimensionless parameters.Then the iterative formula is obtained by fixed-point iteration method.Firstly,selecting the initial preliminary by numerical method based on analysis of the monotonicity and convexity of function and equation root,along with the geometric function image.The direct calculation formula of contracted depth has obtained at last;after initial optimization as the target of the least number of iterations with the minimum relative error.After analyzing the relative error and application examples,the result shows that the direct formula is brief,widely suitable and exact;the maximum relative error is less than 0.43%.