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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%.
In order to facilitate the application of engineering design, a direct calculation formula of the approximate depth of shrinkage of a parabolic cross-section river channel is proposed.Through the mathematical transformation of the basic equation of shrinkage depth, the unknowns and the known quantities are respectively contracted by the dimensionless parameters, Based on the analysis of the monotonicity, the concavity and convexity of the function and the root of the equation, combined with the geometrical image of the function, the initial value of the iteration is selected by numerical calculation , Then the initial value is optimized with the least number of iterations and the least relative error. Finally, the direct calculation formula of shrinkage depth is obtained, and the relative error analysis and application examples are given. The results show that the direct calculation formula is simple in form and applicable to a wide range of applications, The result is accurate with a maximum error of less than 0.43%.