三角形明渠水跃共轭水深的近似解法

摘要

[目的]推求三角形断面共轭水深的近似计算公式,为梯形断面共轭水深的简捷求解提供参考.[方法]根据三角形断面特征,选取无量纲共轭水深作为参数,首先对用于判别三角形明渠水跃类型的弗劳德数计算公式进行推导,然后对三角形明渠水跃共轭水深方程进行化简,得到三角形断面第二共轭水深的迭代计算公式;在工程常用范围内,通过假设无量纲跃前水深,用迭代公式计算跃后水深,再根据对应的无量纲共轭水深值,利用Matlab软件进行函数配线拟合.[结果]在对共轭水深求解方法进行分析研究的基础上,选用了2个二次多项式之比的形式作为三角形明渠共轭水深的近似计算公式,该式在跃后水深与跃前水深比y/x∈(1,85)时,跃前水深最大相对误差小于0.20％,跃后水深最大相对误差小于0.17％.[结论]推导的三角形断面共轭水深近似计算公式可用于跃后与跃前水深的求解,为梯形断面共轭水深的简化计算提供了新的途径.%[Objective] A formula for approximate calculation of conjugate depths in triangle channel was developed to provide basis for further study of simple solution for conjugate depths in trapezoidal channel.[Method] According to the feature of triangular open channel,the dimensionless conjugate depth was selected as parameter.The Froude number calculation formula was deduced and used to distinguish types of hydraulic jump.Then,the iteration formula for the downstream depth was deduced by simplifying conjugate depth equation in triangle channel.With the iteration formula,the downstream depth was calculated by assigning series of the dimensionless upstream depth in the range of project application.Fitting was also conducted with a range of corresponding dimensionless conjugate depths using Matlab.[Result] Based on the analysis of solving scheme of conjugate depth,the final formula was expressed in terms of ratio of two quadratic polynomials.When the ratio of downstream depth to upstream depth was 1-85,the maximum relative error of calculated upstream depth was less than 0.20％ and that of calculated downstream depth was less than 0.17％.[Conclusion] The formula can solve both the downstream depth and upstream depth,which provides a new option for simplified calculation of conjugate depths in trapezoidal channel.