U形渠道正常水深的直接水力计算公式

Direct calculation formula for normal depth of U-shaped channel

针对水力性能优良的U形渠道,依据明渠均匀流基本原理,以过水断面水深恰好等于U形底弧弓高时断面过流量作为分界流量Qb,引入断面特征参数和无量纲相对正常水深,采用麦考特优化法,以离差平方和最小为目标,利用SAS软件编程,通过最优化拟合建立了U形渠道正常水深直接水力计算公式。通过误差分析表明,所建公式在渠道过流量Q小于分界流量Qb时,计算相对误差绝对值均小于0.44%;渠道过流量Q大于分界流量Qb时,计算相对误差绝对值均不超过1%,可见公式具有较高的精度,且物理概念清晰、计算方便快捷;该公式不仅可用于解决宽浅式渠道正常水深水力计算,也适用于窄深式渠道正常水深的水力计算,具有较强的通用性,可为渠道工程设计和运行管理提供理论依据和有益参考。

The normal depth is an important hydraulic element for channel design, operation and management. U-shaped channel has excellent hydraulic performance and strong ability of anti-frost heave, while the normal depth in the governing equations of the current common methods is implicit and have no analytical solutions. We introduced cross section characteristic parameter and dimensionless relative normal depth and took the actual flow rate in channel as the transitional flow rate when the flow depth is exactly equal to the bottom arc height. The direct calculation formula for the normal depth in U-shaped channel was presented by Marquardt method based on the NLIN procedure using SAS software. Marquardt method was used to assess whether the fitting formula reaches the best fitting result by using the iterative program to calculate the residual sum of squares. Fitting formula is the best one when the minimum residual sum of squares exists, having the characteristics of well adaptable to formula, low requirement to iterative initial value and easy to convergence, etc. This method is an effective one to solve the question of fitting non-linear equation and one of the most popular method in the field of solving non-linear equation at present. The error analysis showed the absolute value of the relative error for the proposed formula was smaller than 0.44% when the actual flow rate in channel was smaller than the transitional flow rate. Though the relative error increases when the actual flow rate in channel was larger than the transitional flow rate, the absolute value of relative error was less than 1% in which the relative error less than 0.08% when the slope coefficient was zero, and the error increased when the slope coefficient was not equal zero, but the absolute error was below 1%. It was thus indicated that the proposed formula had high precise and satisfied the need of Engineering practice. In addition, the proposed formula, with the simple form and the clear physical concept, exhibit both simplicity, easy-to-use and strong generality because it can be used to solve the normal depth of wide-shallow channel and the narrow-deep channel. Therefore, the efficient computational tools presented in this paper will be useful in the design and management of channel.