【作者】 冷畅俭；

【导师】 王正中；

【作者基本信息】 西北农林科技大学 ， 农业水土工程， 2013， 博士

【摘要】 明渠各种特征水深的计算是进行渠道断面设计、渠道闸门运行、渠道消能防冲的重要依据。这些特征水深一般包括明渠恒定均匀流正常水深、恒定非均匀流临界水深、共轭水深和收缩断面水深,其基本方程一般为含参变量的非线性方程,理论上多无解析解。应用数学方法探求含参变量的非线性方程的解法,提出形式简捷、理论性强、计算精度高、适用范围广的计算方法,不仅对解决工程实际问题具有非常重要的作用,而且对进一步完善水力学计算方法及计算理论体系有重要意义。含参变量非线性方程的常规解法有试算法、图表法、迭代法、计算机编程求解法。试算法具有盲目性,图表法不便于应用且精度不高；随着计算机技术的飞速发展,计算机编程求解几乎可以解决一切复杂的计算问题,但是在含参变量非线性方程的求过程中,有许多科学规律和方法值得探索,如果只依靠编制计算机程序进行求解,那么无论从数学思想还是对计算水力学的发展而言就失去了科学意义；同时,水力学各种特征水深中的典型水深---共轭水深、收缩断面水深的基本方程均为复杂的含参变量的非线性方程,采用计算机编程计算,工程设计单位在应用中,常因物理概念不明确或编程复杂而受到限制。在众多求解非线性方程的方法中,迭代法是最常用最基础的方法,但常因迭代初值不合理或迭代方程收敛慢,导致迭代计算中存在较多困难。本文首先根据计算迭代理论对求解含参变量的非线性方程的迭代法进行了分析,以参变量定义域内收敛速度最慢处方程的解为一次初值,并将该含参变量的超越方程或高次方程在此处进行二阶泰勒级数展开,舍去高阶余量,进一步求解该二次方程得到初值,将初值代入迭代公式仅需迭代一次即可得到精度较高的近似计算公式；同时,通过对迭代方程收敛过程的进一步分析和分类,分别对交错迭代和单调迭代提出了相应的快速迭代方法；对迭代初值选取策略及迭代加速技术进行了深入研究,提出了该类方程求解的新理论及方法,给出了高效迭代公式及与之配套的合理迭代初值。其次,在全面分析总结各种断面共轭水深、收缩断面水深计算方法的基础上,应用本文提出的含参变量非线性方程求解方法——合理迭代初值及高效迭代方程配套,通过对特征水深基本方程进行变换处理,得到了快速收敛的迭代公式,再通过采用最小二乘法及优化计算得到合理的迭代初值,创新性地提出了二次抛物线形渠道共轭水深的计算公式、以及标准U形渠道和三次抛物线形渠道收缩断面水深的直接计算公式；提出的这三套公式的误差分析表明：在工程常用范围内,跃前水深、跃后水深直接计算公式的最大相对误差分别为0.47%和0.55%；三次抛物线形渠道收缩断面水深的最大相对误差小于0.16%；标准U形渠道收缩断面水深的最大相对误差小于0.55%：三套公式形式比较简捷、精度高、适用范围涵盖了工程常用范围。最后,对近年来工程中常用的7种几何断面(即矩形断面、梯形断面、圆形断面、U形断面、二次抛物线形断面、三次抛物线形断面、城门洞形断面)的2种特征水深(即明渠共轭水深、收缩断面水深)的水力计算研究成果进行了分析、归类和总结,分析评价各家公式的推理过程及优越性、误差和工程实用性,通过比较评价分析推荐了具有理论性强、公式简捷、精度高、适用范围广的计算公式,使各种断面2种特征水深---共轭水深和收缩断面水深的计算理论体系更加完善系统,计算更加简便实用。更多还原

【Abstract】 The calculation of characteristic depths of open channels is the important reference in the design of channel sections, operation of channel gates, and energy dissipation and erosion control of channels. Characteristic depths consist of four types of water depths, i.e., normal depth for uniform flow in open channel, critical depth for non-uniform flow, conjugate depths, and contracted depth. Governing equations for characteristic depths are normally nonlinear equations with parameters and no analytical solutions exist. By using mathematical methods, solving methods for nonlinear equations with parameters were developed. Explicit equations for characteristic depths with simple forms, high accuracy, and wide application ranges were obtained. These equations will play an important role in practical engineering and in the development of hydraulics and also will improve the calculation method and theory system of hydraulics.The common solving methods for nonlinear equations with parameters include trial and error method, chart method, iterative method, and computer programming method. Trial and error method is time-consuming while chart method has low accuracy. With the development of the computer technology, computer programming can solve almost all the complex problems in the world. However, in the solving process of nonlinear equations with parameters, there exist a large number of science laws and methods to examine. Therefore, it maybe make none sense from viewpoint of hydraulics and mathematical system only for application of computer programming. Moreover, the governing equations for conjugate and contracted depths are complicated nonlinear equations with parameters. Thus, the computer programming is restricted without clear physical concept in practical engineering. Among these methods, iterative method is the most fundamental way to obtain equations of nonlinear equations. However, it is not easy to select appropriate initial value for iterative method which will lead to slow converging rates.First of all, this paper presents the iterative methods for nonlinear equations with parameters on the basis of iterative theory. The initial guess was given by solving the equations with parameters to obtain the solution with the lowest convergence, where we expanded the transcendental equations or high orders equations in the form of Taylor series and removed high order quantities. The final initial guess was obtained by solving the second order equation. Approximate equations will be developed by carrying out only the first iteration. In the meantime, the acceleration methods for staggered and monotony iteration were developed by analyzing the converging processes of iterative equation. Hence the selected method of initial guess and the accelerative technology were discussed in detail. New theory and method for the equations of this type were developed. Also, the iterative equations with high efficiency and the corresponding initial guess were given.Secondly, based on the summary of present calculation methods for conjugate depths and contracted depths of various channel sections, the theory provided in this paper for nonlinear equations with parameters were applied to carrying out the transformation of fundamental equations for characteristic depths. The transformations result in iterative equations with high efficiency. The appropriate initial guess was obtained by means of least square method and optimal calculation. Based on the theory described above, the method for computing the conjugate depths of quadric parabola section were developed, and as well as the methods for the contracted depths of U-shaped section and cubic parabola section. Error analysis of these three sets of equations shows that the maximum relative errors of the depth before the jump and the depth after the jump are0.47%and0.55%, respectively in the practical range; the maximum errors in the equations for the contracted depth of cubic parabola and U-shaped are less than0.16%and0.55%respectively. The recommended equations in this study have simple forms and high accuracy and can be used in wide application ranges.Finally, this paper presents previous results of explicit equations of two types of characteristic depths (conjugate depths and contracted depth) of seven types of sections including rectangular, trapezoidal, circular, U-shaped, quadric parabola, cubic parabola, and city-gate sections. The derivation processes of these equations were presented and relative errors and advantages were also analyzed. By means of comprehensive analysis, the explicit equations with simple forms, high accuracy, and wide application range were recommended. The theory presented in this paper will improve the theoretical system of two types of characteristic depths and also make the determination of conjugate and contracted depths much simpler.更多还原