明渠正常水深数值求解方法研究
发布时间：2018-10-19 18:15
【摘要】：明渠正常水深是渠道断面设计、运行管理的基础数据。其求解方程多为高次或超越方程,无解析解。随着施工技术的发展,明渠断面的种类增加,结构形式更为复杂。求解这些断面正常水深,采用常规计算方法费时、精度低,因此,提出形式简捷、精度高、适用范围广的数值计算公式,显得尤为重要。研究明渠正常水深计算,不仅对解决工程实际具体问题具有非常重要的作用,而且还能完善水力学计算方法及计算理论体系,有利于分析正常水深的影响因素,使水力学计算的理论性更加增强。由于断面形状不同,正常水深求解方程所含的参变量不同,造成引入的无量纲参数和已知参变量的整合方式不同,因此,数值求解方法和数值计算公式不同。国内外研究成果和我国现有渠道断面调查,目前较为普遍出现的断面有8类(即矩形、梯形、U形、弧底梯形、城门洞形、圆形、马蹄形、抛物线形)16种形式。在这些断面正常水深计算上,部分断面已经提出了一套或多套数值求解公式,部分断面还停留在常规试算法和迭代法上,因此,归纳和总结现有数值计算公式,补充未涉及数值计算的断面的数值计算公式,具有工程实际意义。首先根据迭代理论对方程进行分析,采用牛顿迭代等方法提高收敛阶,加快收敛速度；其次,合理的迭代初值配合高效的迭代公式,可以提高计算公式的计算精度和简捷性,扩大公式的适用范围。但是,由于合理初值选取的难度较大,在迭代初值的选取技巧上进行深入研究,以参变量定义域内收敛速度最慢处方程的解为一次初值,并将此值代入超越方程或高次方程,按照二阶级数展开,求解二次方程,按照工程实际选取方程的解作为迭代初值,代入迭代公式后可得到精度较高的数值计算公式,或以曲线拟合、逐次逼近等方法确定迭代初值函数,给出迭代初值函数与高效迭代公式配套使用的数值计算公式。本文的主要内容和创新包括：(1)不等腰梯形断面的正常水深计算,目前以试算法求解为主。本文推导出迭代方程并进行了收敛性证明。以本文提出的等价边坡系数为参变量,分析无量纲正常水深与综合参数之间的关系,补充提出了不等腰梯形正常水深迭代方程与初值函数高效配套的数值计算公式,最多经过两次迭代,最大相对误差不大于0.5%。(2)对于任意普通型城门洞形断面的数值计算公式,目前只有一套,并且只适用于中心角为π。本文采用逐次逼近原理,以函数替代的方式补充提出适用不同中心角的数值计算公式,公式适用范围的上限为明满交替的起始点。水深位于直线段,公式适用于任意中心角,最大相对误差不超过0.1%;水深位于圆弧段,适用于中心角为π、2/3π、5/6π,最大相对误差为0.73%。克服了现有公式只适用于中心角为π的缺点。(3)对于马蹄形断面,标准Ⅰ、Ⅱ型研究相对成熟,但是对于标准Ⅲ型,目前尚未有数值计算公式,本文以分段函数的方式,采用最优拟合法补充提出了该段断面的正常水深数值计算公式,最大相对误差不大于0.79%,完善了马蹄形系列断面水力计算体系。(4)对迭代公式重新改造,提高收敛阶,寻求合理的初值函数。对矩形、弧底梯形、圆形三种断面的迭代公式进行改造,提出新的数值计算公式,不仅扩大了取值范围,并且提高了计算精度。特别是圆形断面,计算精度比现有公式的最高精度高出3倍多。(5)对U形断面采用函数替代法,提出新的数值计算公式,最大相对误差为0.24%,提高了计算精度和扩大适用范围。合理初值函数配合迭代公式,对抛物线形断面提出2套公式,适用于二次和半立方抛物线,最大相对误差不超过0.4%。(6)以近30年来研究成果为基础,对现有的82套正常水深数值计算公式按照断面的种类,进行归类和误差验算,分析评价各家公式的简捷性、精度、适用性。通过综合评价,推荐了20套简捷、精度高、适用范围广的数值计算公式,完善了正常水深计算体系。本文所提出或推荐的数值计算公式,满足工程常用范围,有利于基层单位的使用,计算精度高,具有实用价值,为工程设计和运行管理提供帮助;在理论体系上,对特征水深的影响因素以及断面优化分析提供基本参数,具有重要的意义。
[Abstract]:Normal water depth of open channel is the basic data of channel section design and operation management. The solution equation is a high order or transcendental equation with no analytical solution. With the development of construction technology, the type of open channel section increases and the structure form is more complicated. It is very important to solve the normal water depth of these sections, which is time-consuming and low in precision. Therefore, it is very important to put forward a numerical formula with simple form, high precision and wide application range. The calculation of normal water depth in open channel not only plays an important role in solving practical problems of engineering, but also improves hydraulic calculation method and calculation theory system, which is beneficial to analyzing the influencing factors of normal water depth. Because the cross-section shape is different, the normal water depth solves the difference of the parameters included in the equation, which leads to the difference between the introduced dimensionless parameter and the known parametric variable. Therefore, the numerical solution method and the numerical calculation formula are different. At present, there are 8 categories (i.e. rectangle, trapezoid, U shape, arc bottom trapezoid, city door opening shape, round shape, horseshoe shape, parabolic shape) in 16 forms. In the calculation of the normal water depth of these sections, a set or sets of numerical solution formulas have been put forward in some sections, and some sections are also on the conventional trial algorithm and the iterative method. Therefore, the calculation formulas of the existing numerical values are summarized and summarized, and the numerical formulas of the sections that do not involve numerical calculation are supplemented. It has practical significance. First, according to the iterative theory, the equation is analyzed, Newton iteration and other methods are adopted to improve the convergence order, and the convergence speed is accelerated; secondly, the reasonable iteration initial value is matched with the efficient iterative formula, so that the calculation accuracy and the simplicity of the calculation formula can be improved, and the application range of the formula can be enlarged. However, because the difficulty of selecting the initial initial value is large, in-depth study is carried out on the selection technique of the initial value of the iteration, the solution of the equation at the slowest convergence speed in the domain of the parametric variable domain is the initial initial value, and the value is substituted into the transcendental equation or the high-order equation to expand according to the number of the two classes, solving the quadratic equation, according to the solution of the actual selecting equation of the engineering as the initial value of iteration, substituting the iterative formula to obtain a numerical formula with higher precision, or determining an iterative initial value function by curve fitting, successive approximation and the like, The numerical formulas of iterative initial value function and efficient iteration formula are given. The main contents and innovations of this paper are as follows: (1) the calculation of the normal water depth of the non-isosceles trapezoid section is mainly based on the trial algorithm. In this paper, the iterative equation is derived and the convergence proof is carried out. Based on the equivalent slope coefficient proposed in this paper, the relation between the normal water depth and the comprehensive parameters of dimensionless normal water depth is analyzed, and the numerical calculation formula of the non-isosceles trapezoid normal water depth iterative equation and the initial value function is supplemented, and the most two iterations are obtained. The maximum relative error is not greater than 0.5%. (2) For the numerical calculation formula of the shape section of any common city door opening, only one set is present and only applicable to the central angle. In this paper, the successive approximation principle is used to supplement the numerical formulas for applying different central angles in a function substitution mode, and the upper limit of the formula application range is the starting point of the full-full alternation. The water depth is located in straight section, the formula is suitable for any central angle, the maximum relative error is not more than 0. 1%, the water depth is located in the circular arc section, it is applicable to the central angle of 0.1, 2/ 3, 5/ 6, and the maximum relative error is 0. 73%. overcomes the defect that the existing formula is only applicable to the central angle as the central angle. (3) For the horseshoe-shaped section, the standard 鈪，
本文编号：2281954
[Abstract]:Normal water depth of open channel is the basic data of channel section design and operation management. The solution equation is a high order or transcendental equation with no analytical solution. With the development of construction technology, the type of open channel section increases and the structure form is more complicated. It is very important to solve the normal water depth of these sections, which is time-consuming and low in precision. Therefore, it is very important to put forward a numerical formula with simple form, high precision and wide application range. The calculation of normal water depth in open channel not only plays an important role in solving practical problems of engineering, but also improves hydraulic calculation method and calculation theory system, which is beneficial to analyzing the influencing factors of normal water depth. Because the cross-section shape is different, the normal water depth solves the difference of the parameters included in the equation, which leads to the difference between the introduced dimensionless parameter and the known parametric variable. Therefore, the numerical solution method and the numerical calculation formula are different. At present, there are 8 categories (i.e. rectangle, trapezoid, U shape, arc bottom trapezoid, city door opening shape, round shape, horseshoe shape, parabolic shape) in 16 forms. In the calculation of the normal water depth of these sections, a set or sets of numerical solution formulas have been put forward in some sections, and some sections are also on the conventional trial algorithm and the iterative method. Therefore, the calculation formulas of the existing numerical values are summarized and summarized, and the numerical formulas of the sections that do not involve numerical calculation are supplemented. It has practical significance. First, according to the iterative theory, the equation is analyzed, Newton iteration and other methods are adopted to improve the convergence order, and the convergence speed is accelerated; secondly, the reasonable iteration initial value is matched with the efficient iterative formula, so that the calculation accuracy and the simplicity of the calculation formula can be improved, and the application range of the formula can be enlarged. However, because the difficulty of selecting the initial initial value is large, in-depth study is carried out on the selection technique of the initial value of the iteration, the solution of the equation at the slowest convergence speed in the domain of the parametric variable domain is the initial initial value, and the value is substituted into the transcendental equation or the high-order equation to expand according to the number of the two classes, solving the quadratic equation, according to the solution of the actual selecting equation of the engineering as the initial value of iteration, substituting the iterative formula to obtain a numerical formula with higher precision, or determining an iterative initial value function by curve fitting, successive approximation and the like, The numerical formulas of iterative initial value function and efficient iteration formula are given. The main contents and innovations of this paper are as follows: (1) the calculation of the normal water depth of the non-isosceles trapezoid section is mainly based on the trial algorithm. In this paper, the iterative equation is derived and the convergence proof is carried out. Based on the equivalent slope coefficient proposed in this paper, the relation between the normal water depth and the comprehensive parameters of dimensionless normal water depth is analyzed, and the numerical calculation formula of the non-isosceles trapezoid normal water depth iterative equation and the initial value function is supplemented, and the most two iterations are obtained. The maximum relative error is not greater than 0.5%. (2) For the numerical calculation formula of the shape section of any common city door opening, only one set is present and only applicable to the central angle. In this paper, the successive approximation principle is used to supplement the numerical formulas for applying different central angles in a function substitution mode, and the upper limit of the formula application range is the starting point of the full-full alternation. The water depth is located in straight section, the formula is suitable for any central angle, the maximum relative error is not more than 0. 1%, the water depth is located in the circular arc section, it is applicable to the central angle of 0.1, 2/ 3, 5/ 6, and the maximum relative error is 0. 73%. overcomes the defect that the existing formula is only applicable to the central angle as the central angle. (3) For the horseshoe-shaped section, the standard 鈪，
本文编号：2281954
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