【摘要】：明渠正常水深是渠道斷面設計、運行管理的基礎數(shù)據(jù)。其求解方程多為高次或超越方程,無解析解。隨著施工技術的發(fā)展,明渠斷面的種類增加,結構形式更為復雜。求解這些斷面正常水深,采用常規(guī)計算方法費時、精度低,因此,提出形式簡捷、精度高、適用范圍廣的數(shù)值計算公式,顯得尤為重要。研究明渠正常水深計算,不僅對解決工程實際具體問題具有非常重要的作用,而且還能完善水力學計算方法及計算理論體系,有利于分析正常水深的影響因素,使水力學計算的理論性更加增強。由于斷面形狀不同,正常水深求解方程所含的參變量不同,造成引入的無量綱參數(shù)和已知參變量的整合方式不同,因此,數(shù)值求解方法和數(shù)值計算公式不同。國內外研究成果和我國現(xiàn)有渠道斷面調查,目前較為普遍出現(xiàn)的斷面有8類(即矩形、梯形、U形、弧底梯形、城門洞形、圓形、馬蹄形、拋物線形)16種形式。在這些斷面正常水深計算上,部分斷面已經(jīng)提出了一套或多套數(shù)值求解公式,部分斷面還停留在常規(guī)試算法和迭代法上,因此,歸納和總結現(xiàn)有數(shù)值計算公式,補充未涉及數(shù)值計算的斷面的數(shù)值計算公式,具有工程實際意義。首先根據(jù)迭代理論對方程進行分析,采用牛頓迭代等方法提高收斂階,加快收斂速度；其次,合理的迭代初值配合高效的迭代公式,可以提高計算公式的計算精度和簡捷性,擴大公式的適用范圍。但是,由于合理初值選取的難度較大,在迭代初值的選取技巧上進行深入研究,以參變量定義域內收斂速度最慢處方程的解為一次初值,并將此值代入超越方程或高次方程,按照二階級數(shù)展開,求解二次方程,按照工程實際選取方程的解作為迭代初值,代入迭代公式后可得到精度較高的數(shù)值計算公式,或以曲線擬合、逐次逼近等方法確定迭代初值函數(shù),給出迭代初值函數(shù)與高效迭代公式配套使用的數(shù)值計算公式。本文的主要內容和創(chuàng)新包括：(1)不等腰梯形斷面的正常水深計算,目前以試算法求解為主。本文推導出迭代方程并進行了收斂性證明。以本文提出的等價邊坡系數(shù)為參變量,分析無量綱正常水深與綜合參數(shù)之間的關系,補充提出了不等腰梯形正常水深迭代方程與初值函數(shù)高效配套的數(shù)值計算公式,最多經(jīng)過兩次迭代,最大相對誤差不大于0.5%。(2)對于任意普通型城門洞形斷面的數(shù)值計算公式,目前只有一套,并且只適用于中心角為π。本文采用逐次逼近原理,以函數(shù)替代的方式補充提出適用不同中心角的數(shù)值計算公式,公式適用范圍的上限為明滿交替的起始點。水深位于直線段,公式適用于任意中心角,最大相對誤差不超過0.1%;水深位于圓弧段,適用于中心角為π、2/3π、5/6π,最大相對誤差為0.73%�？朔爽F(xiàn)有公式只適用于中心角為π的缺點。(3)對于馬蹄形斷面,標準Ⅰ、Ⅱ型研究相對成熟,但是對于標準Ⅲ型,目前尚未有數(shù)值計算公式,本文以分段函數(shù)的方式,采用最優(yōu)擬合法補充提出了該段斷面的正常水深數(shù)值計算公式,最大相對誤差不大于0.79%,完善了馬蹄形系列斷面水力計算體系。(4)對迭代公式重新改造,提高收斂階,尋求合理的初值函數(shù)。對矩形、弧底梯形、圓形三種斷面的迭代公式進行改造,提出新的數(shù)值計算公式,不僅擴大了取值范圍,并且提高了計算精度。特別是圓形斷面,計算精度比現(xiàn)有公式的最高精度高出3倍多。(5)對U形斷面采用函數(shù)替代法,提出新的數(shù)值計算公式,最大相對誤差為0.24%,提高了計算精度和擴大適用范圍。合理初值函數(shù)配合迭代公式,對拋物線形斷面提出2套公式,適用于二次和半立方拋物線,最大相對誤差不超過0.4%。(6)以近30年來研究成果為基礎,對現(xiàn)有的82套正常水深數(shù)值計算公式按照斷面的種類,進行歸類和誤差驗算,分析評價各家公式的簡捷性、精度、適用性。通過綜合評價,推薦了20套簡捷、精度高、適用范圍廣的數(shù)值計算公式,完善了正常水深計算體系。本文所提出或推薦的數(shù)值計算公式,滿足工程常用范圍,有利于基層單位的使用,計算精度高,具有實用價值,為工程設計和運行管理提供幫助;在理論體系上,對特征水深的影響因素以及斷面優(yōu)化分析提供基本參數(shù),具有重要的意義。
[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，

本文編號：2281954