Numerical Solution To Normal Depth Of Open Channals

Normal depth of open channels is the basic data of section design and management, whose equations are mostly high or transcendental ones without analytical solutions. With the development of construction technology, there are a variety of cross sections which have more complicated structures. It is a waste of time and of low precision that conventional methods are used to solve the normal depth of these sections, so it is very necessary to put forward numerical formulas of simplicity, high accuracy and wide applicability. Studying normal depth calculation of open channels is not only important for solving engineering problems, helpful for analyzing the factors affecting the normal depth, but also perfects hydraulics calculation methods and calculation theory systems, which enhances the theory of hydraulics calculation.Because of the difference in section shape, parameters in the solving equations of normal depth are varied, leading to different integration modes between the introduced dimensionless parameters and the parameters already known. Based on the study at home and abroad, normal depth methods of the common 8 types of shapes (namely triangle, rectangle, trapezoid, U-shaped, trapezoid-arc, city-gate, horseshoe and power function) and 16 kinds of cross sections are concluded in the paper. On the normal depth calculation for these sections, there are one or more sets of numerical solving formula for some section forms while for some other forms the methods still rest on tentative calculation and integration. Therefore it is of engineering practical value to summarize current numerical methods and then add up numerical methods for sections that are not considered in the past. Firstly, equations are analyzed with iteration theory, and Newton iteration is used to improve the convergence order and speed up convergence; Secondly, rational iterative initial values together with highly effective integration equations can help to enhance the precision and applicability. But it is hard to choose the right initial values, so the skills of selecting iterative initial values are studied:the solution of the equation at the lowest speed of convergence within the domain of parameter is used as an initial value which is brought into a transcendental or high order equation and expanded according to the second order series to solve the quadratic equation whose solution is selected based on the engineering practice as the iterative initial value, which is brought into the iterative formula, then the numerical calculation formula with higher precision can be got. And after curve fitting and successive approximation are used to determine iterative initial value function, the numerical calculation formula is obtained by combining iterative initial value function with efficient iterative formula. The following are the main and innovative points in the paper:(1)Normal depth calculation of non-isosceles trapezoid section is mainly based on tentative calculation. This paper deduces the iterative equation and verifies its convergence. Based on the equivalent slope parameter, relationship between dimensionless normal depth and comprehensive parameter is analyzed. Efficient matching numerical calculation formula for non-isosceles trapezoid section is put forward by combining the iterative equation of normal depth of non-isosceles trapezoid section with the iterative function, and the maximum relative error is less than 0.5% after two iterations.(2) There is only one set of numerical calculation formula of any ordinary city-gate section, which is only suitable for central angles π. This paper adopts successive approximation theory and puts forward numerical calculation formula for any angle by function substitution. The upper limit range for the formula is the start point of mixed free-surface-pressure flow. When water depth is in the straight line segment, the formula is applicable to any central angle and the maximum relative error is less than 0.1%; When water depth is in the circular arc segment, the formula is applicable to central angles being π、5/6π、 2/3π, and the maximum relative error is 0.73%. The formula is preferable to the existing one that is only applicable to the central angle being π and the ratio of height to width being 1.(3) Study on horseshoe section of Standard Type Ⅰ and Ⅱ is ripe, but there is no numerical calculation formula for Standard Type Ⅲ. The numerical calculation formula of normal depth of horseshoe section of Standard Type Ⅲ is put forward by piecewise function and optimal fitting method and the maximum relative error is less than 0.79%, which supplements the system of hydraulic calculation of horseshoe sections.(4) The iterative formula is innovated and convergence order is improved to seek the reasonable initial value function. New numerical calculation formula is got by innovating iterative formulas of rectangle, trapezoid-arc and circle sections, which expands the value range and improves the calculation precision. The highest precision of the new formula is three times higher than that of the past one for circular sections.(5)New numerical calculation formula of U-shaped section is got by using function substitution method and the maximum relative error is 0.24%, which is of higher precision and wider applicability. Two sets of formulas for parabolic sections are put forward by combining reasonable initial value functions with iterative formulas, which are applicable to quadratic and cubic parabola and whose maximum relative error is less than 0.4%.(6)Based on the researches in the past 30 years, the existing 82 sets of numerical formulas of normal depth are classified and error-checked according to the types of sections, and the simplicity, precision and applicability of each set is evaluated.20 sets of numerical formulas with simplicity, high precision and wide applicable scope are recommended, which perfects the calculation system of normal depth.The numerical calculation formula proposed or recommended by this paper can satisfy the common use in engineering practice, suitable for grassroots units. The calculation precision is high and the formulae are of pragmatic value, which can provide assistance to engineering design, operation, and management. Theoretically, the formulae can provide important information such as the factors affecting the feature normal depth and basic parameters for section optimization analysis.