Numerical integration method for calculating water surface profile of gradually varied steady flow in parabola shaped channel

Abstract: The parabola shaped channel has an excellent hydraulic performance and strong ability of anti-frost heave, and is used widely in the field of spillway and irrigation channels. The water surface profile of gradually varied steady flow is an important hydraulic element for the channel design and the operational management, but the differential equation of water surface profile is the transcendental equation with no analytic solution. The current differential test algorithm, iterative method and graphical method for solving the water surface profile has complex calculation process, large errors and low efficiency, so on. To obtain the simplified calculation formula of water surface profile of parabolic cross-section, we introduced the concepts of the parabolic cross-sectional characteristic water depth (that is the product of the water depth and the parabolic shaped perimeter) and characteristic wetted perimeter (2 times of the product of parabolic shaped perimeter and wetted perimeter), and did the identical deformation for the basic differential equations of the gradually varied steady water surface profile of the parabolic cross-sectional channel. The optimal hydraulic cross-sectional parabolic cross-sectional characteristic water depth is 0.947 m. In practical engineering, the design of the parabolic cross-sectional channel is required to be close to the optimal hydraulic cross-section as far as possible under permissive conditions so that it would obtain good hydraulic conditions, save the substantial cultivated land, reduce project expense and achieve better economic benefits. The study here focused on the common parabolic cross-sectional channel in the engineering practice with the range of characteristics water depth of 0.6-1.5 m. Since the characteristic wetted perimeter of the original differential equations is a complex non-integrable function within the most commonly used scope in practical engineering, we analyzed the interrelation between characteristic water depth and characteristic wetted perimeter, plotted a curve with characteristics water depth as ordinate against characteristic wetted perimeter. The results showed that the relationship between characteristics water depth and characteristic wetted perimeter followed the power function. The equation coefficients were fitted by the least squares method, and then a simple integral expression of characteristics wetted perimeter was obtained. The error analysis showed that the absolute value of the relative error of the proposed formula was smaller than 1.08%, indicating that this method is effective to solve the unintegrable equation of characteristic wetted perimeter. As a result, the direct numerical integration method had been deduced by the beginning and final section water depth for determination of the flow distance of the water surface profiles in parabola shaped channel. To test the feasibility of the proposed integration method in calculating the flow distance of water surface profile, it was used to compare with the difference method for 2 cases. Results showed that the maximum relative error was less than 0.2% between both methods, indicating the high efficiency and high precision of the proposed method. In addition, the proposed formula also has simple form, clear physical concept, easiness to use and wide applications. The direct integration formula proposed here is useful in the channel design and management.