Simple calculation method for normal depth of a kind of water supply tunnel for irrigation and drainage
Zhao Yanfeng1, Wang Zhengzhong1, Fang Xing2, Liu Jiliang1, Hong Anyu1
(1.Institute of Water Resources and Hydropower Engineering, Northwest A & F University, Yangling, Shaanxi 712100, China; 2.Department of Civil Engineering, Auburn University, Auburn, Alabama 36849, U.S.A.)
In order to obtain an explicit formula for normal depth of standard citygate section,first,the identical transformation of the governing equation for normal depth of standard citygate section was made,including the transcendental equation for normal depth in the bottom corner arc and the top arc,and the higher degree equation for normal depth in the lateral side.The single variable equation was obtained for dimensionless normal depth and the comprehensive parameter of the known quantities.Second,the concept of quasilinear function was introduced and treated as the standard template.The standard template for quasilinear function was applied to the single variable equation for normal depth.Using optimization and approximation to quasilinear function,the estimation substitution formula for the transcendental equation and higherdegree equation was developed in the practical range of y∈［0.05,1.80］,and the substitution formula has the same form as the linear function,i.e.,the quasilinear explicit expression for normal depth.Finally,error analysis of the formula was made.The result indicates that the maximum relative errors of the formula for normal depth are less than 0.36%,0.31%,and 0.39% in the bottom arc,the lateral side,and the top arc,respectively.The relative errors indicate that the quasilinear equation has high accuracy in the practical range,which can provide simple method for computing the normal depth of the uniform flow in the design of water supply tunnel for drainage and irrigation and controlling the water level.
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