Abstract Comparisons reveal that the 2.5, 10/3 and 3 order parabola-shaped channels improve flow conveyance discharges by 0.68%–7.86% and save construction costs by −0.95 to −27.99% over the common rectangular, triangular, trapezoidal sections, the extensively studied quadratic, semi-cubic and horizontal-bottom semi-cubic parabola-shaped sections. However, few researches on the critical, normal, alternate and conjugate depths, playing key roles in the design, operation and management of open channels, for these three kinds of parabola-shaped channels are available at present. Because the governing equations of normal, alternate and conjugate depths are nonlinear, so these characteristic depths cannot be solved analytically except for the critical depths. The biggest challenges in the calculation of normal depths are the nonintegrable wetted perimeters for the three kinds of parabola-shaped channels. Direct and numerical solutions for wetted perimeters are proposed based on the Gauss-Legendre six points method and the Simpson numerical integral method, respectively. Then, three direct solutions of normal depths are proposed using exact wetted perimeters based on the Simpson numerical integral method for the three kinds of channels. Subsequently, the specific energy equations are deformed into dimensionless forms. Meanwhile, analytical critical depths are skillfully utilized to obtain dimensionless specific force equations of conjugate depths. Then explicit solutions of alternate and conjugate depths have as well been proposed using iterative equations for alternate and conjugate depths based on the fixed-point iterative method by running 1stOpt software in the commonly using range of engineering. Through error analysis, the maximum relative errors of normal depths h , h ' , h ' ' ; alternate depths h c , h c ' , h c ' ' ; and conjugate (initial and sequent) depths h 1 , h 2 , h 1 ' , h 2 ' , h 1 ' ' , h 2 ' ' for 2.5, 10/3 and 3 order parabola-shaped cross-sections are only 0.34%, −0.15%, 0.001%; 0.13%, 0.12%, 0.099%; and 0.098%, −0.20%, 0.116%, −0.24%, 0.11%, 0.23% respectively. These explicit solutions of 2.5, 10/3 order parabola-shaped channels are proposed for the first time. And, explicit solutions of 3 order parabola-shaped channel have a similar range of application and higher precision than that of former studies.

中文翻译：

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三种抛物线形通道的法向、交替和共轭深度的显式公式

摘要 比较表明，2.5、10/3和3阶抛物线形通道比常见的矩形、三角形、梯形截面、广泛使用的矩形、三角形、梯形截面提高了0.68%~7.86%的流量输送流量，节省了-0.95~-27.99%的施工成本。研究了二次、半立方和水平底半立方抛物线形截面。然而，对于这三种抛物线形渠道，在明渠设计、运行和管理中起关键作用的临界深度、正常深度、交替深度和共轭深度的研究很少。由于法向、交替和共轭深度的控制方程是非线性的，因此除了临界深度之外，无法解析求解这些特征深度。法向深度计算的最大挑战是三种抛物线形通道的不可积湿周长。分别基于Gauss-Legendre六点法和Simpson数值积分法提出了湿周的直接解和数值解。然后，基于辛普森数值积分方法，提出了三种通道的三种使用精确湿周长的法向深度的直接解。随后，比能方程变形为无量纲形式。同时，巧妙地利用解析临界深度得到共轭深度的无量纲比力方程。然后，在常用工程范围内运行1stOpt软件，基于定点迭代法，使用交替和共轭深度的迭代方程，提出了交替和共轭深度的显式解。通过误差分析，法线深度h、h'、h''的最大相对误差；交替深度 hc , hc ' , hc ' ' ; 和共轭（初始和后续）深度 h 1 , h 2 , h 1 ' , h 2 ' , h 1 ' ' , h 2 ' ' 对于 2.5、10/3 和 3 阶抛物线形横截面仅为 0.34% , -0.15%, 0.001%; 0.13%、0.12%、0.099%；和 0.098%、-0.20%、0.116%、-0.24%、0.11%、0.23%。这些 2.5、10/3 阶抛物线形通道的显式解是首次提出的。和，