Explicit formulas of normal, alternate and conjugate depths for three kinds of parabola-shaped channels

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 hc,hc',hc''; and conjugate (initial and sequent) depths h1,h2,h1',h2',h1'',h2'' 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.