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Explicit Solutions for the Specific Energy Equation in Open Channels with a Parabolic Cross-section
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This study derived a non-linear specific energy equation for open channels with a parabolic cross-section and proposed an explicit method to solve it with high accuracy. The Newton-Raphson method, renowned for its second-order accuracy, was used to solve the equation, with the initial calculation value derived from a regression equation easily generated using Microsoft Excel. For comparison, when the regression equation was employed, the relative error against the exact solution reached up to 2.46% for subcritical flows and 8.59% for supercritical flows. However, the explicit method proposed in this study significantly reduced the relative error to just 0.26% for subcritical flows and 0.21% for supercritical flows, achieving results that closely matched the exact solution. In other words, the explicit method introduced in this study offers a straightforward yet highly precise way to determine the alternate depths of open channel flows with a parabolic cross-section.
Abstract
This study derived a non-linear specific energy equation for open channels with a parabolic cross-section and proposed an explicit method to solve it with high accuracy. The Newton-Raphson method, renowned for its second-order accuracy, was used to solve the equation, with the initial calculation value derived from a regression equation easily generated using Microsoft Excel. For comparison, when the regression equation was employed, the relative error against the exact solution reached up to 2.46% for subcritical flows and 8.59% for supercritical flows. However, the explicit method proposed in this study significantly reduced the relative error to just 0.26% for subcritical flows and 0.21% for supercritical flows, achieving results that closely matched the exact solution. In other words, the explicit method introduced in this study offers a straightforward yet highly precise way to determine the alternate depths of open channel flows with a parabolic cross-section.