A NOVEL RIGOROUS EXPLICIT FRAMEWORK FOR CRITICAL FLOW DEPTH IN TRAPEZOIDAL OPEN CHANNELS

Abstract

The determination of critical flow depth in trapezoidal open channels remains a classical yet inherently implicit problem in hydraulic engineering, as the governing relationship is strongly nonlinear and does not admit a closed-form solution in terms of elementary functions. This study develops a rigorous, unified, and practically implementable framework that overcomes this limitation by combining exact analysis with highly accurate explicit formulations. Starting from the classical critical-flow condition and trapezoidal geometry, a compact dimensionless governing equation is derived using physically meaningful parameters, namely a shape parameter and a normalized discharge parameter. Through an appropriate change of variables, the problem is recast into a canonical algebraic form equivalent to a trinomial sextic equation with a unique physically admissible root, thereby establishing a rigorous exact foundation.

On this basis, two complementary solution strategies are proposed. First, a quasi-exact computational approach is developed using a normalized formulation combined with a highly efficient initial estimate and a one-shot Newton iteration, yielding a maximum deviation strictly below 3.6 10^-5 % over the entire admissible range. Second, a fully explicit analytical formulation is derived through a Transformed Asymptotic-Taylor Reconstruction Method, producing a closed-form expression for the governing parameter with a maximum deviation of only 7.1 10^-6 %. The proposed formulation preserves the correct asymptotic behavior and rigorously recovers the classical rectangular and triangular channel solutions as limiting cases, ensuring full physical consistency.

A key feature of the study is that the developed models remain valid for unsymmetrical trapezoidal channels through the introduction of an equivalent side-slope parameter, which constitutes an exact transformation rather than an approximation. The resulting formulations eliminate the need for iterative procedures, graphical methods, or empirical fits, while maintaining quasi-exact accuracy. The study thus provides a new generation of analytical tools that combine mathematical rigor, computational efficiency, and broad applicability for hydraulic design and analysis.