HIGHLY ACCURATE EXPLICIT RATIONAL APPROXIMATIONS FOR THE NORMAL FLOW DEPTH PROBLEM IN RECTANGULAR CHANNELS USING MANNING’S EQUATION

Abstract

This paper addresses the classical normal depth problem in rectangular channels governed by Manning’s implicit equation. It proposes and evaluates a set of nine high-accuracy explicit and rational approximation models aimed at estimating the dimensionless normal flow depth variable with exceptional accuracy across the full admissible flow domain.

After presenting the exact solution to the transcendental Manning’s equation, the study develops a hierarchy of eight increasingly accurate explicit approximate models.

First, the classical Lagrange-Burmann series expansion is revisited and shown to diverge even in the practical restricted domain of the dimensionless discharge parameter M ∈ [0, 1.6], disqualifying it as a reliable approximation. Similarly, models based on Laguerre and Legendre polynomials are examined and found inappropriate: the Laguerre model lacks orthogonality over the finite domain, while the Legendre model, despite ensuring orthogonality, relies on a normalized variable that leads to poor accuracy, with relative errors reaching up to 55% for low values of M.

The first rational approach leverages an accurate Padé surrogate model, achieving a maximum deviation below 0.000045%.

The Adaptive Antoulas-Anderson (AAA) rational approximation further improves accuracy, yielding a deviation under 10^-7 %.

A third solution based on Chebyshev polynomial approximation maintains a maximum relative deviation below 0.00035%. Building upon this, a Lawson-refined AAA model significantly enhances performance with a deviation below 3.7×10^-9 %.

To preserve monotonicity and shape, a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) is adopted, maintaining an absolute deviation under 1.7×10^-5 %.

The Achour and Amara accurate two-piece rational model (Model I) demonstrates outstanding performance, producing a deviation of just 0.0000004% in the lower domain and sub-1.32×10^-6 % in the upper domain.

In parallel, an iterated-perturbation analytical model by Amara and Achour yields a robust maximum deviation under 0.006%.

A highly accurate one-piece [3/3] rational model by Achour and Amara is also developed, achieving a deviation below 0.00016%.

Lastly, a rational two-piece [2/2] model (Model II) offers deviations of 0.0062% for the lower piece and 0.00014% for the upper piece.

All proposed models are thoroughly validated through analytical and numerical comparisons against the exact solution derived from Manning’s equation.

The paper also rigorously critiques several classical approximation techniques, including those based on Laguerre and Legendre polynomials, and the Lagrange-Burmann theorem, which are shown to be unsuited for the present problem due to divergence, lack of orthogonality, or excessive error over the target interval.

The comparative assessment highlights that only the rational models specifically constructed to reflect the structure of the underlying implicit equation offer practical and accurate solutions.

The comprehensive suite of proposed models offers a range of efficient, accurate, and easily implementable alternatives for hydraulic engineers and researchers.