Monograph NORMAL FLOW DEPTH IN TRAPEZOIDAL CHANNELS. A DARCY-WEISBACH-RMM SOLUTION FRAMEWORK

Abstract

This monograph establishes a mathematically rigorous framework for determining the normal flow depth in trapezoidal open channels by reformulating the problem within the Darcy-Weisbach-Rough Model Method (RMM) structure. Unlike classical Manning- or Chezy-based approaches, whose resistance parameters depend implicitly on the unknown flow depth, the proposed formulation yields a physically self-consistent, dimensionless implicit governing equation free from circular dependence.

The problem reduces to a nonlinear algebraic fixed-point equation in a reduced flow depth variable, expressed solely in terms of measurable quantities: relative conductivity, geometric parameters, slope, roughness, and viscosity.

The nonlinear fixed-point operator associated with the implicit governing equation satisfies a Lipschitz condition with constant strictly less than unity over the investigated hydraulic domain, i.e., conductivity varying within the range 0.1 ≤ Q* ≤ 4, and wall inclination angle such as 10° ≤ α ≤ 80°, thereby guaranteeing strong contraction, numerical stability, and rapid geometric convergence of the iterative solution.

Several solution strategies are derived and analytically examined, namely: an enhanced fixed-point iteration with an adaptive initial estimate (Initial guess), a controlled cube-root linearization producing an explicit closed-form approximation, an Aitken-Steffensen accelerated one-shot scheme obtained from finite-difference extrapolation, and Newton and secant corrections interpreted through local linearization theory.

Error behavior is analyzed through amplification properties of the nonlinear mapping, revealing a geometry-driven structure of deviation that is independent of the selected numerical strategy. Moderate sidewall inclinations, particularly around 45°, reduce the geometric nonlinearity of the governing equation and enhance the contractive strength of the fixed-point operator; notably, the numerical deviations reach their minimum near this angle, indicating an optimal balance between hydraulic behavior and computational stability.

The results confirm that the implicit governing equation is globally well-conditioned and that appropriately constructed explicit or semi-explicit approximations achieve accuracy levels comparable to fully converged iterative solutions. The proposed framework therefore provides both theoretical clarity and computational efficiency for the normal-flow depth problem in trapezoidal channels.