EFFICIENT COMPUTATION OF NORMAL FLOW DEPTH IN CIRCULAR CONDUITS WITHIN THE DARCY-WEISBACH-RMM FRAMEWORK

Abstract

The determination of normal flow depth in circular conduits remains a fundamental problem in hydraulic engineering, yet most existing methods rely on resistance equations, namely Manning, Chezy, or Darcy-Weisbach, that introduce conceptual inconsistencies when used in dimensionless formulations. These classical approaches require resistance coefficients that, in practice, depend on the unknown normal flow depth itself, thereby compromising both physical coherence and computational reliability. To overcome this limitation, the present study adopts the Darcy-Weisbach relationship within the rigorous framework of the Rough Model Method (RMM), which entirely eliminates the circular dependence on empirical resistance parameters. By transforming the governing hydraulic equation into a dimensionally consistent implicit relationship, the analysis yields a robust expression for the relative normal flow depth ξ_R in the rough reference circular conduit model.

The mathematical arrangement of the final expression ensures numerical stability across the full admissible range of the relative conductivity Q* ∈ [0, π], This makes it ideally compatible not only with conventional fixed-point approaches but also with convergence-enhancing accelerators, including Aitken’s Δ² process, yielding highly accurate results with minimal computational effort.”

Owing to the severe nonlinearity of the relationship, classical analytical tools, such as the Lagrange-Burmann inversion theorem, prove ineffective because the resulting series converges only for extremely small values of the relative conductivity Q*. To circumvent these intrinsic limitations, the present work provides a comprehensive numerical reference Table containing exact solutions of ξ_R for Q* ∈ [0, π], computed with high precision using a reliable bracketing approach. This tabulation, combined with simple linear interpolation, enables practitioners to determine the sought normal flow depth directly and with high accuracy, without iterative computation.

The proposed methodology is conceptually rigorous, computationally efficient, and entirely free from empirical assumptions. The resulting tool is therefore well suited for engineering design, performance evaluation, and operational hydraulic analysis involving circular conduits under uniform flow conditions.