Development of a Generalized One-Dimensional Hydrodynamic Equation via the Dual Approach

This study aims to develop a more general form of the one-dimensional hydrodynamic equation by employing the dual approach through direct integration of the three-dimensional hydrodynamic equations. Unlike previous formulations that combine two-horizontal and two-vertical components, this method allows for the derivation of a one-dimensional equation that inherently incorporates the effects of geometric deformation of the river cross-section. The proposed equation enhances the classical Saint–Venant model by introducing additional terms whose orders of magnitude are rigorously evaluated relative to the conventional momentum terms. This methodological innovation offers a more comprehensive representation of real-world flow dynamics in open channels. The novelty of this research lies in the direct derivation of a one-dimensional hydrodynamic equation from the full three-dimensional system via the dual approach, bypassing the need for intermediate two-dimensional simplifications and capturing a broader range of hydrodynamic behavior. Keywords: Classical average; Dual approach; One-dimensional hydrodynamic equation; Three-dimensional hydrodynamic equation; River cross-section deformation, Saint–Venant model.