Fully developed two-dimensional free-surface liquid layer flow over a three-dimensional obstacle

This paper presents a novel asymptotic framework to analyze steady two-dimensional free-surface flows over three-dimensional topography at high Reynolds numbers on an inclined plane. Departing from classical triple-deck theory, the study develops a double-deck scaling that uniquely captures the interaction between fully developed viscous flow and free-surface dynamics. The proposed methodology bridges a critical gap in existing analyses that predominantly focus on the depth averaged equations, or thin layer approximations. Through matched asymptotic expansions, the analysis derives a system of nonlinear partial differential equations governing the viscous-inviscid interaction, explicitly incorporating surface tension effects through Weber number dependence. Linearized analytical solutions are obtained for small obstacles via Fourier transforms and special functions, while numerical solutions of the full nonlinear system are obtained for larger obstacles using spectral methods. The nonlinear solutions reveal distinct and complex free-surface patterns controlled by obstacle size, Weber number (We), and inclination parameter (s), demonstrating the mechanisms governing the interaction between viscous effects and free-surface wave dynamics.