Turbulent Shear Flow over a Downstream-Skewed Wavy Bed: Analytical Model Based on the RANS Equations with Boussinesq Approximation

Abstract

This study presents an analytical model of a steady turbulent flow over a two-dimensional downstream-skewed wavy bed of small amplitude. The mathematical framework rests on the time-averaged continuity and Reynolds-averaged Navier–Stokes (RANS) equations. The streamwise velocity profile is considered to follow a self-similar power law, whereas the Reynolds normal stresses are founded on the turbulent diffusivity hypothesis. The curvilinearity in flow streamlines induced by the wavy bed is introduced into the analysis via Boussinesq approximation. The analysis provides solutions to the free-surface, bed shear stress, and Reynolds shear stress profiles. As the flow Froude number varies, the free-surface and bed shear stress profiles change their phases. The phase shift of the free-surface profile with respect to the bed profile decreases with an increase in skewness factor, whereas the phase shift of the bed shear stress profile with respect to the bed profile increases with an increase in skewness factor attaining a constant value. The convex and concave shapes of the Reynolds shear stress profiles on the downslope and upslope of the bed profile are attributed to the decelerated and accelerated flows, respectively. The implementation of the key findings of this work to study the hydrodynamics of fluvial bedforms is discussed from the standpoint of sediment transport, as a future scope of research.