Linear and Weakly Nonlinear Instabilities of Sand Waves Caused by a Turbulent Flow

Abstract

This study examines the instability of sand waves (dunes and antidunes) from both linear and weakly nonlinear perspectives. The linear and weakly nonlinear analyses use the standard linearization and the center manifold-projection technique, respectively. The mathematical framework includes the depth-averaged continuity and momentum equations, the advection–diffusion equation for suspended sediment concentration, and Exner’s equation for bed evolution. The streamline curvature is treated using the Boussinesq approximation. The model considers the departure of the pressure distribution from the hydrostatic law. Both modes of sediment transport, as bedload and as suspended load, are taken into consideration. The perturbations characterize the maximum growth rate for a selected wave number, called the resonant wave number, which is the most favorable wave number for the formation of sand waves. As the flow Froude number and the relative roughness number increase, the dimensionless resonant wave number decreases. The dimensionless amplitude of sand waves increases as the flow Froude number and the relative roughness number increase to achieve a maximum, and subsequently it decreases. The predicted wave number and amplitude of sand waves satisfactorily match the available experimental data.