Instability theory of sand ripples formed by turbulent shear flows

Abstract

A theory of turbulent shear flow over a sand bed is developed addressing the instability principle of the fluid-granular bed interface leading to the formation of ripples. The Reynolds averaged Navier-Stokes (RANS) equations and the time-averaged continuity equation are analyzed using a 1/7-th power law of the time-averaged streamwise velocity and treating the curvilinear streamlines by the Boussinesq approximation. The integration of the RANS equations leads to a governing dynamical equation of flow over a mobile bed. A near-bed flow layer of 3.5 times the ripple height is considered being affected by the ripples. The dynamical equation of the mobile sand bed is based on the Exner’s sediment continuity equation in conjunction with the Meyer-Peter and Müller bed-load transport formula as modified to account for the effect of local bed slope due to bedforms. The coupled dynamical equations are then analyzed to estimate the parameters for the instability that results in the formation of ripples on the bed. The nondimensional ripple length (ratio of ripple length to sand size) increases with an increase in Shields parameter. The theoretical results have an agreement with the experimental data.