Theory of Turbulent Flow over a Wavy Boundary

Abstract

A new mathematical framework of the steady unseparated turbulent flow over a wavy (sinusoidal) boundary is derived by treating the Reynolds averaged Navier-Stokes (RANS) equations. Theoretical formulations for the flow profile, boundary shear stress and Reynolds shear stress distributions are obtained assuming a power law of streamwise velocity and accounting for the effects of curvilinear streamlines induced by the wavy boundary. The flow profiles, and distributions of boundary shear stress and Reynolds shear stress are computed for different flow parameters and presented in graphical forms. In subcritical flow, the flow profile is out of phase with the wavy boundary; while the boundary shear stress distribution is almost in phase with the wavy boundary. Instead, the flow profile and the boundary shear stress distribution in supercritical flow are opposite to those in subcritical flow. Upstream of the crest of wavy boundary the vertical distribution of the Reynolds shear stress is characterized by a concave shape because of an accelerated flow; while downstream of the crest it is a convex shape because of a decelerated flow. The theoretical results show a satisfactory agreement with the experimental data. The analysis is then extended to the mobile boundary flow to determine the phase lag distance between the locations of the maximum sediment flux and the maximum boundary shear stress. The influence of the flow Froude number on the phase lag is more pronounced than the resistance parameter.