Theory of free surface flow over rough seeping beds

Abstract

A new theory is developed for the steady free surface flow over a horizontal rough bed with uniform upward seepage normal to the bed. The theory is based on the Reynolds averaged Navier–Stokes (RANS) equations applied to the flow domain that is divided into a fully turbulent outer layer and an inner layer (viscous sublayer plus buffer layer), which is a transition zone from viscous to turbulent regime. In the outer layer, the Reynolds stress far exceeds viscous shear stress, varying gradually with vertical distance. Near the free surface, the velocity gradient in vertical direction becomes lesser giving rise to wake flow. On the other hand, in the composite inner layer close to the bed, the viscous shear stress exists together with the turbulent stress. Thus, for the outer layer, a logarithmic law having modified coefficients from the traditional logarithmic law is obtained for the streamwise velocity, whereas for the inner layer, a fifth-degree polynomial including effective height of protrusions holds. The exact velocity expressions for inner and outer layer, which contain principal terms in addition to infinitesimally small terms, are in agreement with the experimental data obtained from laboratory measurements through an acoustic Doppler velocimeter. The experiments were run on two conditions of no seepage and a low upward seepage. Expressions for the Reynolds stress are also derived and computed for validation by the experimental data.