An ordinary differential equation for velocity distribution and dip-phenomenon in open channel flows

An ordinary differential equation (ODE) for velocity distribution in open channel flows is presented based on an analysis of the Reynolds-averaged Navier–Stokes equations and a log-wake modified eddy viscosity distribution. This proposed equation allows to predict the velocity-dip-phenomenon, i.e. the maximum velocity below the free surface. Two different degrees of approximations are presented, a semi-analytical solution of the proposed ODE, i.e. the full dip-modified-log-wake law (fDMLW-law) and a simple dip-modified-log-wake law (sDMLW-law). Velocity profiles of the two laws and the numerical solution of the ODE are compared with experimental data. This study shows that the dip correction is not efficient for a small Coles' parameter, accurate predictions require larger values. The sDMLW-law shows reasonable agreement and seems to be an interesting tool of intermediate accuracy. The fDMLW-law, with a parameter for dip-correction obtained from an estimation of dip positions, provides accurate velocity profiles.