Near‐Bed Velocity Distribution

A continuous velocity distribution is derived for the laminar, transition, and logarithmic regions of the near‐bed portions of boundary layers for smooth and rough surfaces and drag‐reducing fluids. The solution is based on the equations of motion for uniform flow and a parabolic distribution of eddy viscosity over the turbulent portion of the boundary layer. Effects of increased roughness are accounted for by a shift of the distribution of eddy viscosity toward the flow boundary. The resulting velocity distribution agrees with published data and has an advantage over the classical logarithmic distribution for flow over rough surfaces in that the lower velocity limit is zero at the assigned origin for distance measurements near the boundary. The distribution function approaches the logarithmic expression in the outer regions of the boundary layer and has an advantage that the predicted slope of the velocity distribution is zero at the top of the boundary layer. Although the parabolic distribution of eddy diffusivity does not account for velocity distributions with wake components, other distributions of eddy diffusivity to account for these effects are suggested.