On the definition of the shear velocity in rough bed open channel flows.

Shear velocity u(*) is a well-known fundamental velocity scale of the wall turbulence, widely used for scaling various flow quantities such as mean velocity and turbulent intensities. The definition of shear velocity is closely related to the universal Logarithmic law of the wall, which was originally derived for constant shear stress boundary layers. Because the shear stress is constant, the momentum flux rho u(*)(2) is equal to the wall shear stress, so u(*) is often understood as some kind of synonym for the wall shear stress. The universal Log-law is also used for the flows with linear shear stress such as pipe flows and uniform open channel flows. In such flows the turbulent momentum flux delivered to the roughness differs from the wall shear stress, but the difference is negligible for the rough walls where boundary layer thickness (flow depth in case of uniform open channel flow), is much larger than roughness height. In rough wall flows where the flow depth and the roughness height are of the similar order, the difference between the turbulent momentum flux representative of the free outer flow and the wall shear stress may become significant. For such flows it is important to distinguish between the wall shear stress and the shear stress used for evaluation of the shear velocity. This paper highlights the problems associated with the definition of the shear velocity in shallow open channel flows and its use for evaluation of the bed shear stress. The influence of the definition of u(*) on the data analysis is illustrated using experimental results.