Spatially Averaged Log-Law for Flows over Rough Bed in Zero- and Non-Zero-Pressure Gradient Boundary Layers

Theoretical bases for building a logarithmic law for non-uniform flows over a large relative roughness are presented. In order to define the equivalent velocity distribution and to smooth out 3D flow irregularities, a special spatial averaging operation is defined. Basic equations are spatially averaged and double-averaged momentum equations for primary component velocity are derived for uniform flow over a gravel bed as well as for non-uniform flows. A new hypothesis is proposed, and some assumptions are introduced to solve these momentum equations. This results in a new version of the logarithmic velocity distribution (log law). To define this distribution, a full reconstruction of Nikuradse's graph for flows over an irregular gravel riverbed is considered. It is based on very precise measurements of velocity and other hydraulic parameters. In the case of non-uniform flows, the logarithmic velocity profile appears also in accelerating flows in a gravel bed channel, but the friction velocity should be re-defined according to Eq. (24). The same applies to decelerating flow with a positive pressure gradient, but only if the gravitational force exceeds the pressure gradient. For accelerating flows, the additive constant BP depends on the pressure gradient, and its values grow with a growing pressure gradient.