Formulation of the Entropy Parameter Based on Hydraulic and Geometric Characteristics of River Cross Sections

The linear entropic relation between mean flow velocity $\overline{u}$ and maximum velocity $u_{\text{max}}$ is defined through a dimensionless entropy parameter $M$ , which is found constant for gauged river sections. This entropic relation has been tested for many rivers and has been found to be fundamental to addressing velocity measurements during high floods when sampling can be carried out only in the upper portion of flow area where the maximum velocity $u_{\text{max}}$ occurs. It is therefore of considerable interest to investigate the possible dependence of $M$ on hydraulic and geometric characteristics so that it can be determined for ungauged river sites. Thus, this study attempts to define the dependence of $M$ on the geometric and hydraulic characteristics of river cross sections by coupling Manning’s equation expressing $\overline{u}$ with the equation for $u_{\text{max}}$ obtained through a logarithmic velocity distribution, which takes into account the possibility that $u_{\text{max}}$ may occur below the water surface. Analysis shows that $M$ does not depend on the energy or water surface slope $S_f$ , thus justifying why its mean value at a gauged site is always nearly the same, whatever the flood condition. Moreover, the hydraulic and geometric characteristics that permit the determination of $M$ mainly include Manning’s roughness, hydraulic radius, and locations where $u_{\text{max}}$ occurs and the hypothetical zero velocity. Then, a formulation relating Manning’s roughness to $M$ is proposed. Velocity measurements carried out in the past 20 years at two gauged sections along the Tiber River, Italy are used to test the analysis.