Analytical modeling of vertical distribution of streamwise velocity in open channels using fractional entropy

Abstract

Following the work of Chiu based on classical Shannon entropy, several kinds of generalized entropies have been explored in literature for studying open-channel flow. In this work, we explore a new kind of entropy, namely fractional entropy, which is based on the popular fractional calculus, to derive the vertical distribution of streamwise velocity in open channels. The velocity profile is derived analytically using the series approximation for the Lambert function. Also, the entropy index (i.e., the order of the fractional derivative) is considered a varying parameter and is computed along with the Lagrange multipliers by solving a nonlinear system using the second-order moment constraint, i.e., the momentum balance equation. The derived velocity equation is validated with selected laboratory and field data and compared with the Shannon, Tsallis, and Renyi entropy-based velocity profiles. It is observed that the proposed model can predict the measured values well for all the cases and the model corresponding to the entropy-based momentum coefficient formula is superior to all the abovementioned models. Further, the effects of Lagrange multipliers and the entropy index on the velocity profile are discussed. Also, the entropy index values for all the data are not close to zero, which specializes the entropy into the classical Shannon entropy, and hence, the approach justifies the applicability of the fractional entropy over the Shannon entropy in the context of open-channel flow velocity. Moreover, the methodology is developed based on an approximation of the series, which is based on a convergence criterion. This criterion needs to be verified against each of the data sets and, therefore, may become a difficult task for handling large data sets. To address this issue, this study provides a possible way of reformulating the mathematical model in terms of a nonlinear system with inequality constraint as a scope for future research. Finally, it is expected that the study can be further extended to study other kinds of important hydraulic variables such as sediment concentration, shear stress, etc.