Mathematical modelling of streamwise velocity profile in open channels using Tsallis entropy

Abstract

This study derived the vertical distribution of streamwise velocity in wide-open channels by maximizing Tsallis entropy, in accordance with the maximum entropy principle, subject to the total probability rule, and the constraints based on the conservation of mass, momentum, and energy. Entropy maximizing leads to a highly nonlinear differential equation for velocity, which was transformed into a relatively weaker nonlinear equation and then solved analytically using a non-perturbation approach that yielded a series solution. The convergence of the series solution was proved using both theoretical and numerical procedures. For the velocity profile assessment, we calculated the Lagrange multipliers and the entropy index by solving a system of nonlinear equations using the Gauss-Newton method after approximating the constraint integrals using Gauss-Legendre quadrature rule. The derived velocity profile was validated for some selected sets of laboratory and field data and was compared with the existing velocity profiles based on Rényi, Tsallis, and Shannon (with additional constraints) entropies. We found that the incorporation of additional constraints and the effect of the entropy index improved the velocity profile compared to the existing Tsallis and Rényi entropy-based velocity equations. Further, it was observed that the proposed model and the Shannon entropy-based model with additional constraints behaved the same for most of the data sets considered, as the corresponding values of the entropy index were close to 1, which is in agreement with the theoretical consideration. The methodology reported in this study can also be employed for addressing other open channel flow problems, such as sediment concentration and shear stress distribution.