A probabilistic model on streamwise velocity profile in open channels using Tsallis relative entropy theory

The present study uses the concept of information entropy to derive a velocity distribution model in open channel flow. The Tsallis relative entropy theory is explored for that purpose, where the prior probability density function (PDF) is chosen from the maximum Tsallis entropy distribution. Both the cases of the fixed and varying entropy index are considered from the literature. The velocity equations are derived analytically using the homotopy analysis method (HAM) together with the Padé approximation technique in a general framework. The approximations and the HAM-based series solution are verified against laboratory and field data sets. Also, the prediction accuracies of the proposed models are assessed through the measures of statistical errors. It is seen that the model corresponding to the varying index is superior to the other model. The entropy index of Tsallis relative entropy is considered a variable, and the non-unity values of this index justify the applicability of the entropy function. Moreover, it is observed that one of the Lagrange multipliers corresponding to the velocity model with a varying index becomes negligible subject to the data sets considered, and hence, it simplifies the model development mathematically. The proposed approach can be further extended to develop models for estimating the suspended sediment concentration and shear stress distribution.