A Generalized Approach to Model One-Dimensional Nonmonotonous Distributions Using Renyi Entropy Theory with Applications to Open-Channel Turbulent Flows

In open-channel turbulent flows, nonmonotonous distributions (e.g., velocity with dip phenomenon and type-II profiles of suspension concentration distributions) need more careful and appropriate modeling approaches using entropy theory. All existing one-dimensional entropy theory-based models fail to predict such distributions. In this paper, a generalized approach to model the nonmonotonous one-dimensional distributions of certain properties (such as velocity dip phenomenon and type-II suspension distribution) of open-channel turbulent flows is proposed using Renyi entropy theory. This generalized approach is comprised of dividing the whole region of consideration into a finite number of subregions and employing the entropy-based approach to each of these subregions. Using a similar analogy of the principle of maximum entropy (POME), distribution models in one dimension are derived by maximizing the Renyi entropy subject to suitable constraints for all the subregions, and finally, each model of all subregions is combined using an asymptotic matching method to obtain the final single model valid for the whole flow region. This generalized approach is applied to study the vertical velocity distribution with dip phenomenon and type-II profile of sediment concentration distribution for open-channel turbulent flows. These derived models are validated with different sets of experimental data to show the efficiency of the approach. The results show that the proposed model is able to predict the maximum velocity below the free surface, unlike all previous 1D entropy-based velocity models. The results of the error analysis show that on an average, the mean absolute relative error and the root mean square error are reduced by 49% and 54%, respectively. Also, this study derives the very first models of type-II concentration using the entropy theory concept. Error analysis of the proposed type-II model shows that on an average, mean absolute relative error and root mean square error are reduced by 64% and 48%, respectively, compared to the deterministic models. The $R^2$ values of these models were compared, which states that model efficiency can be improved, on average, up to 67% using the proposed methodology. Furthermore, it is found that errors of the velocity prediction from the model with a formula-based dip position are within 2% limit of the errors of velocity prediction with an observed dip position. Apart from these applications, this methodology can also be applied to study other nonmonotonous distributions (such as location of dip position, hindered settling velocity, etc.) that occur in hydrology, environmental engineering, and other fields using entropy theory.