An Entropy Based Model for Velocity-Dip-Position

Abstract

In this study, theoretical models have been developed to predict the velocity-dip-position in steady and uniform turbulent flow through open channels. Unlike the previous works where empirical or semi-empirical models were suggested, the present models are developed from a mathematical approach based on the concept of entropy theory. Considering dimensionless dip-position as a random variable and starting from the Shannon entropy on probability distribution, models are derived by maximizing the entropy function using the principle of maximum entropy. It has been shown that proposed models are applicable over the whole cross section as well as at the central section of any rectangular open channel. The models are validated with a large number of experimental data sets published in literature for a wide variety of flow conditions. Apart from this, the models are also compared with other similar models existing in literature and the prediction accuracy of the present models are confirmed by computing five different errors for all the models. Out of the two proposed models, the model M2c satisfies the required asymptotic boundary conditions. At the end, model M2c is expressed in terms of a damping function. The non-occurrence of maximum velocity at the free surface, commonly known as dip-phenomenon, is explained in the light of the proposed damping concept.

Keywords: velocity-dip-phenomenon, Shannon entropy, Lagrange multiplier, least-square technique, open channel turbulent flow